It is Sunnah for the one who recites the verse Sabbih isma Rabbika al-A‘la to say Subhaana Rabbiy al-A‘la

Firstly:

When praying, if the Muslims recites a verse that mentions tasbeeh (glorifying Allah), it is Sunnah for him to glorify Allah; if he recites a verse that speaks of mercy, it is Sunnah for him to pray for mercy; if he recites a verse that speaks of punishment, it is Sunnah for him to seek refuge with Allah, because of the report narrated by Muslim (772) and Ahmad (22750) from Hudhayfah (may Allah be pleased with him), who said: I prayed with the Prophet (blessings and peace of Allah be upon him) one night. He began to recite al-Baqarah… and if he recited a verse that spoke of mercy, he would ask for mercy, and if he recited a verse that mentioned punishment, he would seek refuge with Allah, and if he recited a verse that declared Allah to be above all shortcomings, may He be glorified and exalted, he would glorify Him.

The scholars of the Standing Committee for Issuing Fatwas said:

If the imam asks for mercy, seeks refuge with Allah or glorifies Allah when reciting the appropriate verses in a supererogatory (naafil) prayer, there is a proven report from the Prophet (blessings and peace of Allah be upon him) which indicates that doing so is recommended (mustahabb), as in the hadith of Hudhayfah that is narrated in Saheeh Muslim, where he described the Prophet’s prayer and recitation at night.

But if he does that in an obligatory prayer, it is better not to do it, because there is no report from the Prophet (blessings and peace of Allah be upon him) to suggest that he did that in an obligatory prayer. End quote.

Fataawa al-Lajnah ad-Daa’imah (5/344)

Secondly:

Abu Dawood (883) and Ahmad (2066) narrated from Ibn ‘Abbaas that when the Prophet (blessings and peace of Allah be upon him) recited the words “Sabbih isma Rabbika al-A‘la (Glorify the Name of your Lord, the Most High)” [al-A ‘la 87:1], he would say: “Subhaana Rabbiy al-A‘la (Glory be to my Lord Most High).”

Abu Dawood said, commenting on this hadith: It was narrated by Abu Wakee‘ and Shu‘bah, from Abu Ishaaq, from Sa‘eed ibn Jubayr, from Ibn ‘Abbaas, in a mawqoof report. End quote.

Classed as saheeh by al-Albaani in Saheeh Abi Dawood in a marfoo‘ report.

Ibn Abi Shaybah (2/247) narrated with a saheeh isnaad from ‘Umayr ibn Sa‘eed who said: I prayed Jumu‘ah with Abu Moosa and he recited “Sabbih isma Rabbika al-A‘la (Glorify the Name of your Lord, the Most High)” [al-A ‘la 87:1], then he said: “Subhaana Rabbiy al-A‘la (Glory be to my Lord Most High)” as he was praying.

‘Umayr ibn Sa‘eed is trustworthy. See: at-Tahdheeb (8/129)

It was narrated from ‘Ali ibn Abi Taalib that he recited “Sabbih isma Rabbika al-A‘la (Glorify the Name of your Lord, the Most High)” [al-A ‘la 87:1], then he said: “Subhaana Rabbiy al-A‘la (Glory be to my Lord Most High),” when he was praying. It was said to him: Are you adding to the Qur’an? He said: No; rather we were instructed to say something, so I said it. Narrated by al-Firyaabi, Ibn Abi Shaybah, ‘Abd ibn Humayd and Ibn al-Anbaari in al-Masaahif. Al-Albaani said: Its isnaad is hasan.

Saheeh Abi Dawood (4/40)

To sum up:

It is prescribed for the one who recites “Sabbih isma Rabbika al-A‘la (Glorify the Name of your Lord, the Most High)” [al-A ‘la 87:1], to say tasbeeh to himself because of that, because it is proven in the Sunnah that the Prophet (blessings and peace of Allah be upon him) would glorify his Lord when he recited a verse that mentioned glorification of Allah, and this is one such verse. And it is proven from more than one of the companions of the Prophet (blessings and peace of Allah be upon him) that he would glorify his Lord in such cases, and especially when reciting this verse. That was also narrated in a marfoo‘ report attributed to the Prophet (blessings and peace of Allah be upon him).

Al-Mannaawi (may Allah have mercy on him) said:

From this it may be understood that whenever the reciter or listener comes across a verse that glorifies Allah, he should glorify Allah, may He be exalted; if he comes across a verse that praises Allah, he should praise Him; if he comes across a verse that magnifies Him, he should magnify Him, and so on.

Fayd al-Qadeer (5/156).

Shaykh Ibn Baaz (may Allah have mercy on him) said:

It is Sunnah for everyone who recites Qur’an, in prayer or otherwise, if he comes to a verse that speaks of mercy, to ask Allah, may He be exalted, for His grace; if he comes to a verse that speaks of punishment, he should seek refuge with Him from the fire; if he comes to a verse that glorifies Allah, may He be glorified, he should glorify Him and say “Subhaanahu wa ta‘aala (may He be glorified and exalted)” and the like. It is mustahabb for anyone who recites “Alaysa Allahu bi ahkaam al-haakimeen (Is not Allâh the Best of judges?)” [at-Teen 95:8] to say: “Bala wa ana ‘ala dhaalika min ash-shaahideen (Yes indeed, and I am one of the witnesses to that).” If he recites “Alaysa dhaalika bi qaadirin ‘ala an yuhyiya al-mawta (Is not He (Allâh Who does that), Able to give life to the dead?)” [al-Qiyaamah 75:40], he should say: “Bala ashhad (Yes indeed, and I bear witness).” If he recites “Fa bi ayyi hadeethin ba‘dahu yu’minoon (In what message after this will they then believe?)” [al-A‘raaf 7:185], he should say: “Aamantu Billah (I believe in Allah).” If he recites, “Fa bi ayyi alaa’i Rabbikuma tukhaddibaan (Then which of the Blessings of your Lord will you both (jinn and men) deny?)” [ar-Rahmaan 55:15], he should say: “Laa nukhadhdhibu bi shay’in min aayaati Rabbina (We do not deny any of the blessings of our Lord).” If he recites “Sabbih isma Rabbika al-A‘la (Glorify the Name of your Lord, the Most High)” [al-A ‘la 87:1], he should say, “Subhaana Rabbiy al-A‘la (Glory be to my Lord Most High).” This is mustahabb for the imam, one who is praying behind him, and one who is praying on his own, because it is supplication and it is required from them, like saying Ameen. The same ruling applies when reading Qur’an outside of prayer. End quote.

Majmoo‘ Fataawa Ibn Baaz (11/76)

But here it is important to note two points:

1. Tasbeeh when reciting this and similar verses is by saying “Subhaana Rabbiy al-A‘la (Glory be to my Lord Most High), as is proven in the reports, and not by saying “Rabbika” as mentioned in the question.

2. If the imam recites and does not say tasbeeh, and does not leave room for the people praying behind him to do so, rather he carries on reciting, then it is not prescribed for the one who is praying behind him to say tasbeeh; rather it is prescribed for him to listen attentively to the recitation of the imam. What is prescribed for the one who is praying behind the imam is to say the tasbeeh to himself and not to say it out loud, lest he disturb other worshippers. It is not prescribed for one who is praying behind the imam to raise his voice in anything except saying Ameen.

See also question no. 150148

And Allah knows best.