Is it permissible for a person to have less than a one-seventh share in the sacrifice (udhiyah) of a cow?

It is permissible for seven people to share in the sacrifice (udhiyah) of a cow or camel. This has been explained previously in the answer to question no. 45757.

It is not permissible for one of the partners to have less than a one-seventh share, unless he simply wants the meat and does not intend to offer an udhiyah, in which case there is nothing wrong with him having whatever share he wants.

Shaykh Ibn ‘Uthaymeen said in his essay Ahkaam al-Udhiyah:

It is acceptable to offer one sheep on behalf of a single individual, and to offer one seventh of a camel or cow on behalf of what one sheep may be offered on behalf of.

If two or more people share ownership of the udhiyah and (both together) offer it as an udhiyah, that is not permissible and the udhiyah is not valid unless it is a camel or cow that is offered on behalf of up to seven only. That is because the udhiyah is an act of worship and a means of drawing closer to Allah, may He be exalted, so it is not permissible to do it as an act of worship except in the manner prescribed in terms of time, numbers and manner.

End quote from Rasaa’il Fiqhiyyah (p. 58, 59)

If one of the partners in the udhiyah has a share of less than one seventh, his udhiyah is not valid, but that does not affect the other partners, because there is nothing wrong with seven people having shares in a cow that is slaughtered as an udhiyah, with some of them intending to offer an udhiyah and the others simply wanting the meat.

This has been explained previously in the answer to question 45771.

And Allah knows best.