Question 28

By Lagrange's Theorem, the order of any subgroup $H$ of a finite group $G$ must divide the order of $G$:

Given $|G| = 35 = 5 \times 7$. The positive divisors of 35 are:

A subgroup of $G$ other than $G$ itself is a proper subgroup ($|H| < 35$). The possible orders of proper subgroups are 1, 5, and 7.

The largest among these divisors is 7. (By Sylow's theorem or Cauchy's theorem, a group of order 35 is cyclic, isomorphic to $\mathbb{Z}_{35}$, and has unique subgroups of orders 1, 5, 7, and 35).