Well, it is either cyclic or not.
Suppose it is not cyclic and choose an element g in G. The order of g, <g>, must divide the order of G, so <g> divides 25 or <g> is 1 or 5 or 25. In every case, the result holds true.
The group is not necessarity cyclic, how about .
If is cyclic then it is clearly true because and .
If is not cyclic then choose so that . Then is a subgroup of . It cannot (subgroup) have order 1 because it is not identity and it cannot be 25 (the full group) because we assumed it is not cyclic. Thus it must be 5 by Lagrange's theorem.