Given x, an element of order 627 in a group G (not necessarily cyclic, finite, or Abelian). How many distinct subgroups of G are contained in $\langle x \rangle$?
I'm not sure if I'm reasoning this out correctly. So since $|x| = 627$, there should be $627$ subgroups of x. To determine distinct subgroups, we can use a theorem.
Thm: Let $|a| = n$, let $k$ be a positive integer. Then $|a^k| = \frac{n}{gcd(n,k)}$ and $\langle a^k \rangle = \langle a^{gcd(n,k)} \rangle$.
As well as the corollary that says $\langle a \rangle = \langle a^j \rangle$ if and only if $gcd(n,j)$ = 1 where $|a| = n$.
So one way that seems obvious is to go through $k \in [0,627]$ and slowly determine each distinct subgroup. But is there a smarter way to do this?