Take the set of integers modulo $n$, written $\Bbb{Z}/(n)$ and any prime $p$ and consider the multiplicative subgroup specified by $H^{\times} = \{ x^p = 1: x \in \Bbb{Z}/(n)^{\times}\}$, then $|H^{\times}| = p^k$ for some $k$. Is this already known?

Generalizing, from what I can tell looking at the results of some simple python code. I think the conjecture could be:

Let $p \gt 0$ be any natural with prime factors $p\_1, \dots, p\_r$, then $|H^{\times}| = p\_1^{e\_1}\cdots p\_r^{e\_r} = $ some composition of powers of the prmes making up $p$.