# What are the closed subgroups of $p$-adic solenoid?

Let $$f\colon S^1 \rightarrow S^1$$ given by $$f(z)=z^p$$, and think $$S^1 = \{z\in \mathbb{C}\colon |z|=1\}$$ as a multiplicative group, so $$f$$ is an homomorphism. Let $$S_n=S^1$$ and $$f_n=f$$ for all $$n$$, and consider the inverse limit

$$\mathcal{S}_p =\varprojlim(S_n, f_n)$$.

Note that $$\mathcal{S}_p$$ is a topological group.

I know that all closed subgroups of $$S^1$$ are $$S^1$$ itself and all finite cyclic subgroups.

I want to classify all closed subgroups of $$\mathcal{S}_p$$ using this fact. So far, I have proved that $$\mathcal{S}_p$$ and all finite cyclic groups $$C_m$$, where $$m$$ is the cardinality of the group with $$(m,p)=1$$, occur in that list. But those are not all subgroups of $$\mathcal{S}_p$$, so I ask you if you know what groups are missing and if you can sketch a proof for getting them, please.