Closed subgroups of $Z_{p}^{\times}$

Question

I was able to prove that any closed subgroup of additive group of $Z_{p}$ is of the form $p^{n}Z_{p}$ for some $n$. I asked the same question for the multiplicative group of units in $Z_{p}$, that is $Z_{p}^{\times}$. Since it has the direct product decomposition ${\mu}_{p-1}{\times}(1+pZ_{p})$ I ran into trouble. I thought of finding closed subgroups of $(1+pZ_{p})$ but I fear that may not help since subgroups of $A{\times}B$ are not always of the form (subgroup of $A$ ${\times}$ subgroup of $B$). I don't know how to proceed. Any hints, suggestions are welcome. Thanks!!