By the Galois Correspondence (for path connected, locally simply connected spaces $X$, say) if $\pi_1 (X)$ is a finite group, then there is a unique isomorphism class of non-based covering space with fundamental group $\cong P$ where $P \in \mathrm{Syl}_p (\pi_1 (X))$ for each prime $p$ dividing $|\pi_1 (X)|$, since by the Sylow theorems any $p$-Sylow subgroup is conjugate.
I was wondering whether there is a geometric reason as to why this is true. What is the geometric relationship between the 'holes' in such spaces and coverings of these maximal prime power order?