The subgroup $O_{\pi}(G)$ in a finite group

Question

For a finite group $G$ the set $O_{\pi}(G)$ is the maximal normal $\pi$-subgroup of $G$. Could there anything said about $G / O_{\pi}(G)$?

And maybe do you know some more properties around $O_{\pi}(G)$ and some exercises around this construction?

Answer 1

Your question is very general, but for starters you could take a look at Rose, "A Course on Group Theory", pg. 56-58 for basics and exercises about $O_\pi$ and $O^\pi$. Throughout the book there are exercises about $O_\pi$.