Remarks on the Supersolvability of a Group with Prime Indices of Some Subgroups

In the paper, a characterization is obtained for a finite group such that, for each prime $p$, every maximal subgroup of any Sylow $p$-subgroup of this group is contained in a subgroup of index $p$; in particular, such groups are supersolvable. It is proved that a group $G$ is supersolvable if and only if, for every prime $p\in\pi(G)$, there is a supersolvable subgroup of index $p$. New properties of groups containing two supersolvable subgroups of different prime indices are established.