Primitive subgroups and PST-groups
All groups are finite. A subgroup H of a group G is called a primitive subgroup if it is a proper subgroup in the intersection of all subgroups of G containing H as its proper subgroup. He, Qiao and Wang  proved that every primitive subgroup of a group G has index a power of a prime if and only if G/Φ(G) is a solvable PST-group. Let X denote the class of groups G all of whose primitive subgroups have prime power index. It is established here that a group G is a solvable PST-group if and only if every subgroup of G is an X-group.
Ballester Bolinches, Adolfo Beidleman, J.C. Esteban Romero, Ramón 2014 Primitive subgroups and PST-groups Bulletin of the Australian Mathematical Society 89 3 373 378