On the solvability of a finite group with seminormal or subnormal Schmidt subgroups of one of its maximal subgroups

A Schmidt group is a finite non-nilpotent group all of whose proper subgroups are nilpotent. A group with a nilpotent maximal subgroup is known to be solvable if the derived subgroup of a Sylow 2-subgroup of a maximal subgroup is contained in the center of the Sylow 2-subgroup. If a maximal subgroup of a group is non-nilpotent, then it has a Schmidt subgroup. The structure of a group and, in particular, its solvability, depend on the properties of Schmidt subgroups of its maximal subgroup. In this paper, we establish the solvability of a finite group such that some Schmidt subgroups of its maximal subgroup are seminormal or subnormal in the group.