Influence of complemented subgroups on the structure of finite groups

Abstract

P‎. ‎Hall proved that a finite group $G$ is supersoluble with elementary abelian Sylow subgroups if and only if every subgroup of $G$ is complemented in $G$‎. ‎He called such groups complemented‎. ‎A‎. ‎Ballester-Bolinches and X‎. ‎Guo‎ ‎established the structure of minimal non-complemented groups‎. ‎We give the classification of finite non-soluble groups‎ ‎all of whose second maximal subgroups are complemented groups‎. ‎We also prove that every finite group with less than‎ ‎$21$ non-complemented non-minimal ${2,3,5}$-subgroups is soluble‎.