Influence of complemented subgroups on the structure of finite groups

P. Hall proved that a finite group G is supersoluble with elementary abelian Sylow sub-groups 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.