influence of complemented subgroups on the structure of finite groups

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.