University of IsfahanInternational Journal of Group Theory2251-765010220210601Influence of complemented subgroups on the structure of finite groups65742426110.22108/ijgt.2019.119105.1570ENIzabela AgataMalinowskaFaculty of Mathematics, University of Bialystok, 15-245 Bialystok, Ciolkowskiego 1M, Poland0000-0002-2772-7379Journal Article20190910P. 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.https://ijgt.ui.ac.ir/article_24261_26abaf2b4bf9f112a26c308aa371615b.pdf