18.68 (2014)

What are the nonabelian composition factors of a finite nonsoluble group all of whose maximal subgroups have complements?
Note that finite simple groups with all maximal subgroups having complements are, up to isomorphism, $L_2(7)$, $L_2(11)$, and $L_5(2)$ (V. M. Levchuk, A. G. Likharev, Siberian Math. J., 47, no. 4 (2006), 659–668). The same groups exhaust finite simple groups with Hall maximal subgroups and composition factors of groups with Hall maximal subgroups (N. V. Maslova, Siberian Math. J., 53, no. 5 (2012), 853–861). It is also proved that in a finite group with Hall maximal subgroups all maximal subgroups have complements (N. V. Maslova, D. O. Revin, Siberian Adv. Math., 23, no. 3 (2013), 196–209).