The intersection of all non-nilpotent maximal subgroups containing the normalizer of some Sylow subgroup is nilpotent in every finite non-solvable group.
Shidov, On maximal subgroups of finite groups, Siberian Mathematical Journal 12 (1971) 488–489
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On an extension of Shlyk's theorem
The intersection of all non-nilpotent maximal subgroups containing the normalizer of some Sylow subgroup is nilpotent in every finite non-solvable group.