The intersection of all non-nilpotent maximal subgroups containing the normalizer of some Sylow subgroup is nilpotent in every finite non-solvable group.
Shlyk, On the intersection of maximal subgroups in finite groups, Mathematical Notes 14 (1973) 804–809
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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.