The intersection of all non-nilpotent maximal subgroups containing the normalizer of some Sylow subgroup is nilpotent in every finite non-solvable group.
Gasch¨ utz,¨Uber die Φ-Untergruppen endlicher Gruppen (German), Mathematis- che Zeitschrift 58(1953) 160–170
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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.