Finite groups with some weakly ℋ-subgroups

Let [Formula: see text] be a finite group. A subgroup [Formula: see text] of [Formula: see text] is said to be an [Formula: see text]-subgroup of [Formula: see text] if [Formula: see text] for all [Formula: see text]. A subgroup [Formula: see text] of [Formula: see text] is called weakly [Formula: see text]-subgroup in [Formula: see text] if there exists a normal subgroup [Formula: see text] of [Formula: see text] such that [Formula: see text] and [Formula: see text] is an [Formula: see text]-subgroup in [Formula: see text]. The aim of this paper is to analyze the structure of a finite group in which the fewer maximal subgroups of Sylow [Formula: see text]-subgroup are weakly [Formula: see text]-subgroups. Our results improve and generalize many known results.