Finite groups with SS -quasinormal maximal subgroup series

Let [Formula: see text] be finite group. A subgroup [Formula: see text] of [Formula: see text] is said to be an [Formula: see text]-quasinormal subgroup of [Formula: see text], if there exists a subgroup [Formula: see text] of [Formula: see text] such that [Formula: see text] and [Formula: see text] permutes with every Sylow subgroup of [Formula: see text]. Let [Formula: see text] be a maximal subgroup series of [Formula: see text], where [Formula: see text] is a maximal subgroup of [Formula: see text] for every [Formula: see text]. In this paper, we investigate the finite groups [Formula: see text] that admit an [Formula: see text]-quasinormal maximal subgroup series, i.e., all [Formula: see text] are [Formula: see text]-quasinormal in [Formula: see text]. First, we prove that if [Formula: see text] possesses an [Formula: see text]-quasinormal maximal subgroup series, then [Formula: see text] is solvable. Furthermore, we show that [Formula: see text] is supersolvable if and only if [Formula: see text] possesses an [Formula: see text]-quasinormal maximal subgroup series which is subnormal in [Formula: see text].