Let G be a finite group. A subgroup H of G is said to be S-permutable in G if H permutes with all Sylow subgroups of G. We denote by \(H^{sG}\) the S-permutable closure of H in G, that is, the intersection of all S-permutable subgroups of G that contain H. A subgroup H of G is called Hall S-permutably embedded (or Hall S-quasinormally embedded) in G if H is a Hall subgroup of its S-permutable closure \(H^{sG}\) . In this paper, we investigate the structure of a finite group G under the assumption that maximal or minimal subgroups of a Sylow subgroup are Hall S-permutably embedded in G.
