Abstract
A subgroup H of a finite group G is quasinormal in G if it permutes with every subgroup of G. A subgroup H of a finite group G is Shq-supplemented in G if G has a quasinormal subgroup N such that HN is a Hall subgroup of G and (H boolean AND N) H-G/H-G <= Z(s) (G/H-G),where H-G is the core of H in G and Z(S)(G/H-G) is the S-hypercenter of G/H-G. This paper concerns the structure of a finite group G under the assumption that some subgroups of G are S-hq-supplemented in G.