Finite groups with C-quasinormal subgroups

Consider some finite group G and a finite subgroup H of G. Say that H is c-quasinormal in G if G has a quasinormal subgroup T such that HT = G and T boolean AND H is quasinormal in G. Given a noncyclic Sylow subgroup P of G. we fix some subgroup D such that 1 < vertical bar vertical bar D < vertical bar P vertical bar and study the structure of G under the assumption that all subgroups H of P of the same order as D, having no supersolvable supplement in G, are c-quasinormal in G.