A subgroup H of the finite group G is said to be quasinormally (resp. Squasinormally) embedded in G if for every Sylow subgroup P of H, there is a quasinormal (resp. S-quasinormal) subgroup K in G such that P is also a Sylow subgroup of K. Groups with certain quasinormally (resp. S-quasinormally) embedded subgroups of prime-power order are studied. For example, if a group G has a normal subgroup H such that G=H A F and such that for each Sylow subgroup P of H, every member in some M d ðPÞ is

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... sinormally embedded in G, then G A F: here M d ðPÞ is a set of maximal subgroups of P with intersection the Frattini subgroup.