Summary: A subgroup of a finite group \(G\) is said to be \(S\)-quasinormal in \(G\) if it permutes with every Sylow subgroup of \(G\). A subgroup \(H\) of a finite group \(G\) is said to be \(s\)-semipermutable in \(G\) if it permutes with every Sylow \(p\)-subgroup of \(G\), where \(p\) and the order of \(H\) are relatively prime. A subgroup \(H\) of a finite group \(G\) is said to be \(S\)-quasinormally embedded in \(G\) if every Sylow subgroup of \(H\) is a Sylow subgroup of some \(S\)-quasinormal subgroup of \(G\). In this paper, we are interested in studying the structure of the finite group \(G\) under the assumption that certain subgroups of prime power order of \(G\) are \(s\)-semipermutable or \(S\)-quasinormally embedded in \(G\). Some recent results are improved and generalized.