According to Li and Zhao (Ukr Math J 64:102–109, 2012), a subgroup H of a finite group G is said to be \(S\Phi \)-supplemented in G if there exists a subnormal subgroup T of G such that \(G=HT\) and \(H\cap T\le \Phi (H)\), where \(\Phi (H)\) is the Frattini subgroup of H. In this paper, we extend the concept of \(S\Phi \)-supplemented subgroups, and give some criteria for (p-)hypercyclically embeddability of normal subgroups of finite groups by using fewer primary subgroups with given order. Our main results not only simplify, but also generalize some known theorems concerning \(S\Phi \)-supplemented subgroups.

据李和赵（UKR数学Ĵ64：102-109，2012），子组ħ有限群G ^据说是\（S \披\）在-supplemented ģ如果存在低于正常的子群Ť的ģ使得\(G=HT\)和\(H\cap T\le \Phi (H)\)，其中\(\Phi (H)\)是H的 Frattini 子群。在本文中，我们扩展了\(S\Phi \) -补充子群的概念，并给出了 ( p-)通过使用较少的具有给定顺序的主要子群，有限群的正常子群的超循环可嵌入性。我们的主要结果不仅简化了，而且还推广了一些关于\(S\Phi \)补充子群的已知定理。