It is well known that all maximal subgroups of a finite solvable group are solvable and have prime power indices. However, the converse statement does not hold. Finite nonsolvable groups in which all local subgroups are solvable were studied by J. Thompson (1968). R. Guralnick (1983) described all the pairs \((G,H)\) such that \(G\) is a finite nonabelian simple group and \(H\) is a subgroup of prime power index in \(G\). Several authors studied finite groups in which every subgroup of non-prime-power index (not necessarily maximal) is a group close to nilpotent. Weakening the conditions, E. N. Bazhanova (Demina) and N. V. Maslova (2014) considered the class \(\mathfrak{J}_{pr}\) of finite groups in which all nonsolvable maximal subgroups have prime power indices and, in particular, described possibilities for nonabelian composition factors of a nonsolvable group from the class \(\mathfrak{J}_{pr}\). In the present note, the authors continue the study of the normal structure of a nonsolvable group from \(\mathfrak{J}_{pr}\). It is proved that a group from \(\mathfrak{J}_{pr}\) contains at most one nonabelian chief factor and, for each positive integer \(n\), there exists a group from \(\mathfrak{J}_{pr}\) such that the number of its nonabelian composition factors is at least \(n\). Moreover, all almost simple groups from \(\mathfrak{J}_{pr}\) are determined.

中文翻译：

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其最大子组可解或具有质数幂指数的有限组

众所周知，有限可解组的所有最大子组都是可解的，并且具有质数指数。但是，相反的语句不成立。J. Thompson（1968）研究了其中所有局部亚组都可求解的有限不可求解组。R. Guralnick（1983）描述了所有对（（G，H）\），使得 \（G \）是有限的非阿贝尔简单群，  \（H \）是 \（G \ ）。几位作者研究了有限群，其中非质力指数的每个子群（不一定是最大值）都是一个接近幂等的群。E. N. Bazhanova（Demina）和N. V. Maslova（2014）削弱了这些条件，认为该类\（\ mathfrak {J} _ {pr} \）其中所有不可解的最大子组都具有素数幂指数的有限组，特别是描述了\（\ mathfrak {J} _ {pr} \）类中不可解组的非阿贝尔组成因子的可能性 。在本文中，作者继续研究\（\ mathfrak {J} _ {pr} \）中不可解基团的正常结构 。证明\（\ mathfrak {J} _ {pr} \）中的一组最多包含一个非阿贝尔主因子，并且对于每个正整数 \（n \），存在一个来自\（\ mathfrak {J } _ {pr} \），使其非阿贝尔构成因子的数量至少为 \（n \）。此外，\（\ mathfrak {J} _ {pr} \）中的所有几乎简单的组 确定。