Finite groups whose n-maximal subgroups are σ-subnormal

Let σ = {σi | i ∈ I} be some partition of the set of all primes P. A set H of subgroups of G is said to be a complete Hallσ-set of G if every member ≠ 1 of H is a Hall σi-subgroup of G, for some i ∈ I, and H contains exactly one Hall σi-subgroup of G for every σi ∈ σ(G). A subgroup H of G is said to be: σ-permutable or σ-quasinormal in G if G possesses a complete Hall σ-set H such that HAx = AxH for all A ∈ H and x ∈ G: σ-subnormal in G if there is a subgroup chain A = A0 ≤ A1 ≤ · · · ≤ At = G such that either Ai−1⊲_Ai\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${A_{i - 1}}\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \triangleleft } {A_i}$$\end{document} or Ai=(Ai-1)Ai is a finite σi-group for some σi ∈ σ for all i = 1;:::; t.