Indices of Maximal Subgroups of Finite Soluble Groups

We look at the structure of a soluble group $G$ depending on the value of a function $m(G)=\max\limits_{p\in\pi(G)}$, where $m_p(G)=\max\{\log_p|G:M|\mid M<_{\max}G,\ |G:M|=p^a\}$, $p\in \pi (G)$.
\medskip Theorem 1. {\it States that for a soluble group $G$, (1) $r(G/\Phi (G))=m(G)$; (2) $d(G/\Phi (G))\leqslant1+\rho(m(G))\leqslant3+m(G)$; (3) $l_p(G)\leqslant1+t$, where $2^{t-1}<m_p(G)\leqslant 2^t$.}
\medskip Here, $\Phi(G)$ is the Frattini subgroup of $G$, and $r(G)$, $d(G)$, and $l_p(G)$ are, respectively, the principal rank, the derived length, and the $p$-length of $G$. The maximum of derived lengths of completely reducible soluble subgroups of a general linear group $GL(n,F)$ of degree $n$, where $F$ is a field, is denoted by $\rho(n)$. The function $m(G)$ allows us to establish the existence of a new class of conjugate subgroups in soluble groups. Namely,
\medskip Theorem 2. {\it Maintains that for any natural $k$, every soluble group $G$ contains a subgroup $K$ possessing the following properties: (1) $m(K)\leqslant k$; (2) if $T$ and $H$ are subgroups of $G$ such that $K\leqslant T<_{\max}H\leqslant G$ then $|H:T|=p^t$ for some prime $p$ and for $t>k$. Moreover, every two subgroups of $G$ enjoying (1) and (2) are mutually conjugate.}