On a characterization of the Frattini subgroup of a finite solvable group

Suppose that $G$ is a finite solvable group, $n$ is the length of a $G$-chief series of the group $F(G)/\Phi(G)$, and $k$ is the number of central $G$-chief factors of this series. We prove that in this case $G$ contains $4n-3k$ maximal subgroups whose intersection is $\Phi (G)$. This result refines V. S. Monakhov's statement that, for any finite solvable nonnilpotent group $G$, its Frattini subgroup $\Phi(G)$ coincides with the intersection of all maximal subgroups $M$ of the group $G$ such that $MF(G)=G$. In addition, it is shown in Theorem 4.2 that the group $G$ contains $4(n-k)$ maximal subgroups whose intersection is $\delta(G)$. The subgroup $\delta(G)$ is defined as the intersection of all abnormal maximal subgroups of $G$ if $G$ is not nilpotent and as $G$ otherwise.