Let $G$ be a group and $H/K$ be a supplemented chief factor of $G$. I have observed one thing:
"If $H/K$ is a Frattini chief factor of $G$ (that means, $H/K\leq \Phi(G/K)$) then $H/K$ can be supplemented by a proper subgroup $M$ of $G$ but never supplemented by a maximal subgroup of $G$ containing $K$".
Is this a correct argument? Also, can you give me an example of a supplemented Frattini chief factor of a group $G$?
What I tried:
Suppose on the contrary, $H/K$ is supplemented by a maximal subgroup $M$ of $G$ containing $K$, therefore G=HM and $K\leq H\cap M$. But $H\leq M$ (since $H/K$ is a Frattini chief factor), It follows $HM=M$ and $M\neq G$, which is a contradiction. Therefore our assumption was wrong. Thus a Frattini chief factor can never be supplemented by a maximal subgroup of $G$.
For reference:
A chief factor $H/K$ of $G$ is a supplemented chief factor if there exists a proper subgroup $M$ of $G$ such that $G=HM$ and $K\leq H\cap M$.
Thanks.