Example of a supplemented chief factor of a group

Question

Let $H/K$ is a chief factor of a group $G$. We say $H/K$ is a supplemented chief factor of $G$ if there exists a proper subgroup $M$ of $G$ such that $G=MH$ and $K\leq H\cap M$ and $M$ is called supplement of $H/K$ in $G$. Can you give an example of a supplemented chief factor of a group? I want to understand it illustratively.

Answer 1

There are many examples and I am not sure which one would be best to illustrate this concept. Anyway, my choice is the quaternion group of order $8$. The subgroup lattice of that group looks like this:

The blue dot is the trivial subgroup and the top one is the whole of $Q_8$. The three subgroups directly below it are the maximal subgroups of $Q_8$, all cyclic and of order $4$. The sole dot above the blue one is the Frattini subgroup of $Q_8$ and also its centre.

Now take as $K$ this subgroup, i.e. $K=\Phi(Q_8)$ and as $H$ any of the three maximal subgroups. Then $H/K$ is clearly a chief factor of $Q_8$ and it is easy to see that any maximal subgroup other than the $H$ you chose is a supplement for $H$, i.e. it is the $M$ you are looking for.