  For a finite group $G$, we investigate the behaviour of four invariants,
$\text{MaxDim}(G),$ $\text{MinDim}(G),$ $\text{MaxInt}(G)$ and
$\text{MinInt}(G),$ measuring in some way the width and the height of the
lattice $\mathcal M(G)$ consisting of the intersections of the maximal
subgroups of $G.$
