Detecting the index of a subgroup in the subgroup lattice

A theorem by Zacher and Rips states that the finiteness of the index of a subgroup can be described in terms of purely lattice-theoretic concepts. On the other hand, it is clear that if $G$ is a group and $H$ is a subgroup of finite index of $G$, the index $\vert G:H\vert$ cannot be recognized in the lattice ${\mathfrak{L}}(G)$ of all subgroups of $G$, as for instance all groups of prime order have isomorphic subgroup lattices. The aim of this paper is to give a lattice-theoretic characterization of the number of prime factors (with multiplicity) of $\vert G:H\vert$.