  Given a finite non-cyclic group $G$, call $\sigma(G)$ the smallest number of
proper subgroups of $G$ needed to cover $G$. Lucchini and Detomi conjectured
that if a nonabelian group $G$ is such that $\sigma(G) < \sigma(G/N)$ for every
non-trivial normal subgroup $N$ of $G$ then $G$ is \textit{monolithic}, meaning
that it admits a unique minimal normal subgroup. In this paper we show how this
conjecture can be attacked by the direct study of monolithic groups.
