A lattice theoretic characterization for the existence of a faithful irreducible representation

In a recent article Sébastien Palcoux formulated a sufficient condition on the subgroup lattice of a finite group G that guarantees the existence of a faithful irreducible complex representation of G, and asked whether his condition is also necessary. In this short note we give an affirmative answer using Kochendörffer’s criterion for the existence of a faithful irreducible representation based on the structure of the socle of G.