Let \(G\) be the elation group of a finite elation generalized quadrangle \({\mathcal Q}\), then \({\mathcal Q}\) can be reconstructed from \(G\) and a certain collection \({\mathcal F}\) of subgroups of \(G\), called a Kantor family. In all known examples \(G\) is a \(p\)-group and it is conjectured that this always holds true. The author investigates the case in which at least one \(A \in {\mathcal F}\) is a normal subgroup of \(G\). Geometrically, this means that there exists a line through the elation center which admits a full group of symmetries. He confirms the conjecture in this case and shows that \(G/A\) is either elementary abelian or nonabelian of exponent \(p\), \(p\) odd. In the proof he uses some results from the theory of translation transversal designs.