  Let $G$ be a group of collineations of a finite thick generalised quadrangle
$\Gamma$. Suppose that $G$ acts primitively on the point set $\mathcal{P}$ of
$\Gamma$, and transitively on the lines of $\Gamma$. We show that the primitive
action of $G$ on $\mathcal{P}$ cannot be of holomorph simple or holomorph
compound type. In joint work with Glasby, we have previously classified the
examples $\Gamma$ for which the action of $G$ on $\mathcal{P}$ is of affine
type. The problem of classifying generalised quadrangles with a
point-primitive, line-transitive collineation group is therefore reduced to the
case where there is a unique minimal normal subgroup $M$ and $M$ is
non-Abelian.
