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Let G be a group of collineations of a finite thick generalised quadrangle τ. Suppose that G acts primitively on the point set P of τ, and transitively on the lines of τ. We show that the primitive action of G on P cannot be of holomorph simple or holomorph compound type. In joint work with Glasby, we have previously classified the examples τ for which the action of G on 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.