Values of the offset of the central galaxy from the optical luminosity weighted centre for each member of the X-ray overluminous (blue), X-ray normal (black), and X-ray underluminous (red) sub-samples. Underluminous groups with X-ray upper limits are denoted with a light red triangle. The main panel shows the values for each group, while the right panel shows the distribution of central galaxy offset for each of the sub-samples. The top panel shows the sub-sample means, both across the full range in L_opt (stars) and split into two bins in L_opt (circles). X-ray underluminous systems have, on average, larger offsets than the X-ray overluminous groups.