We use the ROSAT All-Sky Survey to study the X-ray properties of a sample of 625 groups and clusters of galaxies selected from the Sloan Digital Sky Survey. We stack clusters with similar velocity dispersions and investigate whether their average X-ray luminosities and surface brightness profiles vary with the radio activity level of their central galaxies. We find that at a given value of {sigma}, clusters with a central radio active galactic nucleus (AGN) have more concentrated X-ray surface brightness profiles, larger central galaxy masses and higher X-ray luminosities than clusters with radio-quiet central galaxies. The enhancement in X-ray luminosity is more than a factor of 2, is detected with better than 6{sigma} significance and cannot be explained by X-ray emission from the radio AGN itself. This difference is largely due to a subpopulation of radio-quiet, high velocity dispersion clusters with low-mass central galaxies. These clusters are underluminous at X-ray wavelengths when compared to otherwise similar clusters where the central galaxy is radio-loud, more massive, or both.