Results are presented from a search for radio cores at 5 GHz with the VLA in 18 moderately luminous radio galaxies from the second Bologna catalogue. Maps of several of the ten detected cores indicate structure on arcsecond scales in addition to the unresolved central source. Higher-resolution, multifrequency observations are very desirable in future studies of such cores. A statistical study of these and previously observed samples of radio galaxies was then performed. It was found that the radio core luminosity is much stronger in galaxies that are optically bright than faint, and that strong cores tend to occur only in galaxies with luminous extended sources while weak cores can occur in all ranges of extended source luminosity.