We report the results of a Monte Carlo simulation study of isolated comb copolymer cylindrical brushes. The conformational characteristics of molecules consisting of a flexible backbone densely grafted with side chains strongly depend on the rigidity of the side chains, which is exemplified by using both fully flexible and totally rigid side chains. The short length scale fluctuations of the backbone are not affected by the presence of rigid rod side chains, whereas flexible side chains induce local stretching of the backbone. However, at the large length scale, the persistence length lambda of the cylindrical brush is considerably larger for rigid rod side chains than for flexible side chains. Moreover, for rigid side chains the ratio between the persistence length lambda and the diameter D of the brush increases approximately linearly as a function of the rod length L, thus supporting the possibility of lyotropic behavior. In contrast to this, in the case of flexible side chains, lambda/D is approximately independent of the side chain length M.