A direct numerical simulation was carried out of plane turbulent Couette flow at a Reynolds number of 750, based on half the velocity difference between the walls and half the channel width. Particular attention was paid to choosing a computational box that is large enough to accommodate even the largest scales of the turbulence. In the central region of the channel very large elongated structures were observed, in accordance with earlier findings. The study is focused on the properties of these structures, but is also aimed at obtaining accurate turbulence statistics. Terms in the energy budget were evaluated and discussed. Also, the limiting values of various quantities were determined and their relevance in high Reynolds number flows discussed. The large structures were shown to be very sensitive to an imposed system rotation. They could be essentially eliminated with a stabilizing system rotation (around the spanwise axis) small enough for only minor damping of the rest of the scales. Despite the fact that the large structures dominate the appearance of the flow field their energy content was shown to be relatively small, on the order of 10% of the total turbulent kinetic energy.