The effect of radiative energy transport on the onset and evolution of natural convective flows is studied in a Rayleigh-Benard system. Steady, axisymmetric flows of a radiatively participating fluid contained in a rigid-walled, vertical cylinder which is heated on the base, cooled on top, and insulated on the side wall are calculated by using the Galerkin finite element method. Bifurcation analysis techniques are used to investigate the changes in the flow structure due to internal radiation. The results of this two-parameter study-where the Rayleigh number, Ra and optical thickness $\tau $, are varied-apply to fluids ranging from opaque to nearly transparent with respect to infrared radiation. For any non-opaque fluid, internal radiation eliminates the static state that, without radiation, exists for all values of the Rayleigh number. This heat transfer mechanism also destroys a symmetry of the system that relates clockwise and counter-clockwise flows. The connectivity between characteristic flow families and the range of Ra where families are stable are found to depend greatly on $\tau $. Results demonstrate the inadequacy of characterizing the behaviour of this system using simple notions of radiative transfer in optically thick or thin media; the nonlinear interaction of radiation and flow are far more complicated than these asymptotic limits would imply.