We present images of X-ray emission from hot ps within the Coma cluster of galaxies. These maps, made with the ROSAT satellite, have much higher signal-to-noise ratio than any previous X-ray image of a galaxy duster, and allow cluster structure to be analysed in unprecedented detail. They show greater structural irregularity than might have been anticipated from earlier observations of Coma. Emission is detected from a number of bright cluster galaxies in addition to the two known previously. In four cases, there is evidence that these galaxies lie at the centre of an extended subconcentration within the cluster, possibly the remnant of their associated groups. For at least two galaxies, the images show direct evidence for ongoing disruption of their gaseous atmosphere. The luminosity associated with these galaxies is comparable to that detected around similar ellipticals in much poorer environments. Emission is easily detected to the limit of our field, about 1-degrees from the cluster centre, and appears to become more regular, at large radii. The data show clearly that this archetype of a rich and regular galaxy cluster was, in fact, formed by the merging of several distinct subunits which are not yet fully destroyed.