We classify the conjugacy classes of prime order groups,  acting orientably on the closed orientable surface of genus g,  or equivalently subgroups,  the corresponding mapping class group. This classification is given by 3 canonical actions and the connected sum at fixed points. We also mention application of this classification to finding Hamiltonian cycles in the Cayley graphs of certain finite group presentations and to computing the Farrell cohomology of certain pure mapping class groups.