Any finite group is the group of symmetries of some map (atom--bifurcation)
Maps are studied, i.e. cell decompositions of closed two-dimensional surfaces, or two-dimensional atoms, which encode bifurcations of Liouville fibrations of nondegenerate integrable Hamiltonian systems. Any finite group G is proved to be the symmetry group of an orientable map (of an atom). Moreover one such a map X(G) is constructed algorithmically. Upper bounds are obtained for the minimal genus Mg(G) of an orientable map with the given symmetry group G, and for the minimal number of vertices, edges and sides of such maps.