In this paper we examine Γ0(N)-orbits on Q^and the suborbital graphs for Γ0(N). Each such suborbitalgraph is a disjoint union of subgraphs whose vertices form a blockof imprimitivity for Γ0(N). Moreover, these subgraphs areshown to be vertex Γ0(N)-transitive and edge Γ0(N)-transitive. Finally, necessary and sufficient conditions forbeing self-paired edge are provided