With the rapid advance of wireless communication technologies, the promising space-air-ground integrated networks (SAGINs) are advocated to provide ubiquitous multicast transmission services for the social community constituted by a group of mobile users that have strong social ties and similar content interests. However, due to the limited yet valuable spectrum resources, the network heterogeneity, and diverse service demands of mobile users, it is challenging to efficiently allocate bandwidth for social communities with the objective of achieving satisfactory quality of experience (QoE) in SAGINs. To address this problem, in this paper, we propose a hierarchical bandwidth allocation scheme to enable high-quality multicast services for social communities in SAGINs. Specifically, we first develop a hierarchical bandwidth allocation framework. Wherein, the low earth orbit (LEO) satellite is utilized to provide space-to-air (S2A) unicast bandwidth for unmanned aerial vehicles (UAVs) at a certain price. Each UAV is employed to provide air-to-ground (A2G) multicast bandwidth for ground social communities with a certain A2G multicast bandwidth charge. We then formulate the hierarchical bandwidth allocation problem as a four-stage Stackelberg game, where the target of each participant is to maximize its own utility. Afterward, through the game analysis by the backward induction method, the existence of the Stackelberg equilibrium is proved, where the closed-form solutions on the optimal policies of both the social communities and UAVs are derived by the convex optimization method, and the optimal pricing policies of the LEO satellite is achieved by a proposed gradient descent iteration algorithm. Finally, extensive experiments are conducted to demonstrate that the proposed scheme can greatly increase the utilities of social communities while consuming a less bandwidth compared to other conventional schemes.