Online groups have become increasingly prevalent, providing users with space to share experiences and explore interests. Therefore, user-centric group discovery task, i.e., recommending groups to users can help both users' online experiences and platforms' long-term developments. Existing recommender methods can not deal with this task as modeling user-group participation into a bipartite graph overlooks their item-side interests. Although there exist a few works attempting to address this task, they still fall short in fully preserving the social context and ensuring effective interest representation learning. In this paper, we focus on exploring the intents that motivate users to participate in groups, which can be categorized into different types, like the social-intent and the personal interest-intent. The former refers to users joining a group affected by their social links, while the latter relates to users joining groups with like-minded people for self-enjoyment. To comprehend different intents, we propose a novel model, DiRec, that first models each intent separately and then fuses them together for predictions. Specifically, for social-intent, we introduce the hypergraph structure to model the relationship between groups and members, leading to a richer understanding of the social context. As for interest-intent, we employ novel structural refinement on the interactive graph to uncover more intricate user behaviors and group interests, realizing better representation learning of interests. Furthermore, we also observe the intent overlapping in real-world scenarios and devise a novel self-supervised learning loss that encourages such alignment for final recommendations. Extensive experiments on three public datasets show the significant improvement of DiRec over the state-of-the-art methods.