||It has been recently argued that supergroups emerge from junctions of branes in string theory and M-theory (arXiv:1410.1175, arXiv:1703.00982, arXiv:1512.06646). We study the BPS indices of the supergroup WZW models which may describe certain junctions of M2-branes and M5-branes. We find that the vacuum configurations of the brane system are in one-to-one correspondence with the weight diagrams of the associated Lie superalgebras. For the case arising from the BLG model an explicit expression can be given in terms of a mock modular form. We discuss the physical implications of the mock modularity and its modular completion of the index, at the cost of a holomorphic anomaly. We will also consider the alternative matrix quantum mechanical descriptions of the supergroup WZW models.