Higgs bundles are pairs of holomorphic vector bundles and holomorphic 1-forms taking values in the endomorphisms of the bundle, and their moduli spaces carry a natural hyperkahler structure, through which one can study Lagrangian subspaces (A-branes) or holomorphic subspaces (B-branes). Notably, these A and B-branes have gained significant attention both within mathematics and string theory. In this talk we shall consider novel correspondences between branes lying completely within the singular fibres of the Hitchin fibration, which can be understood through group isomorphisms. By considering a toy mode, the first half of the talk will introduce the audience to Higgs bundles and some of the open problems in the area -- the second half of the talk will be about work in progress with Steve Bradlow and Lucas Branco, and with Sebastian Schulz.