Using homotopy theory, we classify nodal structures of non-Hermitian systems under a range of symmetries. Without any additional symmetries, non-hermitian degenerate operators are related to a braid group-valued topological invariant. This invariant is non-Abelian, which forbids a naïve extension of the fermion-doubling theorem to non-hermitian systems. In art. we show that this can lead to non-Abelian monopole charges. In art. we consider spatial symmetries of crystalline systems in two dimensions, and show that these can enforce degenerate points. Additional anti-unitary symmetries can enforce exceptional lines, a uniquely non-Hermitian phenomenon. In art. we investigate the homotopy structure of 𝓟𝓣-symmetric operators in general, and use it to classify non-Hermitian degeneracies for these systems.