The Hofstadter model exemplifies a large class of physical systems characterized by particles hopping on a lattice immersed in a gauge field. Recent advances in various synthesis platforms have enabled highly controllable simulations of such systems using tuned gauge fields with complex spatial textures. These synthetic gauge fields may introduce synthetic symmetries not found in electronic materials. Here, in the SU(2) non-Abel Hofstadter model, we theoretically show the emergence of multiple non-isomorphic chiral symmetries, combining internal unitary antisymmetry and fractional space translations. Depending on the value of the gauge field, the non-symmorphic chiral symmetry exhibits a non-abelian algebra, preserves the Kramer quartet state in the bulk band structure, and creates a general 4-fold degeneracy at all momentums. These non-isomorphic chiral symmetries guard the double Dirac semimetals with zero energy, and introducing boundaries introduce gaps in the quantum-confined insulating phase. Moreover, system size parity can determine whether the resulting insulating phase is trivial or topological. Our work shows pathways for creating topologies via synthetic symmetries emerging from synthetic gauge fields.
