Non-Abelian fracton order from gauging a mixture of subsystem and global symmetries

We construct a one-step gauging procedure of a pure matter theory on a lattice with a nontrivial mixture of subsystem and global symmetries. Gauging such mixed symmetries can produce non-Abelian fracton orders, which contain immobile excitations with nontrivial fusion rules. Furthermore, by constraining the local Hilbert space, the algebra of charge and flux operators can be identified with a subalgebra of the quantum double models, so that we can classify the species of fractons, lineons, etc., from the gauge group. In particular, if the resulting algebra depends on the constraint, it leads to the fusion of excitations of one geometry into that of another geometry (e.g. loops into lineons) in the Abelian case.