  For a finite-index $\mathrm{II}_1$ subfactor $N \subset M$, we prove the
existence of a universal Hopf $\ast$-algebra (or, a discrete quantum group in
the analytic language) acting on $M$ in a trace-preserving fashion and fixing
$N$ pointwise. We call this Hopf $\ast$-algebra the quantum Galois group for
the subfactor and compute it in some examples of interest, notably for
arbitrary irreducible finite-index depth-two subfactors. Along the way, we
prove the existence of universal acting Hopf algebras for more general
structures (tensors in enriched categories), in the spirit of recent work by
Agore, Gordienko and Vercruysse.
