  Using totally symmetric sets, Chudnovsky, Kordek, Li, and Partin gave a
superexponential lower bound on the cardinality of non-abelian finite quotients
of the braid group. In this paper, we develop new techniques using multiple
totally symmetric sets to count elements in non-abelian finite quotients of the
braid group. Using these techniques, we improve the lower bound found by
Chudnovsky et al. We exhibit totally symmetric sets in the virtual and welded
braid groups, and use our new techniques to find superexponential bounds for
the finite quotients of the virtual and welded braid groups.
