  We show that the Witt class of a weakly group-theoretical non-degenerate
braided fusion category belongs to the subgroup generated by classes of
non-degenerate pointed braided fusion categories and Ising braided categories.
This applies in particular to solvable non-degenerate braided fusion
categories. We also give some sufficient conditions for a braided fusion
category to be weakly group-theoretical or solvable in terms of the
factorization of its Frobenius-Perron dimension and the Frobenius-Perron
dimensions of its simple objects. As an application, we prove that every
non-degenerate braided fusion category whose Frobenius-Perron dimension is a
natural number less than 1800, or an odd natural number less than 33075, is
weakly group-theoretical.
