The hunt for anyonic excitations in quantum magnets is frustrated by the absence of any order parameter that could be used to detect such phases. Consequently a very important ally is the Hastings-Oshikawa-Lieb-Schultz-Mattis theorem for 2D quantum magnets, which guarantees that a fully symmetric gapped Mott insulator must be topologically ordered, though is silent on which topological order is permitted. After introducing the HOLSM theorem,  I will explain a new line of argument that constrains which topological order is permitted in a symmetric gapped Mott insulator. For example, I'll show that a fully symmetric magnet with S = 1/2 per unit cell cannot be in the double-semion topological phase. An application of our result is to the Kagome lattice quantum antiferromagnet where recent numerical calculations of entanglement entropy indicate a ground state compatible with either toric code or double semion topological order. Our result rules out the latter possibility.