  We investigate unbounded domains in hierarchically hyperbolic groups and
obtain constraints on the possible hierarchical structures. Using these
insights, we characterise the structures of virtually abelian HHGs and show
that the class of HHGs is not closed under finite extensions. This provides a
strong answer to the question of whether being an HHG is invariant under
quasiisometries. Along the way, we show that infinite torsion groups are not
HHGs.
  By ruling out pathological behaviours, we are able to give simpler, direct
proofs of the rank-rigidity and omnibus subgroup theorems for HHGs. This
involves extending our techniques so that they apply to all subgroups of HHGs.
