  We introduce hybrid fracton orders: three-dimensional gapped quantum phases
that exhibit the phenomenology of both conventional three-dimensional
topological orders and fracton orders. Hybrid fracton orders host both (i)
mobile topological quasiparticles and loop excitations, as well as (ii)
point-like topological excitations with restricted mobility, with non-trivial
fusion rules and mutual braiding statistics between the two sets of
excitations. Furthermore, hybrid fracton phases can realize either conventional
three-dimensional topological orders or fracton orders after undergoing a phase
transition driven by the condensation of certain gapped excitations. Therefore,
they serve as parent orders for both long-range-entangled quantum liquid and
non-liquid phases. We study the detailed properties of hybrid fracton phases
through exactly solvable models in which the resulting orders hybridize a
three-dimensional $\mathbb Z_2$ topological order with (i) the X-Cube fracton
order, or (ii) Haah's code. The hybrid orders presented here can also be
understood as the deconfined phase of a gauge theory whose gauge group is given
by an Abelian global symmetry $G$ and subsystem symmetries of a normal subgroup
$N$ along lower-dimensional sub-regions. A further generalization of this
construction to non-Abelian gauge groups is presented in forthcoming work.
