  We propose a family of Abelian quantum Hall states termed the non-diagonal
states, which arise at filling factors $\nu=p/2q$ for bosonic systems and
$\nu=p/(p+2q)$ for fermionic systems, with $p$ and $q$ being two coprime
integers. Non-diagonal quantum Hall states are constructed in a coupled wire
model, which shows an intimate relation to the non-diagonal conformal field
theory and has a constrained pattern of motion for bulk quasiparticles,
featuring a non-trivial interplay between charge symmetry and translation
symmetry. The non-diagonal state is established as a distinctive
symmetry-enriched topological order. Aside from the usual $U(1)$ charge sector,
there is an additional symmetry-enriched neutral sector described by the
quantum double model $\mathcal{D}(\mathbb{Z}_p)$, which relies on the presence
of both the $U(1)$ charge symmetry and the $\mathbb{Z}$ translation symmetry of
the wire model. Translation symmetry distinguishes non-diagonal states from
Laughlin states, in a way similar to how it distinguishes weak topological
insulators from trivial band insulators. Moreover, the translation symmetry in
non-diagonal states can be associated to the
$\mathbf{e}\leftrightarrow\mathbf{m}$ anyonic symmetry in
$\mathcal{D}(\mathbb{Z}_p)$, implying the role of dislocations as two-fold
twist-defects. The boundary theory of non-diagonal states is derived
microscopically. For the edge perpendicular to the direction of wires, the
effective Hamiltonian has two components: a chiral Luttinger liquid and a
generalized $p$-state clock model. Importantly, translation symmetry in the
bulk is realized as self-duality on the edge. The symmetric edge is thus either
gapless or gapped with spontaneously broken symmetry. For $p=2,3$, the
respective electron tunneling exponents are predicted for experimental probes.
