  In five dimensions, we consider a model described by the Einstein gravity
with a source given by a scalar field and various Abelian gauge fields with
dilatonic-like couplings. For this model, we are able to construct two dyonic
black holes whose three-dimensional horizons are modeled by two nontrivial
homogeneous Thurston's geometries. The dyonic solutions are of Lifshitz type
with an arbitrary value of the dynamical exponent. In fact, the first gauge
field ensures the anisotropy asymptotic while the remaining Abelian fields
sustain the electric and magnetic charges. Using the Hamiltonian formalism, the
mass, the electric and magnetic charges are explicitly computed. Interestingly
enough, the dyonic solutions behave like Chern-Simons vortices in the sense
that their electric and magnetic charges turn to be proportional. The extension
with an hyperscaling violating factor is also scrutinized where we notice that
for specific values of the violating factor, purely magnetic solutions are
possible.
