  We study $N$-component interacting particles (hardcore bosons and fermions)
loaded in topological lattice models with SU$(N)$-invariant interactions based
on density matrix renormalization group method. By tuning the interplay of
interspecies and intraspecies interactions, we demonstrate that a class of
SU$(N)$ fractional quantum Hall states can emerge at fractional filling factors
$\nu=N/(N+1)$ for bosons ($\nu=N/(2N+1)$ for fermions) in the lowest Chern
band, characterized by the nontrivial fractional Hall responses and the
fractional charge pumping. Moreover, we establish a topological
characterization based on the $\mathbf{K}$ matrix, and discuss the close
relationship to the fractional quantum Hall physics in topological flat bands
with Chern number $N$.
