  As a topological insulator, the quantum Hall (QH) effect is indexed by the
Chern and spin-Chern numbers $\mathcal{C}$ and $\mathcal{C}_{\text{spin}}$. We
have only $\mathcal{C}_{\text{spin}}=0$ or $\pm \frac{1}{2}$ in conventional QH
systems. We investigate QH effects in generic monolayer honeycomb systems. We
search for spin-resolved characteristic patterns by exploring Hofstadter's
butterfly diagrams in the lattice theory and fan diagrams in the low-energy
Dirac theory. The Chern and spin-Chern numbers are calculated based on the
bulk-edge correspondence in the lattice theory and on the Kubo formula in the
Dirac theory. It is shown that the spin-Chern number can takes an arbitrary
high value for certain QH systems in coexistence with buckled structure and
magnetic order. This is a new type of topological insulators. Samples may be
provided by silicene with ferromagnetic order and transition-metal oxide with
antiferromagnetic order.
