  The topological properties in topological superconductors are usually
characterized by the bulk Chern numbers, edge-state spectra, and Majorana zero
modes. Whether they are equivalent or inequivalent is not well understood.
Here, we investigate this issue with focus on a checkerboard-lattice model
combining the Chern insulator and chiral $p$-wave superconductivity. Multiple
topologically superconducting phases with Chern numbers up to $\mathcal{N}=4$
are produced. We explicitly demonstrate the mismatch between the Chern numbers,
edge states and Majorana zero modes in this two-dimensional
topological-superconductor model. The intrinsic reason is that some edge states
in the superconducting phases inherited from the Chern-insulator phase are not
protected by the particle-hole symmetry. We further check the mismatches in
vortex states. Our results therefore clarify these different but complementary
topological features and suggest that further considerations are required to
characterize various topological superconductors.
