We investigate the fractal energy spectrum and quantum Hall response of a two-dimensional 1/5-depleted square lattice subjected to a perpendicular magnetic field. Using a tight-binding model that includes both nearest-neighbor $\left(t_1\right)$ and next-nearest-neighbor $\left(t_2\right)$ hopping, we compute the Hofstadter butterfly and extract quantized Hall conductivities via Chern number calculations. In the absence of diagonal hopping $(t_2=0)$, the spectrum exhibits exact particle-hole and flux-inversion symmetries, and the total Chern number across all bands vanishes. When $t_2$ is introduced, these symmetries are broken, the butterfly becomes deformed, new gaps open, and remarkably a nonzero total Chern sum can emerge, signaling unconventional topological phases. By systematically varying $t_1$ and $t_2$, we identify regimes with large individual Chern indices and parameter windows where gap stability and Hall plateaus are optimized. Our results demonstrate that lattice depletion combined with diagonal hopping provides a tunable route to engineer robust Chern insulators in both artificial and oxide-based square-lattice systems.

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• Chern insulators
• Quantum Hall effect
• Square lattice
• Tight-binding model