  A spin-1/2 Heisenberg model on honeycomb lattice is investigated by doing
triplon analysis and quantum Monte Carlo calculations. This model, inspired by
Cu$_2$(pymca)$_3$(ClO$_4$), has three different antiferromagnetic exchange
interactions ($J_A$, $J_B$, $J_C$) on three different sets of nearest-neighbour
bonds which form a kagome superlattice. While the model is bipartite and
unfrustrated, its quantum phase diagram is found to be dominated by a quantum
paramagnetic phase that is best described as a spin-gapped hexagonal-singlet
state. The N\'eel antiferromagnetic order survives only in a small region
around $J_A=J_B=J_C$. The magnetization produced by external magnetic field is
found to exhibit plateaus at 1/3 and 2/3 of the saturation value, or at 1/3
alone, or no plateaus. Notably, the plateaus exist only inside a bounded region
within the hexagonal-singlet phase. This study provides a clear understanding
of the spin-gapped behaviour and magnetization plateaus observed in
Cu$_2$(pymca)$_3$(ClO$_4$), and also predicts the possible disappearance of 2/3
plateau under pressure.
