  The orbital magnetization of the electron gas on a two-dimensional kagome
lattice under a perpendicular magnetic field is theoretically investigated. The
interplay between the lattice geometry and magnetic field induce nontrivial
$k$-space Chern invariant in the magnetic Brillouin zone, which turns to result
in profound effects on the magnetization properties. We show that the
Berry-phase term in the magnetization gives a paramagnetic contribution, while
the conventional term brought about by the magnetic response of the magnetic
Bloch bands produces a diamagnetic contribution. As a result, the superposition
of these two components gives rise to a delicate oscillatory structure in the
magnetization curve when varying the electron filling factor. The relationship
between this oscillatory behavior and the Hofstadter energy spectrum is
revealed by selectively discussing the magnetization and its two components at
the commensurate fluxes of $f$=1/4, 1/3, and 1/6, respectively. In particular,
we reveal as a typical example the fractal structure in the magnetic
oscillations by tuning the commensurate flux around $f$=1/4. The
finite-temperature effect on the magnetization is also discussed.
