  We study the topological phase in dipolar-coupled two-dimensional breathing
square lattice of magnetic vortices. By evaluating the quantized Chern number
and $\mathbb{Z}_{4}$ Berry phase, we obtain the phase diagram and identify that
the second-order topological corner states appear only when the ratio of
alternating bond lengths goes beyond a critical value. Interestingly, we
uncover three corner states at different frequencies ranging from sub GHz to
tens of GHz by solving the generalized Thiele's equation, which has no
counterpart in condensed matter system. We show that the emerging corner states
are topologically protected by a generalized chiral symmetry of the
quadripartite lattice, leading to particular robustness against disorder and
defects. Full micromagnetic simulations confirm theoretical predictions with a
great agreement. A vortex-based imaging device is designed as a demonstration
of the real-world application of the second-order magnetic topological
insulator. Our findings provide a route for realizing symmetry-protected
multi-band corner states that are promising to achieve spintronic higher-order
topological devices.
