  We propose a network-model realization of magnetic higher-order topological
phases (HOTPs) in the presence of the combined space-time symmetry
$C_4\mathcal{T}$ -- the product of a fourfold rotation and time-reversal
symmetry. We show that the system possesses two types of HOTPs. The first type,
analogous to Floquet topology, generates a total of $8$ corner modes at $0$ or
$\pi$ eigenphase, while the second type, hidden behind a weak topological
phase, yields a unique phase with $8$ corner modes at $\pm\pi/2$ eigenphase
(after gapping out the counterpropagating edge states), arising from the
product of particle-hole and phase rotation symmetry. By using a bulk
$\mathbb{Z}_4$ topological index ($Q$), we found both HOTPs have $Q=2$, whereas
$Q=0$ for the trivial and the conventional weak topological phase. Together
with a $\mathbb{Z}_2$ topological index associated with the reflection matrix,
we are able to fully distinguish all phases. Our work motivates further studies
on magnetic topological phases and symmetry protected $2\pi/n$ boundary modes,
as well as suggests that such phases may find their experimental realization in
coupled-ring-resonator networks.
