  Discoveries of marked similarities to high-$T_{\text{c}}$ cuprate
superconductors point to the realization of superconductivity in the doped
$J_{\text{eff}} = 1 / 2$ Mott insulator Sr$_2$IrO$_4$. Contrary to the mother
compound of cuprate superconductors, several stacking patterns of in-plane
canted antiferromagnetic moments have been reported, which are distinguished by
the ferromagnetic components as $-++-$, $++++$, and $-+-+$. In this paper, we
clarify unconventional features of the superconductivity coexisting with $-++-$
and $-+-+$ structures. Combining the group theoretical analysis and numerical
calculations for an effective $J_{\text{eff}} = 1 / 2$ model, we show unusual
superconducting gap structures in the $-++-$ state protected by nonsymmorphic
magnetic space group symmetry. Furthermore, our calculation shows that the
Fulde-Ferrell-Larkin-Ovchinnikov superconductivity is inevitably stabilized in
the $-+-+$ state since the odd-parity magnetic $-+-+$ order makes the band
structure asymmetric by cooperating with spin-orbit coupling. These unusual
superconducting properties are signatures of magnetic multipole order in
nonsymmorphic crystal.
