  We demonstrate theoretically a large transverse magneto-optical Kerr effect
(TMOKE) in subwavelength gratings consisting of alternating magneto-insulating
and nonmagnetic dielectric nanostripes. The reflectivity of the grating reaches
$96\%$ at the frequencies corresponding to the maximum of the TMOKE response.
The combination of a large TMOKE response and high reflectivity is important
for applications in $3$D imaging, magneto-optical data storage, and magnonics.
