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We demonstrate both experimentally and numerically multifold enhancement of magneto-optical effects in subwavelength dielectric nanostructures with a magnetic surrounding exhibiting localized magnetic Mie resonances. We employ amorphous silicon nanodisks covered with a thin nickel film and achieve the 5-fold enhancement of the magneto-optical response of the hybrid magnetophotonic array of nanodisks in comparison with a thin nickel film deposited on a flat silica substrate. Our findings allow for a new basis for active and nonreciprocal photonic nanostructures and metadevices, which could be tuned by an external magnetic field.