No. They are separate genera and species from the same family, Brassicaceae, informally known as the mustards, mustard flowers, the crucifers, cruciferous vegetables or the cabbage family.
The family contains well-known species such as Brassica oleracea (broccoli, cabbage, cauliflower, kale, collards, Brussels sprouts, etc.), Brassica rapa (turnip, mizuna, napa cabbage, etc.), Brassica napus (canola, etc.), Raphanus sativus (common radish), Armoracia rusticana(horseradish), and many others.
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Horseradish a plant, the root of which is grated to make a pungent sauce to accompany meats and sometimes fish. It is a member of the Brassica family of plants - the cabbages.