The Fiberglass Menagerie

My daughter has started a zoo on our refrigerator door. She has 20 animals, and wants to put them in cages, with an equal number of animals in each cage. (She's not concerned about some animals eating others; she assures me they're all vegetarians.) Which arrangement will afford her the greatest number of possible groupings of the animals?

Assume that the cages themselves are indistinguishable. So the same grouping of animals placed in different cages would not be considered a different grouping.

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