Order of product of subgroups

Algebra Level 3

Let $$G$$ be a group and let $$H$$ and $$K$$ be two subgroups of $$G$$.

If both $$H$$ and $$K$$ have $$12$$ elements, which of the following numbers cannot be the the number of elements in the set $$HK$$ ?

Notation:

• $$HK = \{hk\mid h\in H~\textrm{and}~k\in K\}$$
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