Solution:

Let the numbers be $a,$ $b,$ $c.$ Since $a = 4(b + c),$ we get $4(b + c) + (b + c) = 20,$ so $b + c = 4$ and $a = 16.$

With $b = 7c,$ we have $7c + c = 4,$ so $c = \dfrac{1}{2}$ and $b = \dfrac{7}{2}.$

The product is $16 \cdot \dfrac{7}{2} \cdot \dfrac{1}{2} = 28.$

Thus, the correct answer is A.