jmc

Since $2020=2^2\cdot 5\cdot 101$, we can simply list its factors: $$1,2,4,5,10,20,101,202,404,505,1010,2020.$$ There are $12$ factors; only $1,2,4,5,101$ don't have over $3$ factors, so the remaining $12-5=\boxed{7}$ factors have more than $3$ factors.