Eccoci.

A unitary divisor $d$ of a number $n$ is a divisor of $n$ that has the property $gcd(d,n/d)=1$.
The unitary divisors of $4!=24$ are $1,3,8$ and $24$.
The sum of their squares is $1^2+3^2+8^2+{24}^2=650$.

Let $S(n)$ represent the sum of the squares of the unitary divisors of $n$. Thus $S(4!)=650$.

Find $S(100000000!)$ modulo $1000000009$.