Triangle numbers $T_k$ are integers of the form $\frac{k(k+1)}{2}$.
A few triangle numbers happen to be perfect squares like $T_1=1$ and $T_8=36$, but more can be found when considering the product of two triangle numbers. For example, $T_2\cdot T_{24}=3\cdot 300={30}^2$.

Let $S(n)$ be the sum of $c$ for all integers triples $(a,b,c)$ with $0<c\le n$, $c^2=T_a\cdot T_b$ and $0<a<b$. For example, $S(100)=\sqrt{T_1{T}_8}+\sqrt{T_2{T}_{24}}+\sqrt{T_1{T}_{49}}+\sqrt{T_3{T}_{48}}=6+30+35+84=155$.

You are given $S({10}^5)=1479802$ and $S({10}^9)=241614948794$.

Find $S({10}^{35})$. Give your answer modulo $136101521$.