High Five, Pythagoras!

A Pythagorean triple is a set of positive integers $a < b < c$ such that $a^2 + b^2 = c^2$. Some examples are $\begin{aligned} 3^2 + 4^2 &= 5^2 \\ 5^2 + 12^2 &= 13^2 \\ 8^2 + 15^2 &= 17^2. \end{aligned}$ Note that each of these Pythagorean triples contains a multiple of 5.

How many Pythagorean triples $\{a, b, c\}$ are there for which none of the three numbers is a multiple of 5 and $c \leq 1000?$