Group Actions · Superchallenge problem

$GL(2,\mathbb Z_5)$, the invertible $2\times2$ matrices mod $5$, has order $(5^2-1)(5^2-5)=480$. It acts transitively on the $6$ lines through the origin in $\mathbb Z_5^2$ by $M\cdot L=\{Mv:v\in L\}$. Find the order of the stabilizer of the line through $(1,0)$ (the matrices sending this line to itself, not necessarily fixing $(1,0)$ pointwise).