Here is how far I am: $1000 = 5^{3}\cdot 2^{3}.$ Number of $5$-Sylows is $1$ mod $5$ and has to divide $8,$ so its $1$ making it normal. Lets call it $F.$ Then $G/F$ has order $2^{3}$ and is thus solvable and $F$ has order $5^{3}$ so is solvable implying $G$ is solvable.
Is this correct? Thanks