The football went flying (threw, through) the window

It would be through. Threw is the past of throw. Through means that it goes into something,

It is through.

Threw: If it was threw then the sentence would mean: The football went flying and it would throw the window. Or something like that.

Through: If it was though the sentence would mean: The football went flying out the window.

If you use common sense, through not threw is the answer.