Another way of looking at the subtract twice the last removed digit technique others have described for 7 is that it's equivalent to dividing by 50, and seeing whether the remainder and quotient add up to a known multiple of 7.
Let's try this with 11172686, as you suggested.
Diving that by 50, you get 223453, remainder 36. Add the quotient and the remainder together, and you get 223489. Not sure whether that's a multiple of 7? Then we repeat the process with this new answer.
223489 divided by 50 equals 4469 remainder 39. The quotient and the remainder add up to 4508. Still not sure? Let's try it again.
4508 divided by 50 is 90 remainder 8. Quotient plus remainder gives us 98, and you should know that this is divisible by 7.
Why does this work? It's described in more detail here. That link also generalizes the technique for divisibility by any number ending in 9, or any prime number which can be scaled up to end in 9 (such as 7, which can be scaled up to 49).