$|G|=112$, prove that $G$ is solvable.
I already proved that $G$ is not simple so I know that it has normal subgroup but I don't know from what order. Another solution that I tried was looking at the sylow subgroups. In that case, if 2-sylow subgroup is normal or 7-sylow subgroup is normal then the solution is obvious, but in case that there are 7 2-sylow subgroups and 8 7-sylow subgroups, I don't know how to continue.
On
The number $112=2^4\cdot 7$ has prime factors $2$ and $7$. By Burnside's $p^aq^b$-theorem, all groups of such order, having only two different prime divisors, are solvable. For order $p^aq$ the proof is not difficult, see here.
In case you do not want to use Burnside's result, we already know from the other question,
Proving that a group of order $112$ is not simple
about the Sylow subgroups. For example, one possibility is that the group has a normal Sylow $7$-subgroup $N$ of order $7$. Hence the quotient has order $2^4$ and is nilpotent (since it is a $p$-group), hence solvable. Now, if $N$ and $G/N$ are solvable, so is $G$, and we are done.
Another approach is to show that all the composition factors of $G$ are of prime order. $A_5$ is the only non-abelian simple group of order $<100$: $G$ is non abelian simple group of order $<100$ then $G\cong A_5$
Since $G$ is not simple as you've claimed, the composition factors of $G$ have order not greater than $56$. Conclude that they are all abelian, and thus cyclic of prime order.
Well... the upper bound $100$ is more than enough. You only need to show that there is no non abelian simple group of order not greater than $56$.