It seems like proving that groups of a certain size are never simple is usually done with Sylow theorems, showing that a Sylow subgroup of a particular size must be normal. But is there an example group size $n$ where no groups of size $n$ are simple, yet there is a group of size $n$ which has no normal Sylow subgroups? If so, how does the proof go for that size $n$ that no groups of size $n$ are simple?
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The smallest example is $n = 24$.
The symmetric group $S_4$ is a group of order $24$ with no normal Sylow subgroups, and there are no simple groups of order $24$.
I would suggest that the symmetric group $S_5$ of order $120$ might fit the bill. It is not simple, but the only non-trivial normal subgroup is $A_5$ of order $60$ which is not a Sylow subgroup.
There are no simple groups of order $120$.