Determine, up to isomorphism, the possible chief factors of a group of order $336$.
I have answered this but would like to know if I am right or have missed anything?
So I know the possible chief factors are the possible characteristically simple groups whose order divides $336$. These are either simple or direct products of isomorphic simple groups.
The only non-abelian simple group whose order divides $336$ is a group, call it $H$ whose order is $168$. Then we are left with groups of prime order for primes dividing $336$, giving $C_2$, $C_3$ and $C_7$.
We can also take direct products of copies of these simple groups, as long as the order of the direct product still divides $336$. So does that leave us with the only possible chief factors as
$H$,
$C_2$,
$C_2\times C_2$,
$C_2\times C_2\times C_2$,
$C_2\times C_2\times C_2\times C_2$,
$C_3$,
$C_7$.