Chief factors of a group, are they always normal in G. Chief factors of a solvable group G, are they always normal in G? I know Chief factors are characteristically simple. Also, characteristic subgroups are normal in G. So I think Chief factors are normal in G. But I found an example, 1,V4,A4,S4 chief series, where Z2 is a chief factor, which is not normal in S4. Now I am confused.Please clarify this.

I think your example is

$$1\lhd C_2\lhd V_4\lhd A_4\lhd G$$

But this is **not** a normal series since, as you mentioned, $\;C_2\rlap{\;\,/}\lhd S_4\;$ and thus we have no chief factors here.

Where we do have a _normal series_ is with

$$1\lhd V_4\lhd A_4\lhd S_4\;\;\;(**)$$

and for this to be a **chief series** it must be that every factor has no subfactor **that is normal in** $\;S_4\;$ ...and this is true! The only non-trivial factor a quotient in the above series has is $\;C_2\lhd V_4\;$, but since $\;C_2\;$ is not normal in $\;S_4\;$ we don't care, and thus $\;(**)\;$ is a _chief series_ ...