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I will be talking about joint work with Phillip Wesolek. A chief factor of a topological group $G$ is a factor $K/L$, where $K$ and $L$ are closed normal subgroups such that no closed normal subgroup of $G$ lies strictly between $K$ and $L$. We show that a compactly generated locally compact group admits an 'essentially chief series', that is, a finite normal series in which each of the factors is compact, discrete or a chief factor. In the totally disconnected case, the proof is based on the fact that $G$ acts vertex-transitively on a locally finite connected graph with compact open stabilizers. I will also indicate why totally disconnected chief factors can have a complicated normal subgroup structure as groups in their own right, in contrast to semisimple groups.