 Subgroups of polycyclic groups are polycyclic

Given a cyclic-factor subnormal series $1=G_0⊴\cdots ⊴G_n=G$ and $H\le G$, intersect each term with $H$. The induced map $(H\cap G_{i+1})/(H\cap G_i)\to G_{i+1}/G_i$ is injective. Since every subgroup of a cyclic group is cyclic, the intersection chain is again a cyclic-factor subnormal series, after removing repetitions. Hence $H$ is a polycyclic group. In particular, every subgroup is finitely generated.