  We show that a non-solvable, torsion-free, virtually solvable group $S$ must
have Hirsch length $h(S)\geq10$. If $h(S)<14$ then $A_5$ is the only simple
factor. If $S$ is virtually nilpotent and $h(S)\leq14$ then its Fitting
subgroup has nilpotency class $\leq3$. (There are virtually abelian examples
with $h(S)=15$, and this is known to be best possible for such groups.)
