Given N, what is a finite non-abelian (and preferably non-solvable) group G in whose subgroup lattice Sub[G] there is an interval that is a chain of length at least N?


Since N can be arbitrarily large (but fixed), perhaps there is no easy answer. In that case, can someone suggest which sorts of groups to look at to find intervals that are chains (say, on the order of 10 subgroups in the chain)?


Thanks in advance!


Edit: Thanks to Carnahan's answer, I see that I should have ruled out direct products of cyclic groups with nonsolvable groups. What I'm interested in are intervals in the subgroup lattice of the form:


$\{ K : H \leq K \leq G \}$


where $H$ is a *corefree* subgroup of $G$.



Which finite nonabelian groups have long chains of subgroups as intervals in their subgroup lattice?
You can construct groups with chains of arbitrary length with the help of wreath products. Since you want to assume that the subgroup $H$ is corefree, you get a *faithful* action of $G$ on the cosets of $H$, so one may identify $G$ with a permutation group. Now assume that $G$ is a transitive permutation group on the set $\Gamma$ and $X$ is a transitive permutation group on the set $\Omega$. The wreath product of $G$ with $X$, that is the semi-direct product $W=X \ltimes G^{\Omega}$, acts on $\Omega\times \Gamma$ by
$$ (\omega, \gamma) (x, f) = (\omega x, \gamma ((\omega x) f ) )
 \quad \text{(where $f\colon \Omega\to G$ and thus $(\omega x) f\in G$.)}
$$
The stabilizer of $(\omega, \gamma)$ is then
$$ W\_{(\omega, \gamma)} = X\_{\omega} \ltimes (G\_{\gamma}\times G \times \dotsm \times G)
$$
(where the component $G\_{\gamma}$ occurs, strictly speaking, at position $\omega$).
Now it is not difficult to see, that if a subgroup $K$ with $W\_{(\omega, \gamma)}\leq K$
contains an element $(x, f) $ with $x\notin X\_{\omega}$, then $K$ contains $G\times G \times \dotsm \times G$. So either $K$ has the form $Y \ltimes (G^{\Omega})$ with $X\_{\omega} < Y \leq X$, or it has the form $X\_{\omega} \ltimes (I \times G \times \dotsm \times G)$ with $G\_{\gamma}\leq I \leq G$.  

So the interval $[W\_{(\omega, \gamma)}, W]$ is lattice isomorphic to the lattice obtained by putting $[X\_{\omega}, X]$ on top of $[G\_{\gamma}, G]$. If the latter are chains, then you get a chain, where the lengths add. Starting with a primitive permutation group (non-solvable, if you wish) and repeating this contruction, you get arbitrarily large chains. Even if you are interested in non-solvable groups, I mention that the Sylow $p$-subgroup of $S\_{p^n}$ is a special case of this contruction.


