I think in the subgroup lattice of the non-solvable group $A_5 \times \mathbb{Z}/2^N\mathbb{Z}$, the interval between $\{1\} \times \{1\}$ and $\{ 1 \} \times \mathbb{Z}/2^N\mathbb{Z}$ is a chain of length $N$.
Edit: For any $N$, you can also find primes $p$ and $q$ such that $q \equiv \pm 1$ mod $p^N$, so there are cyclic groups of order $p^N$ inside any group of Lie type over $\mathbb{F}_q$ that has a split torus (or just a copy of where you use $\mathbb{G}_m$). +1$in the split case and$-1$in the non-split case). These will yield intervals that are chains of length$N$. Similarly, the alternating group $A_{p^N}$ contains a cyclic group of order$p^N$. 2 added 395 characters in body; added 82 characters in body; deleted 79 characters in body I think in the subgroup lattice of the non-solvable group$A_5 \times \mathbb{Z}/2^N\mathbb{Z}$, the interval between $\{1\} \times \{1\}$ and $\{ 1 \} \times \mathbb{Z}/2^N\mathbb{Z}$ is a chain of length$N$. Edit: For any$N$, you can also find primes$p$and$q$such that$q \equiv 1$mod$p^N$, so there are cyclic groups of order$p^N$inside any group of Lie type over $\mathbb{F}_q$ that has a split torus (or just a copy of$\mathbb{G}_m$). These will yield intervals that are chains of length$N$. Similarly, the alternating group $A_{p^N}$ contains a cyclic group of order$p^N$. 1 I think in the subgroup lattice of the non-solvable group$A_5 \times \mathbb{Z}/2^N\mathbb{Z}$, the interval between $\{1\} \times \{1\}$ and $\{ 1 \} \times \mathbb{Z}/2^N\mathbb{Z}$ is a chain of length$N\$.