Re-occuring subgroups of $S_n$

Question

The symmetric group has a number of families of subgroups also indexed by n such as $A_n$, $D_n$, or $AG(1,p)$. Are there other notable groups such as these which for any $m \in \mathbb{Z}$ exist for some $n > m$? I am particularly interested in such families of groups which satisfy the following condition:

1. The group contains no elements of order $n$
2. The group cannot be embedded in $S_m$ for $m<n$