"Novelty" maximal subgroups in $S_n$ What are the maximal subgroups $M < S_n$ such that $M \cap A_n$ is not maximal in $A_n$?
Maximal subgroups of $S_n$ are described by the O'Nan-Scott theorem and very extensively studied in many papers. So I am pretty sure this is known, what I am interested in is a list of examples and a reference.
Is there a general explanation for where these types of maximal subgroups come from?
 A: The best reference for this topic seems to be this paper:
A classification of the maximal subgroups of the finite alternating and symmetric groups,
Martin W Liebeck, Cheryl E Praeger, Jan Saxl,
Journal of Algebra
Volume 111, Issue 2, December 1987, Pages 365-383,
which I'll denote by LPS. Unfortunately this does not provide a complete answer to the question, and I am not aware of any more recent papers that provide more information.
The main theorem of LPS divides the maximal subgroups of $A_n$ and $S_n$ into six categories (a)-(f). (a) and (b) consist respectively of intransitive and inprimitive subgroups, (c)-(e) are respectively primitive groups of affine, diagonal and wreath (also known as product) types, and (f) the almost simple primitive groups.
The Tables I-VI of LPS list groups $G$ of these types that are exceptionally not maximal in $A_n$ or $S_n$. The five groups in Table 1, with  $n=8,7,11,17,23$ are the only examples in which $G$ has one of the types (a)-(e). In fact one is of Type (b), and the other 4 of Type (c). These all correspond to novelty maximals of $S_n$ (i.e the groups are maximal in $S_n$, but their intersections with $A_n$ are non-maximal).
The remaining tables contain the non-maximal examples $G$ of type (f), almost simple. Unfortunately the tables do not indicate which of them correspond to novelty maximals of $S_n$. Presumably it did not occur to the authors to attempt to provide this information, which I expect they could have done. Most of them do not. I calculated that in Table II (non-maximal groups $G$ contained in a larger subgroup of type (a)-(e)), only the first group $L_2(7).2$ of degree $8$ is a novelty maximal. I haven't tried to decide this for the groups in Tables III-VI.
Another infinite family of novelty maximals of $S_n$ is mentioned in Remark 2 of the paper. These are the wreath type groups $S_m \wr S_2$ with $m \equiv 2 \bmod 4$ and $n=m^2$. In these examples, $G \cap A_n$ is imprimitive, so it is contained in a maximal imprimitive example of Type (b).
Unfortunately these tables do not list novelty subgroups in which $G$  primitive almost simple and $G \cap A_n$ is imprimitive, and I am not aware of any attempt to classify such groups. I suspect that there are infinitely many examples, but that's just a guess.
Magma can find the maximal subgroups of $S_n$ for $n \le 1000$, and I used this to find all novelty maximals. (This could be extended to degree 4095 without much effort, using the database of primitive permutation groups.) There are $24$ examples up to degree 1000 of degrees
$$7, 8, 8, 11, 12, 17, 21, 23, 24, 36, 55, 55, 100, 105, 136, 196, 324, 425, 465, 484, 676, 750, 775, 900.$$
Of these, five (degrees $7,8,11,17,23$) are listed in Table I, one (degree $8$) in Table II, three (degrees $12, 24,55$) in Table IV, one (degree 136) in Table VI, seven (degrees $36, 100, 196, 324, 484, 676, 900$) are examples descibed in Remark 2 of LPS, and the remaining seven (degrees $21,55, 105,425, 465, 750,775$) are primitive groups $G$ of Type (f) with $G \cap A_n$ imprimitive.
