There is an equivalence relation between inclusion of finite groups coming from the world of [subfactors](http://en.wikipedia.org/wiki/Subfactor):


**Definition**: $(H\_{1} \subset G\_{1}) \sim(H\_{2} \subset G\_{2})$ if $(R^{G\_{1}} \subset R^{H\_{1}})\cong(R^{G\_{2}} \subset R^{H\_{2}})$ as subfactors.


Here, $R$ is the hyperfinite $II\_1$ factor (a particular von Neumann algebra), and the groups $G\_1$ and $G\_2$ act by outer automorphisms.
The notation $R^G$ refers to the fixed-point algebra.


**Theorem**: Let $(H \subset G)$ be a subgroup and let $K$ be a normal subgroup of $G$, contained in $H$, then:  

$(H \subset G) \sim (H/K \subset G/K)$. In particular, if $H$ is itself normal: $(H \subset G) \sim (\{1\} \subset G/K) $  

**Theorem** : $(\{1\} \subset G\_{1}) \sim(\{1\} \subset G\_{2})$ iff $G\_1 \simeq G\_2$ as groups.


**Remark** : the relation $\sim$ remembers the groups, but not necessarily the subgroups:  

**Exemple** ([Kodiyalam-Sunder](http://www.mscand.dk/article/view/14281) p47) : $(\langle (1234) \rangle \subset S\_4) \sim (\langle (13),(24) \rangle \subset S\_4)$



> 
> Is there a purely group-theoretic reformulation of the relation $\sim$ ?
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**Motivations**: See [here](https://mathoverflow.net/questions/136171/an-upper-bound-for-the-maximal-subgroups-at-fixed-index) and [here](https://mathoverflow.net/questions/135806/are-subfactor-planar-algebras-hard-to-classify-at-index-6/135994#135994).




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**Some definitions:** A *subfactor* is an inclusion of factors. A *factor* is a von Neumann algebra with a trivial center. The *center* is the intersection with the commutant. A *von Neumann algebra* is an algebra of bounded operators on an Hilbert space, closed by taking bicommutant and dual. Here, $R$ is the hyperfinite $II\_{1}$ factor. $R^{G}$ is the subfactor of $R$ containing all the elements of $R$ invariant under the natural action of the finite group $G$. In its [thesis](http://www.ams.org/books/memo/0237/), Vaughan Jones shows that, for all finite group $G$, this action exists and is unique (up to outer conjugacy, see [here](https://perswww.kuleuven.be/%7Eu0018768/artikels/bourbaki-popa.pdf) p8), and the subfactor $R^{G} \subset R$ completely characterizes the group $G$. See the book [*Introduction to subfactors*](http://www.cambridge.org/us/academic/subjects/mathematics/abstract-analysis/introduction-subfactors) (1997) by Jones-Sunder.



Is there a purely group-theoretic reformulation of an equivalence of subgroups?
For finite groups, the answer was given by Izumi in his paper "Characterization of isomorphic group-subgroup subfactors" (MR1920326). There he looks at the crossed product subfactor, but you can always take duals.


Edit after @Andre's comment:


The actual condition between the two pairs of subgroups is quite technical, and it would basically require reproducing an entire page of a 10 page article. Here is a link to the article:
<http://imrn.oxfordjournals.org/content/2002/34/1791.short> 


See also [this video](https://www.youtube.com/watch?v=I52MOU9F-sg&index=5&list=LLhntpxxSKIETTxsQMN8nywg) (27:30) of a talk of M. Izumi on this subject, at the [Sunder Fest 2012](https://www.youtube.com/playlist?list=PL706CCF11D806FAA0).


