Let $G$ be a finite group then the [fundamental theorem of cyclic groups](http://en.wikipedia.org/wiki/Subgroups_of_cyclic_groups) can be formulated as follows:  

*Theorem*: $G$ is cyclic iff it admits no two different subgroups with the same order.  

*proof*: see [here](https://books.google.co.in/books?id=eWkqG0aiVsMC&lpg=PP1&dq=.%20Fundamentals%20of%20Group%20Theory%3A%20An%20Advanced%20Approach&hl=fr&pg=PP1#v=onepage&q&f=false) p44 together with [Lagrange theorem](http://en.wikipedia.org/wiki/Lagrange%27s_theorem_%28group_theory%29). $\square$


But there is an other characterization of the finite cyclic groups, using the lattice theory:  

*Theorem*: $G$ is cyclic iff its subgroups lattice $\mathcal{L}(G)$ is [distributive](http://en.wikipedia.org/wiki/Distributive_lattice).  

*proof*: see [here](http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.dmj/1077490633) theorem 4 p267. $\square$


So we get immediately the following statement for a finite group $G$:  

If $G$ admits no two different subgroups with the same order, then its subgroups lattice is distributive. 


We ask about a generalization of this statement for an inclusion $(H \subset G)$ of finite groups: 


**Question**: Is it true that if $(H \subset G)$ admits no two different intermediate subgroups $H \subset K \subset G$ with the same order, then its intermediate subgroups lattice $\mathcal{L}(H \subset G)$ is distributive? 


*Remark*: It is checked (by GAP) for $[G:H] \le 31$.  

The converse is false because the inclusion $(S\_2 \times S\_2 \subset S\_3 \times S\_3)$ is a counter-example (see [here](https://mathoverflow.net/a/157860/34538)). 



Generalization of the fundamental theorem of cyclic groups
Say $G=S\_n$ and $H$ is a Young subgroup with three orbits, no two of which have the same size and no two of which have sizes summing to $n/2$. Then the only subgroups between $H$ and $G$ should be three Young subgroups with two orbits, no one of which contains any other and no two of which have the same order.


