Let $G$ be a finite group. Several recent papers (see e.g. http://www.jstor.org/discover/10.2307/2695441) deal with the following notion: $G$ is called a group with perfect order subsets or briefly, a POS-group if the number of elements of any possible order in $G$ is a divisor of $|G|$. Note that the symmetric group $S_n$ is not a POS-group for any $n\ge 4$ by http://arxiv.org/pdf/1007.0568.pdf.

Inspired by the above notion, we will call $G$ a group with perfect subgroup order subsets or briefly, a PSOS-group if the number of subgroups of any possible order in $G$ is a divisor of $|G|$. It is easy to see that a finite nilpotent group is a PSOS-group if and only if it is cyclic. Also, there are many examples of non-nilpotent PSOS-groups, such as the dihedral groups $D_{2n}$ with $n$ odd.

My question is whether $S_n$ is a PSOS-group, more precisely which are the positive integers $n$ such that $S_n$ is a PSOS-group?

Let $G$ be a finite group. Several recent papers (see e.g. http://www.jstor.org/discover/10.2307/2695441) deal with the following notion: $G$ is called a group with perfect order subsets or briefly, a POS-group if the number of elements of any possible order in $G$ is a divisor of $|G|$. Note that the symmetric group $S_n$ is not a POS-group for any $n\ge 4$ by http://arxiv.org/pdf/1007.0568.pdf.

Inspired by the above notion, we will call $G$ a group with perfect subgroup order subsets or briefly, a PSOS-group if the number of subgroups of any possible order in $G$ is a divisor of $|G|$. Obviously, every finite cyclic group is a PSOS-group. Also, there are many examples of non-cyclic PSOS-groups, such as the dihedral groups $D_{2n}$ with $n$ odd.

My question is whether $S_n$ is a PSOS-group, more precisely which are the positive integers $n$ such that $S_n$ is a PSOS-group?

Is it true that the number of subgroups of any possible order of $S_n$ divides $n!$?

Let $G$ be a finite group. Several recent papers (see e.g. http://www.jstor.org/discover/10.2307/2695441) deal with the following notion: $G$ is called a group with perfect order subsets or briefly, a POS-group if the number of elements of any possible order in $G$ is a divisor of $|G|$. Note that the symmetric group $S_n$ is not a POS-group for any $n\ge 4$ by http://arxiv.org/pdf/1007.0568.pdf.

Inspired by the above notion, we will call $G$ a group with perfect subgroup order subsets or briefly, a PSOS-group if the number of subgroups of any possible order in $G$ is a divisor of $|G|$. It is easy to see that a finite nilpotent group is a PSOS-group if and only if it is cyclic. Also, there are many examples of non-nilpotent PSOS-groups, such as the dihedral groups $D_{2n}$ with $n$ odd.

My question is whether $S_n$ is a PSOS-group, more precisely which are the positive integers $n$ such that $S_n$ is a PSOS-group?