# How can I find the subgroups of $S_n$? [duplicate]

As the title says: how can I find the subgroups of $S_n$? I know that $S_n = <(1 2), (1 2 .. n)>$ and $S_n = <(1 i): i = 2, 3, .. n>$. But how do I find all subgroups? And how do I know their orders?

## marked as duplicate by Namaste abstract-algebra StackExchange.ready(function() { if (StackExchange.options.isMobile) return; $('.dupe-hammer-message-hover:not(.hover-bound)').each(function() { var$hover = $(this).addClass('hover-bound'),$msg = $hover.siblings('.dupe-hammer-message');$hover.hover( function() { $hover.showInfoMessage('', { messageElement:$msg.clone().show(), transient: false, position: { my: 'bottom left', at: 'top center', offsetTop: -7 }, dismissable: false, relativeToBody: true }); }, function() { StackExchange.helpers.removeMessages(); } ); }); }); Apr 11 '18 at 12:31

• See also Is there any efficient algorithm for finding subgroups of a given finite group?, which addresses also, $S_n$, a finite group of size $n$. – Namaste Apr 11 '18 at 12:32
• Wilson's "Finite Simple Groups" deals with the the subgroups of $S_n$ and much else of interest besides. amazon.co.uk/Finite-Simple-Groups-Graduate-Mathematics/dp/… – Mark Bennet Apr 11 '18 at 12:41
• @amWhy I have voted to reopen this, because the specific $S_n$ case has been studied, and is more tractable, as you will see from the reference I posted in my other comment (check the index - there is a chapter on the problem - also coverage in Dixon and Mortimer "Permutation Groups"). I imagine the analysis would be a stretch for OP though - but an answer to the $S_n$ case would, I think, be valuable for the site. – Mark Bennet Apr 11 '18 at 12:47