# How many non-isomorphic groups are of order n? [duplicate]

How many non-isomorphic groups are of order $n$?
## marked as duplicate by Derek Holt group-theory 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(); } ); }); }); Nov 12 '15 at 16:24
• Depends on $n$. – Rocket Man Nov 12 '15 at 16:21
• While I understand the gist of your question, it would be better worded as "How many groups of order $n$ are there that are unique up to isomorphism?" At least, that's how I was taught. (To be isomorphic, it has to be isomorphic to something, and while it's implicit that the groups in this case are not isomorphic to each other, it just sounds bad). – Kevin Long Nov 12 '15 at 16:23
• It is known for $n$ up to $2000$. – Derek Holt Nov 12 '15 at 16:24
• There is no known formula. For some classes of groups it is known (prime order, or product of two primes...), but not for general $n$. See OEIS for a partial list (there is a list up to 2047 in the links on this page), and the "duplicate answer" for a bound. In the list, you'll see the number of groups of order $n$ can be very large when $n$ has many small prime factors, especially $2^k$. – Jean-Claude Arbaut Nov 12 '15 at 16:35