# Classify the non-abelian subgroups of order 20. [duplicate]

$20=2^2\times5$

My initial reaction was to write direct products between an abelian and non abelian groups — for example, $Z_5\times D_2$. However, I realized they are both abelian so I am a little stuck. I can't think of any subgroup that is not abelian. Could it be a semidirect product or something? If so, could you give me a hint? My knowledge of semidirect products is sketchy to say the least.

## 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 11 '15 at 8:45

• $D_4$ is not abelian – Kushal Bhuyan Nov 11 '15 at 5:21
• Pardon me I meant to write $D_2$ because $|D_2|$=4 – user139985 Nov 11 '15 at 5:31
• One obvious one is the dihedral group of order 20, which is the semidirect product of $Z_5$ with $Z_2\times Z_2$. There ought to be a couple of semidirect products of $Z_5$ with $Z_4$ depending on whether the (nontrivial) map from $Z_4$ to Aut $Z_5$ is an isomorphism or maps to a subgroup of order 2. – John Brevik Nov 11 '15 at 5:35
• To classify all nonabelian groups of order $20$, you should use the Sylow Theorems. What can you say about the number of Sylow $2$- and $5$-subgroups? – André 3000 Nov 11 '15 at 5:45
take $D_5\times Z_2$. $D_5$ is non-abelian.
• that I am not sure but ...every group of order $2,3,4,5$ is abelian – Black-horse Nov 11 '15 at 5:42