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The different isomorphism classes of subgroups of $S_3$ is:
trivial, $Z_2$, $Z_3$ and $S_3$ itself - that is $4$ different types

The number of isomorphism classes of subgroups of $S_4$ is $9$ and $S_5$ is $16$.

Do anyone know this number for $S_6$?

Lehs
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    I was going to say "Groupprops does!", but in fact they do not (or if they do, they're not telling). It is evidently at most $56$ (are you hoping it's $25$?) – pjs36 Mar 10 '16 at 19:11
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    GAP should be able to do this fairly easily. First ask for the conjugacy classes of the subgroups, then list the ID's of a representative of each, then convert to a set and take the size. – Tobias Kildetoft Mar 10 '16 at 19:56
  • @TobiasKildetoft: One of the answers to the question I indicated as a duplicate used GAP :-) (The number is $29$, a few more than $25$.) – joriki Mar 10 '16 at 20:17
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    @joriki Ahh, nice (and it does it in very close to the way I described). – Tobias Kildetoft Mar 10 '16 at 20:19
  • @jorki is the number $29$ or $\geq 29$? – Lehs Mar 10 '16 at 21:04
  • I would recommend getting GAP so you can do the calculation yourself. It will be a good tool for similar problems later on as well. – Tobias Kildetoft Mar 10 '16 at 21:14
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    @pjs36 I'm testing a pseudo isomorphism test and I admit I was a little bit curious about $25$ or not. – Lehs Mar 10 '16 at 21:39
  • @TobiasKildetoft I have GAP but I'm to old to learn such a complex system https://www.reddit.com/r/Forth/comments/45e55g/sick_and_tired_of_gap/ – Lehs Mar 10 '16 at 21:41

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As so often OEIS has the answer:

http://oeis.org/A174511

There are exactly 29 isomorphism types of subgroups of $S_6$.

ahulpke
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  • Thank you! I found http://schmidt.nuigalway.ie/subgroups/s6.pdf as a reference of the arxiv paper in the first question to which mine is a duplicate, but have difficulties to understand this table. – Lehs Mar 10 '16 at 21:27
  • The link (the OEIS entry) is simply a list of the numbers of isomorphism types of subgroups for different $n$, together with relevant links and references. – ahulpke Mar 11 '16 at 14:09