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I've managed to find all of the subgroups of D14 (6 of order 7, 7 of order 2 and 1 of order 1) but I'm really struggling to put this into a lattice as I've never done it for dihedral groups before, only for symmetric groups. How do I find which of my subgroups are subgroups of the other subgroups? It's very easy with symmetric groups but I think I'm missing something here. Thanks in advance.

Edit: by D14 I'm using D2n notation so in this case the shape in question is a heptagon.

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    $\begingroup$ How are you counting 6 distinct subgroups of order 7? $\endgroup$
    – Kevin Arlin
    Dec 3, 2014 at 17:09
  • $\begingroup$ To clarify, the shape here is a heptagon and when I said D14 I was using the D2n notation (I've edited this into my first post). So it's the 6 rotations (σ^2 through σ^6) that have an order of 7, isn't it? $\endgroup$
    – tom982
    Dec 3, 2014 at 17:13
  • $\begingroup$ But they're not all in distinct subgroups! $\endgroup$
    – Kevin Arlin
    Dec 3, 2014 at 17:15
  • $\begingroup$ Oh, so it's just the one subgroup (σ) that has an order of 7? $\endgroup$
    – tom982
    Dec 3, 2014 at 17:30
  • $\begingroup$ Yes, that's right. $\endgroup$
    – Kevin Arlin
    Dec 3, 2014 at 17:31

2 Answers 2

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Once you get the count of $7$-element subgroups right, it'll actually be quite easy to finish. None of the $2$-element subgroups can be in a $7$-element subgroup, because a group of order $7$ has no element of order $2$, and none of the $7$-element subgroups can be contained in each other, since then they'd be equal.

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  • $\begingroup$ My apologies for the late reply. I finally managed to understand and complete this question, thanks so much for your help! $\endgroup$
    – tom982
    Dec 8, 2014 at 4:26
  • $\begingroup$ Sure, glad to have been useful. $\endgroup$
    – Kevin Arlin
    Dec 8, 2014 at 5:18
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I could be wrong but I think the only subgroups in the lattice will be D14, <σ>, <ρ> and 1 as anything else will generate the entire group. Not certain though... isn't overly similar to any other Theriault stuff

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