# Subgroup Lattice of D14

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.

• How are you counting 6 distinct subgroups of order 7? Dec 3, 2014 at 17:09
• 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? Dec 3, 2014 at 17:13
• But they're not all in distinct subgroups! Dec 3, 2014 at 17:15
• Oh, so it's just the one subgroup (σ) that has an order of 7? Dec 3, 2014 at 17:30
• Yes, that's right. Dec 3, 2014 at 17:31

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.