## Use the subgroup lattice of Dih(16) to answer questions about subgroups

In each of 1-4 list all the subgroups of $D_{16}$ that satisfy the given condition.

1. Subgroups that are contained in $\langle sr^2, r^4 \rangle$
2. Subgroups that are contained in $\langle sr^7, r^4 \rangle$
3. Subgroups that contain $\langle r^4 \rangle$
4. Subgroups that contain $\langle s \rangle$

We assume the subgroup lattice of $D_{16}$.

1. $\langle sr^2,r^4 \rangle$ contains $\langle sr^2,r^4 \rangle$, $\langle sr^6 \rangle$, $\langle sr^2 \rangle$, $\langle r^4 \rangle$, and $1$.
2. Note that $\langle sr^7,r^4 \rangle = \langle sr^3,r^4 \rangle$ since $sr^7 = sr^3 \cdot r^4$ and $sr^3 = r^4 \cdot sr^7$. Thus the subgroups contained in $\langle sr^7,r^4 \rangle$ are $\langle sr^3,r^4 \rangle$, $\langle r^4 \rangle$, $\langle sr^3 \rangle$, $\langle sr^7 \rangle$, and $1$.
3. $\langle r^4 \rangle$ is contained in $\langle r^4 \rangle$, $\langle sr^2,r^4 \rangle$, $\langle s,r^4 \rangle$, $\langle r^2 \rangle$, $\langle sr^3, r^4 \rangle$, $\langle sr^5,r^4 \rangle$, $\langle s,r^2 \rangle$, $\langle r \rangle$, $\langle sr,r^2 \rangle$, and $D_{16}$.
4. $\langle s \rangle$ is contained in $\langle s \rangle$, $\langle s,r^4 \rangle$, $\langle s,r^2 \rangle$, and $D_{16}$.

• Samantha  On September 30, 2010 at 5:13 pm

I believe that for number 1 < r^4 > is also contained within < sr^2,r^4 >

• nbloomf  On September 30, 2010 at 6:58 pm

You’re right. Thanks!

• Jake  On December 7, 2011 at 7:05 am

I think $\langle r^{4} \rangle$ is contained in $\langle s, r^{4} \rangle$ and not $\langle sr^{4} \rangle$

• nbloomf  On December 9, 2011 at 1:04 am

Thanks!