Exercise 2.5.2 (Subgroup of $D_{16}$ that satisfy a given condition)

Proof. We use the lattice given p. 70.

(a)

Subgroups that are contained in $⟨s{r}^2,r^4⟩$: $$⟨s{r}^2,r^4⟩,⟨s{r}^6⟩,⟨s{r}^2⟩,⟨r^4⟩,⟨1⟩.$$

(b)

Since $⟨s{r}^7,r^4⟩=⟨s{r}^3,r^4⟩$, we obtain the subgroups that are contained in $⟨s{r}^7,r^4⟩$: $$⟨s{r}^7,r^4⟩,⟨r^4⟩,⟨s{r}^3⟩,⟨s{r}^7⟩,⟨1⟩.$$

(c)

Subgroups that contain $⟨r^4⟩$: $$⟨r^4⟩,⟨s{r}^2,r^4⟩,⟨s,r^4⟩,⟨r^2⟩,⟨s{r}^3,r^4⟩,⟨s{r}^5,r^4⟩,⟨s,r^2⟩,⟨r⟩,⟨\mathit{sr},r^2⟩,D_{16}.$$

(d)

Subgroups that contain $⟨s⟩$: $$⟨s⟩,⟨s,r^4⟩,⟨s,r^2⟩,D_{16}.$$