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**Primenumber** 1. Draw the lattice of subgroups (up to conjugacy) of $\displaystyle S_{3} \times \mathbb{Z}_{2}$.

The symmetry of a system of balancing balls is given by

$\displaystyle S_{3} \times \mathbb{Z}_{2}$ are $\displaystyle S_{3} \times \mathbb{Z}_{2}$ which can be thought of as the group $\displaystyle \{id,(123),(321),(12),(13),(23),-id,-(123),-(321),-(12),-(13),-(23)\}$

Now I know that, up to conjugacy, the subgroups of $\displaystyle S_{3} \times \mathbb{Z}_{2}$ are $\displaystyle S_{3} \times \mathbb{Z}_{2}$, $\displaystyle \mathbb{Z}_{3} \times \mathbb{Z}_{2}=\{id,-id,(123),(321),-(123),-(321)\}$, $\displaystyle \mathbb{Z}_{2} \times \mathbb{Z}_{2}=\{id,-id,(12),-(12)\}$, $\displaystyle S_{3}=\{id,(123),(321),(12),(13),(23)\}$, $\displaystyle \mathbb{Z}_{3}=\{id,(123),(321)\}$, $\displaystyle \mathbb{Z}_{2}^{'}=\{id,(12)\}$, $\displaystyle \mathbb{Z}_{2}=\{id,-id\}$, $\displaystyle \widetilde{\mathbb{Z}_{2}}=\{id,-(12)\}$ and $\displaystyle I={\id\}$.

However, I don't know how to construct a lattice of subgroups from this. How should it be structured?