# Subgroup of group of order $44$

Pick the correct statement(s) below:

• $(a)$ There exists a group of order $44$ with a subgroup isomorphic to $\Bbb Z_2 \oplus \Bbb Z_2$.
• $(b)$ There exists a group of order $44$ with a subgroup isomorphic to $\Bbb Z_4$.
• $(c)$ There exists a group of order $44$ with a subgroup isomorphic to $\Bbb Z_2 \oplus \Bbb Z_2$ and a subgroup isomorphic to $\Bbb Z_4$.
• $(d)$ There exists a group of order $44$ without any subgroup isomorphic to $\Bbb Z_2 \oplus \Bbb Z_2$ or to $\Bbb Z_4$.
• For $(1)$ take $G=\mathbb{Z}/2\oplus \mathbb{Z}/2\oplus \mathbb{Z}/11$, for $(2)$ take $G=\mathbb{Z}/4\oplus \mathbb{Z}/11$. – Dietrich Burde Apr 18 '14 at 17:33

This is false. We know that there is only $1$ Sylow 11-subgroup of $G$. And so this must be a normal subgroup of $G$. Call it $H$. Then consider $G/H$. This has order 4. As such is must be isomorphic to either $\mathbb{Z}_2\times \mathbb{Z}_2$ or $\mathbb{Z}_4$. Hence $G \cong H \times \mathbb{Z}_4$ or $G \cong H \times \mathbb{Z}_2\times \mathbb{Z}_2$ or $G \cong H \rtimes \mathbb{Z}_4$ or $G \cong H \rtimes (\mathbb{Z}_2\times \mathbb{Z}_2)$.