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If you are interested in the compact real form then "On Non-Linear Realizations of the group $SU(2)$" by Mickelsson and Niederle lists the conjugacy classes of closed proper subgroups of SU(2) as a recap before going on the nonlinear cases. They are

i) The unitary subgroup $U(1)$

ii) The subgroup $N[U(1)]$ (normalizer of $U(1)$)

iii) $C_n$, the cyclic subgroups of order $n$

iv) The subgroups $\tilde{D_{2n}}$ where $\tilde{D_{2n}}/Z_2$ is isomorphic to the dihedral group $D_n$ of order $2n$.

v) The subgroup $\tilde{T}$, where $\tilde{T}/Z_2$ is isomorphic to the tetrahedral group T of order 12.

vi) The subgroup $\tilde{O}$, where $\tilde{O}/Z_2$ is isomorphic to the octahedral group O of order 24.

vii) The subgroup $\tilde{Y}$, where $\tilde{Y}/Z_2$ is isomorphic to the icosahedral group Y of order 60.

They attribute this result to a 'method of Murnaghan' whose book is "The Theory of group representations" and from memory it is in the back as an appendix.

They go on to say which of these lead to homogeneous spaces that are 3-manifolds. An interesting read and possibly of some relevance to your notes.

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If you are interested in the compact real form then "On Non-Linear Realizations of the group $SU(2)$" by Mickelsson and Niederle lists the conjugacy classes of closed proper subgroups of SU(2) as a recap before going on the nonlinear cases

i) The unitary subgroup $U(1)$

ii) The subgroup $N[U(1)]$ (normalizer of $U(1)$)

iii) $C_n$, the cyclic subgroups of order $n$

iv) The subgroups $\tilde{D_{2n}}$ where $\tilde{D_{2n}}/Z_2$ is isomorphic to the dihedral group $D_n$ of order $2n$.

v) The subgroup $\tilde{T}$, where $\tilde{T}/Z_2$ is isomorphic to the tetrahedral group T of order 12.

vi) The subgroup $\tilde{O}$, where $\tilde{O}/Z_2$ is isomorphic to the octahedral group O of order 24.

vii) The subgroup $\tilde{Y}$, where $\tilde{Y}/Z_2$ is isomorphic to the icosahedral group Y of order 60.

They attribute this result to a 'method of Murnaghan' whose book is "The Theory of group representations" and from memory it is in the back as an appendix.

They go on to say which of these lead to homogeneous spaces that are 3-manifolds. An interesting read and possibly of some relevance to your notes.