Geodesic
Canonical subgroups of $\mathbb{H}_1\rtimes SL(2,\mathbb{R})$
Bollettino della Unione matematica italiana, Série 8, 5B (2002) no. 2, pp. 405-430

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We classify, up to conjugation, all subgroups of the semidirect products $\mathbb{H}_1 \rtimes SL(2, \mathbb{R})$ and $\mathbb{R}^{2} \rtimes SL(2, \mathbb{R})$. Our methods can also be applied to all Lie groups that are locally isomorphic to them. 
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     author = {De Mari, Filippo and Nowak, Krzysztof},
     title = {Canonical subgroups of $\mathbb{H}_1\rtimes SL(2,\mathbb{R})$},
     journal = {Bollettino della Unione matematica italiana},
     pages = {405--430},
     publisher = {mathdoc},
     volume = {Ser. 8, 5B},
     number = {2},
     year = {2002},
     zbl = {1098.22007},
     mrnumber = {MR1911198},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/BUMI_2002_8_5B_2_a7/}
}
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De Mari, Filippo; Nowak, Krzysztof. Canonical subgroups of $\mathbb{H}_1\rtimes SL(2,\mathbb{R})$. Bollettino della Unione matematica italiana, Série 8, 5B (2002) no. 2, pp. 405-430. http://geodesic.mathdoc.fr/item/BUMI_2002_8_5B_2_a7/