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The group of rotations of Euclidean space in $N$ dimensions is the special orthogonal group $\text{SO}(N)$. It is simple and all its Lie subgroups are (semi)simple as well.

The conformal group of Euclidean space is simple too; it's the indefinite special orthogonal group $\text{SO}(N+1,1)$. But one of its subgroups is the Euclidean isometry group, also called the inhomogeneous special orthogonal group $\text{ISO}(N)$. This subgroup is not simple: $\text{ISO}(N) = \mathbb R^N \rtimes \text{SO}(N)$.

The above was just an example. In general, is there any way I can know if a simple group will or will not have nonsimple subgroups?

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  • $\begingroup$ I expect this should happen a lot; should it only be for Lie groups of dimension $0$ , of which finite groups are special cases (try $\mathfrak{A}_n$, it contains many nonsimple subgroups). I don't know if there's an easy criterion though. Any simple Lie group that contains a free group will do $\endgroup$ – Max Jan 22 '19 at 10:54
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    $\begingroup$ Your claim "all Lie subgroups of $SO(N)$ are semisimple" is false (for $N\ge 2$). Every nontrivial connected Lie group has closed connected non-simple (abelian) subgroups. $\endgroup$ – YCor Jan 22 '19 at 18:40

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