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Does there exist an infinite group G, such that for any two non trivial subgroups H and K, H is not a subset of K and K is not a subset of H?

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    Yes (assuming you also require the subgroups to be proper). This is the same as all proper nontrivial subgroup having prime order, and such groups are called Tarski monsters. – 2017-01-06
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    @TobiasKildetoft Do you have an example of one of these Tarski monsters? – 2017-01-06
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    [Tarski monsters](https://en.wikipedia.org/wiki/Tarski_monster_group) are horribly complicated. This comment space is too small to contain their truly remarkable construction. – 2017-01-06
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    I am not sure whether there is a book containing the proof. You may try to find the original papers, but they're deep. – 2017-01-06
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    [Wikipedia](https://en.wikipedia.org/wiki/Tarski_monster_group) has references - basically Olshanskii's papers. – 2017-01-06

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