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I was just curious to know how many ways there are in which we can partition a group we know that left cosets right cosets and the double cosets can be used to partition are there any other equivalence relations which can be used to partition a group ??

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    Any equivalence relation on the underlying set is an equivalence relation on the group. Without imposing any restrictions on the equivalence relations that is the answer to your question. – 2017-02-04
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    What do you mean? Every equivalence relation yields a partition and vice versa. – 2017-02-04
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    @freakish yes, exactly. That being said, [Stirling numbers of the second kind](https://en.wikipedia.org/wiki/Stirling_numbers_of_the_second_kind) and [Bell Numbers](https://en.wikipedia.org/wiki/Bell_number) can be used to count how many partitions an $n$ element set has. – 2017-02-04
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    When you talk of "the left-right cosets" you mean, even without meaning to, with respect to some subgroup of the group. There are no left/right cosets just like that, without referring to some subgroup. – 2017-02-04
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    I obviously meant that there existed a subgroup. Even if there is none I can create a cyclic subgroup and I will be able to define my cosets accordingly. – 2017-02-05
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    I would like to clarify myself. I know of only the conjugation as an equivalence relation. Are there any other commonly used equivalence relations that can be used to partition my group ?? – 2017-02-05

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I assume you want relations that depend on the group structure in some way (see @MattSamuel 's comment). Here are several. Other answers may suggest more.

  • conjugacy classes

  • classes determined by the orders of their elements

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    yes thank you that really helped. – 2017-02-06