example of a Jordan-Hölder decomposition

A group that has a composition series will often have several different composition series.

For example, the cyclic group $C_{12}$ has $(E,C_2,C_6,C_{12})$, and $(E,C_2,C_4,C_{12})$, and $(E,C_3,C_6,C_{12})$ as different composition series.However, the result of the Jordan-Hölder Theorem is that any two composition series of a group are equivalent, in the sense that the sequence of factor groups in each series are the same, up to rearrangement of their order in the sequence $A_{i+1}/A_i$. In the above example, the factor groups are isomorphic to $(C_2,C_3,C_2)$, $(C_2,C_2,C_3)$, and $(C_3,C_2,C_2)$, respectively.

This is taken from the http://en.wikipedia.org/wiki/Solvable_groupWikipedia article on solvable groups.