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Continuous variable entanglement between magnon modes in Heisenberg antiferromagnets with Dzyaloshinskii-Moriya (DM) interaction is examined. Different bosonic modes are identified, which allows us to establish a hierarchy of magnon entanglement. We argue that entanglement between magnon modes is determined by a simple lattice-specific parameter, together with the ratio of the strengths of the DM and Heisenberg exchange interactions, and that magnon entanglement can be detected by means of quantum homodyne techniques. As an illustration of the relevance of our findings for possible entanglement experiments in the solid state, a typical antiferromagnet with the perovskite crystal structure is considered, and it is shown that long wave length magnon modes have a maximal degree of entanglement.