Is every abelian group solvable?

I'm curious to know, does the concept of solvability apply to all abelian groups? Considering that abelian groups possess a certain level of symmetry and simplicity in their structure, does this inherently mean that they can always be decomposed into simpler subgroups in a finite number of steps? Or are there specific conditions or properties that an abelian group must possess in order to be classified as solvable? I'm eager to understand the nuances and implications of this question within the realm of group theory and its applications to cryptography, algebra, and other fields of mathematics.