  In 1934, Garrett Birkhoff has shown that the number of isomorphism classes of
finite metabelian groups of order $p^{22}$ tends to infinity with $p$. More
precisely, for each prime number $p$ there is a family
$(M_\lambda)_{\lambda=0,...,p-1}$ of indecomposable and pairwise nonisomorphic
metabelian $p$-groups of the given order. In this manuscript we use recent
results on the classification of possible embeddings of a subgroup in a finite
abelian $p$-group to construct families of indecomposable metabelian groups,
indexed by several parameters, which have upper bounds on the exponents of the
center and the commutator subgroup.
