  We consider the problem of finding lower bounds on the number of unlabeled
$n$-element lattices in some lattice family. We show that if the family is
closed under vertical sum, exponential lower bounds can be obtained from
vertical sums of small lattices whose numbers are known. We demonstrate this
approach by establishing that the number of modular lattices is at least
$2.2726^n$ for $n$ large enough.
  We also present an analogous method for finding lower bounds on the number of
vertically indecomposable lattices in some family. For this purpose we define a
new kind of sum, the vertical 2-sum, which combines lattices at two common
elements. As an application we prove that the numbers of vertically
indecomposable modular and semimodular lattices are at least $2.1562^n$ and
$2.6797^n$ for $n$ large enough.
