Subgroup-quotient duality for groups

There are many loose forms of subgroup-quotient duality, i.e., ways of transforming a definition or fact that is about subgroups to a similar fact about quotients, or more generally, ways of interchanging the roles played by subgroups and quotients in an existing definition. One way of thinking about this is that we are trying to "reverse the arrows" or go to the "opposite category" in a crude sense.

The quick glossary of this duality is given below:

Situations where the duality is rigorous
The main situation where the duality is rigorous is the case of a finite abelian group. See subgroup lattice and quotient lattice of finite abelian group are isomorphic.

Subgroup properties
Below are listed some very basic subgroup properties along with their duals:

Basic verbal-marginal duality
The verbal-marginal duality is a loose, not very precise duality between statements that can be made about verbal subgroups and their factor groups versus marginal subgroups and their factor groups. The duality is difficult to work with rigorously, but it helps as an intuitive guide. The rough idea is that verbal subgroups start constructively by looking at images of word maps, so that the quotient groups are nice and "small" while marginal subgroups start by solving equations, so it is the subgroups themselves that are nice and "small."