Lattice of subgroups embeds in partition lattice

Statement
Suppose $$G$$ is a group, $$P(G)$$ is the partition lattice of $$G$$ (i.e., the lattice whose elements are partitions of $$G$$, with $$\alpha \le \beta$$ if $$\alpha$$ is a finer partition than $$\beta$$. Let $$L(G)$$ denote the fact about::lattice of subgroups of $$G$$.

Then, there is an injective homomorphism of lattices from $$L(G)$$ to $$P(G)$$, given as follows: a subgroup $$H$$ of $$G$$ is sent to the partition of $$G$$ given by the fact about::left cosets of $$H$$ in $$G$$ (as per left cosets partition a group). In particular:


 * 1) For subgroups $$H, K$$ the left coset partition arising from the fact about::join of subgroups $$\langle H, K \rangle$$ is the same as the join in the partition lattice of the left coset partitions arising from $$H$$ and $$K$$ respectively.
 * 2) For subgroups $$H, K$$, the left coset partition arising from the fact about::intersection of subgroups $$H \cap K$$ is the same as the meet in the partition lattice of the left coset partitions arising from $$H$$ and $$K$$ respectively.

Related facts

 * Variety of groups is congruence-uniform
 * Variety of groups is congruence-permutable