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 $\alpha \leq \beta$ if $\alpha$ is a finer partition than $\beta$. Let $L(G)$ denote the 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 Left coset (?)s 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 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 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