A notion of normal submonoid of a monoid M is introduced that generalizes the normal subgroups of a group. When ordered by inclusion, the set $\mathsf {NorSub}(M)$ of normal submonoids of M is a complete lattice. Joins are explicitly described and the lattice is computed for the finite full transformation monoids $T_n$, $n\geq ~1$. It is also shown that $\mathsf {NorSub}(M)$ is modular for a specific family of commutative monoids, including all Krull monoids, and that it, as a join semilattice, embeds isomorphically onto a join subsemilattice of the lattice $\mathsf {Cong}(M)$ of congruences on M. This leads to a new strategy for computing $\mathsf {Cong}(M)$ consisting of computing $\mathsf {NorSub}(M)$ and the so-called unital congruences on the quotients of M modulo its normal submonoids. This provides a new perspective on Malcev’s computation of the congruences on $T_n$.

Let $\unicode[STIX]{x1D70E}=\{\unicode[STIX]{x1D70E}_{i}\mid i\in I\}$ be a partition of the set of all primes $\mathbb{P}$. Let $\unicode[STIX]{x1D70E}_{0}\in \unicode[STIX]{x1D6F1}\subseteq \unicode[STIX]{x1D70E}$ and let $\mathfrak{I}$ be a class of finite $\unicode[STIX]{x1D70E}_{0}$-groups which is closed under extensions, epimorphic images and subgroups. We say that a finite group $G$ is $\unicode[STIX]{x1D6F1}_{\mathfrak{I}}$-primary provided $G$ is either an $\mathfrak{I}$-group or a $\unicode[STIX]{x1D70E}_{i}$-group for some $\unicode[STIX]{x1D70E}_{i}\in \unicode[STIX]{x1D6F1}\setminus \{\unicode[STIX]{x1D70E}_{0}\}$ and we say that a subgroup $A$ of an arbitrary group $G^{\ast }$ is $\unicode[STIX]{x1D6F1}_{\mathfrak{I}}$-subnormal in $G^{\ast }$ if there is a subgroup chain $A=A_{0}\leq A_{1}\leq \cdots \leq A_{t}=G^{\ast }$ such that either $A_{i-1}\unlhd A_{i}$ or $A_{i}/(A_{i-1})_{A_{i}}$ is $\unicode[STIX]{x1D6F1}_{\mathfrak{I}}$-primary for all $i=1,\ldots ,t$. We prove that the set ${\mathcal{L}}_{\unicode[STIX]{x1D6F1}_{\mathfrak{I}}}(G)$ of all $\unicode[STIX]{x1D6F1}_{\mathfrak{I}}$-subnormal subgroups of $G$ forms a sublattice of the lattice of all subgroups of $G$ and we describe the conditions under which the lattice ${\mathcal{L}}_{\unicode[STIX]{x1D6F1}_{\mathfrak{I}}}(G)$ is modular.