(Normal) subgroups of different orders of a group of order $20.$

We have a group $G$ whose class equation is $1+4+5+5+5=20$.

(A) Every normal subgroup is a disjoint union of conjugacy classes. Each normal subgroup contains the conjugacy class of order $1$, since it contains the identity. A non-trivial normal subgroup must contain at least one other conjugacy class. The other conjugacy classes have sizes $4$, $5$, $5$, $5$. Hence, the order of a non-trivial normal subgroup must be at least $5$.

(B) Note that a group of order $20$ has a unique Sylow-$5$ subgroup $N$. Also note that a group of order $20$ must have an element $x$ of order $2$. Let $H$ be the subgroup generated by $x$. Note that $HN$ is a subgroup of $G$, since $N$ is normal in $G$, and $HN$ is order $10$, since $H$ and $N$ intersect trivially.

The centralizer is normal in the normalizer, but needn't be normal in $G$.