If the number of subgroups $H$ of some finite group $G$ is mentioned, as far as I understood, all subgroups are considered, no matter whether two or more of them are isomorphic, whereas the number of conjugacy classes is the number of different (non-isomorphic) subgroups of $G$. Is that right ?
What is the easiest way to check whether a subgroup $H$ of a group $G$ is normal ? Is it best to check, whether there is an element $a\in G$ and an element $b\in H$, such that $aba^{-1}$ is not element of $H$, or is there a better way ?
MAGMA allows to determine the conjugacy classes and the normal subgroups of a finite group $G$. How can I determine the number of subgroups with MAGMA ?
Magma has a
IsNormal(G,H)command to tell you if $H$ is normal in $G$.If you want to know the number of subgroups of $G$ having some specific property, having a specified order, or just in general all the subgroups, use the
Subgroups(G)command. It will list the conjugacy classes, and for each conjugacy class it will tell you the "length"; this is the number of subgroups in that conjugacy class. So you can just add them up.