>>14251073nope came up with it myself
another thing you can do with the second theorem is identify subgroups as not being normal
first note that you can tell where HK and H intersect K are from the subgroup diagram -- they are the least point above both subgroups and the greatest point below both subgroups, respectively
This gives you a diamond shape in the lattice of subgroups, like the red and blue rectangles I put on the picture
By the second isomorphism theorem, if H is normal, then HK / H is isomorphic to K/(H int K
By one of the other isomorphism theorems, the subgroup lattice of HK/H is just the part of the isomorphism lattice of G between H and HK, and similarly for K / (K int H)
Therefore if we have a diamond of subgroups and one is normal, then the segment between the normal subgroup and the top should look like the segment between the other subgroup and the bottom.
So we can see that neither of the middle red-rectangle subgroups is normal, because both of the top legs of the diamond have an intermediate subgroup and neither of the bottom legs do.
On the other hand, look at the blue rectangles. The segment between the blue rectangle on the left and the top looks like the segment between the blue rectangle on the bottom and the right; a consequence of the left subgroup being normal
This isn't sufficient to show normality -- look at the subgroup lattice of S3 -- but it is sufficient to rule it out
