How's this for an answer? If G is of order 150 with a non-normal subgroup of 25, then G has 237 elements of order 5!! This is a true statement because there is no group of order 150 with a non-normal subgroup of order 25.
Let G be a group of order 150. Then the Sylow 5 subgroup of G is normal.
The order of G is 75*2. So G has a normal subgroup N of order 75 -- see Huppert Endliche Gruppen I, page 30 or prove it yourself.
If you don't read German, I could supply the proof if necessary.
Thus by Sylow, N has a normal Sylow 5 subgroup which is then normal in G.
By the way, everything you say is true. This is typically the way you analyze groups of small order.