Every subgroup of order 74 in a group of order 148 is normal
We will prove a much general claim: if index of a subgroup is 2, then that subgroup must be normal.
Suppose and .
Now, if then gH = Hg = H
Otherwise if then gH = G H (why? because we know that any two cosets are completely distinct or entirely equal, and number of elements in gH is same as number of elements in H; prove this)
But Hg = GH as well.
So gH = Hg when g is not in H as well.
Hence H is normal subgroup of G.