# Expressing dihedral group as internal direct product of normal subgroups

Can dihedral groups be expressed as internal direct product of two normal subgroups? I think no, since an element of a normal subgroup must commute with every other element of other normal subgroup, and if one has a generator of rotation subgroup as element of one normal subgroup, other normal subgroup cannot have reflection elements in it. Meaning both must have either rotation or reflection elements in common, which is also not allowed. Am I missing anything?? Can it written as internal product of more than 2 normal subgroups?

Hint: if $n=2m$, with $m$ odd, then $D_n \cong D_m \times C_2$.