An abundance of information regarding the structure of a finite group can be obtained by studying its irreducible characters. Of particular interest are monomial characters – those induced from a linear character of some subgroup – since Brauer has shown that any irreducible character of a group can be written as an integral linear combination of monomial characters. Our primary focus is the class of M-groups, those groups all of whose irreducible characters are monomial. A classical theorem of Taketa asserts that an M-group is necessarily solvable, and Dade proved that every solvable group can be embedded as a subgroup of an M-group. After discussing results related to M-groups, we will construct explicit families of solvable groups that cannot be embedded as subnormal subgroups of any M-group. We also discuss groups possessing a unique non-monomial irreducible character, and prove that such a group cannot be simple.