Groups of order $28$

Sylow gives \[ n_7\equiv1\pmod7, \qquad n_7\mid4, \] so \(n_7=1\). Thus \(P\trianglelefteq G\) and \(P\cong C_7\). A Sylow \(2\)-subgroup \(Q\) has order \(4\), hence \[ Q\cong C_4 \quad\text{or}\quad Q\cong C_2\times C_2. \] Since \(P\cap Q=1\) and \(|P||Q|=28\), we have \(G=PQ\), so \[ G\cong C_7\rtimes_\theta Q \] for some action \[ \theta:Q\longrightarrow\Aut(C_7)\cong C_6. \]

The image of a homomorphism \(C_4\to C_6\) has order dividing both \(4\) and \(6\), hence has order \(1\) or \(2\).

For the trivial action, \[ G\cong C_7\times C_4\cong C_{28}. \] For a nontrivial action, a generator of \(C_4\) acts by the unique involution in \(\Aut(C_7)\), namely inversion. Up to automorphisms of \(C_4\) and \(C_7\), there is only one such action. It gives \[ C_7\rtimes C_4 =\langle a,t\mid a^7=t^4=1,\ tat^{-1}=a^{-1}\rangle. \]

Any homomorphism \[ C_2^2\longrightarrow C_6 \] has image contained in the unique subgroup of order \(2\) of \(C_6\).

The trivial action gives \[ G\cong C_7\times C_2^2\cong C_{14}\times C_2. \] Any nontrivial action is a surjection \(C_2^2\to C_2\). All nonzero homomorphisms \(C_2^2\to C_2\) are equivalent under \(\Aut(C_2^2)\cong GL_2(\FF_2)\), so there is one nontrivial isomorphism type. Choosing generators \(s,u\) with \(s\) acting by inversion and \(u\) acting trivially gives \[ G\cong (C_7\rtimes C_2)\times C_2. \]

The first two listed direct products are the two abelian groups of order \(28\), one cyclic and one noncyclic. The two nonabelian groups have nonisomorphic Sylow \(2\)-subgroups: one has a cyclic Sylow \(2\)-subgroup \(C_4\), while the other has Sylow \(2\)-subgroup \(C_2^2\). Hence all four types are distinct.