Fe-based superconductors have attracted research interest because of their rich structural variety, which is due to their layered crystal structures. Here we report the new-structure-type Fe-based superconductors CaAFe(4)As(4) (A = K, Rb, Cs) and SrAFe(4)As(4) (A = Rb, Cs), which can be regarded as hybrid phases between AeFe(2)As(2) (Ae = Ca, Sr) and AFe(2)As(2). Unlike solid solutions such as (Ba1-xKx)Fe2As2 and (Sr1-xNax)Fe2As2, Ae and A do not occupy crystallographically equivalent sites because of the large differences between their ionic radii. Rather, the Ae and A layers are inserted alternately between the Fe2As2 layers in the c-axis direction in AeAFe(4)As(4) (AeA1144). The ordering of the Ae and A layers causes a change in the space group from I4/mmm to P4/mmm, which is clearly apparent in powder X-ray diffraction patterns. AeA1144 is the first known structure of this type among not only Fe-based superconductors but also other materials. AeA1144 is formed as a line compound, and therefore, each AeA1144 has its own superconducting transition temperature of approximately 31-36 K.