We investigate the breakups of an encapsulated conducting aqueous droplet under a direct-current electric field via extensive experiments and theoretical analysis. The encapsulating shell phase and the ambient phase consist of leaky dielectric liquids. We change the surface tension by using an aqueous core with different surfactant (Tween 80) concentrations. Moreover, we vary the core size under different electric-field conditions and observe the core dynamics. We present three different breakup modes of the encapsulated droplet. In the first mode, the encapsulated core forms asymmetric Janus shapes after breakup. In the second and third breakup modes, stable and unstable ternary droplets are formed, respectively. We show that the surfactant molecules significantly alter the dynamics of core stretching. According to the theoretical analysis, we identify the critical conditions of instability leading to breakup. We plot the breakup modes in the form of a phase diagram in the electric capillary number (Ca23 = ε3rsEo2/γ23; ratio of interfacial electric to capillary stresses) vs. radius ratio of the core to the shell (β = rc/rs) parametric space at different nondimensional surfactant concentrations (C* = CTween 80/CCMC, where CCMC represents the critical micellar concentration). The study provides essential physical insight into encapsulated emulsions and is useful for their application in various areas of science and technology.