We investigate the plasmonic properties of a symmetric silver sphere septamer and show that the extinction spectrum exhibits a narrow Fano resonance. Using the plasmon hybridization approach and group theory we show that this Fano resonance is caused by the interference of two bonding dipolar subradiant and superradiant plasmon modes of E-1u symmetry. We investigate the effect of structural symmetry breaking and show that the energy and shape of the Fano resonance can be tuned over a broad wavelength range. We show that the wavelength of the Fano resonance depends very sensitively on the dielectric permittivity of the surrounding media with one of the highest LSPR sensitivities reported for a finite nanostructure.