The giant planets in the Solar System each have two groups of satellites. The regular satellites move along nearly circular orbits in the planet's orbital plane, revolving about it in the same sense as the planet spins. In contrast, the so-called irregular satellites are generally smaller in size and are characterized by large orbits with significant eccentricity, inclination or both. The differences in their characteristics suggest that the regular and irregular satellites formed by different mechanisms: the regular satellites are believed to have formed in an accretion disk around the planet, like a miniature Solar System, whereas the irregulars are generally thought to be captured planetesimals. Here we report the discovery of 12 irregular satellites of Saturn, along with the determinations of their orbits. These orbits, along with the orbits of irregular satellites of Jupiter and Uranus, fall into groups on the basis of their orbital inclinations. We interpret this result as indicating that most of the irregular moons are collisional remnants of larger satellites that were fragmented after capture, rather than being captured independently.