It is both straightforward, and practical, to divide most species, hybrids and cultivars. But first a warning: do not simply detach pieces from the edge of your cherished clump and replant it elsewhere or give it away. The youngest and most vigorous growth is always at the edge of the clump, while the tired, woody and unproductive growth is at the centre. By removing the most vigorous growth and leaving the tired old centre behind, you will so weaken the original plant that it may cease flowering and fade away altogether.
There are two approaches to division: the simple approach which is only modestly productive, and a more careful and thorough approach which yields more plants. In both cases the best time to divide the Lenten roses, H. hybridus, and also the hybrids with H. niger such as H. x nigercors and H. x ericsmithii, is in late summer and early autumn, towards the end of August or in September in Britain. These hellebores initiate a burst of root growth in autumn and the optimum time to divide them is immediately prior to this spurt of root development. This allows the growth of new roots to settle in the newly divided plants at once.
The Christmas rose, H. niger, and the many relatively demure species from the Balkans - H. torquatus, H. atrorubens etc - are best divided in early spring. Brave souls wishing to divide H. vesicarius should do so in the autumn - but my advice is dont; a mature plant is so impressive and the chances of establishing divisions are not good.
It has to be said at the outset that not all hellebores can be propagated by division. Those in the caulescent group - H. argutifolius, H. lividus, H. x sternii and H. foetidus can, theoretically, be divided but their tendency to a short life will usually be exacerbated by the disruption of division and there are practical problems associated with dividing a deep-rooted plant, with such a very compact rootstock, and with such tall top growth.