We investigate the association between X-class flares and sunspot groups (SGs) of various classes in Solar Cycles (SCs) 22 and 23 and find the following results. (1) During SCs 22 and 23, 96.2 per cent and 94.3 per cent of X-class flares are associated with large SGs (Zurich classes D, E, and F), respectively; the percentages of large SGs associated with X-class flares are significantly higher than those of the other Zurich class SGs; however, the total number of large SGs in SC 22 is about 80 per cent of that in SC 23, while, the total number of X-class flares in SC 22 is 1.22 times that of SC 23. (2) During SCs 22 and 23, 83.1 per cent and 79.4 per cent of X-class flares are associated with penumbral class k SGs, respectively; the percentages of penumbral class k SGs associated with X-class flares are significantly higher than those of the other penumbral class SGs; however, the ratio of the maximum number of penumbral class k SGs during SC 22 to SC 23 is 1.10, while, the ratio of the maximum number of X-class flares during SC 22 to SC 23 is 2.73. (3) 49.3 per cent and 55.2 per cent of X-class flares occur in beta delta or beta gamma delta or gamma delta SGs in SCs 22 and 23, respectively; the total (maximum) numbers of beta gamma delta and gamma delta SGs in SC 22 are about 63 per cent (53 per cent) and 33 per cent of those in SC 23, respectively; the total (maximum) number of beta delta SGs in SC 22 is 3.03 (5.13) times that of SC 23.