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In this paper, a sample of 451 blazars (193 flat spectrum radio quasars (FSRQs), 258 BL Lacertae objects) with corresponding X-ray and Fermi γ-ray data is compiled to investigate the correlation both between the X-ray spectral index and the γ-ray spectral index and between the spectral index and the luminosity, and to compare the spectral indexes αX, αγ, αXγ, and αγXγ for different subclasses. We also investigated the correlation between the X-ray and the γ-ray luminosity. The following results have been obtained. (1) Our analysis indicates that an anti-correlation exists between the X-ray and the γ-ray spectral indexes for the whole sample. However, when we considered the subclasses of blazars (FSRQs, the low-peaked BL Lacertae objects (LBLs) and the high-peaked BL Lacertae objects (HBLs)) separately, there is not a clear relationship for each subclass. For the average values, , we have ; and (2) For the correlations between the X-ray and the γ-ray luminosities, we have log νLγ = (1.033 ± 0.002)log νLX + (0.213 ± 0.094) for the whole sample, log νLγ = (0.741 ± 0.004)log νLX + (12.378 ± 0.162) for HBLs, and log νLγ = (1.032 ± 0.003)log νLX + (0.643 ± 0.138) for the LBLs and FSRQs; the correlation slope for HBLs is different from that for FSRQs and LBLs. For the average values, , we have , , and for the X-ray luminosity; and , , and for the γ-ray luminosity. (3) There is an anti-correlation between the effective spectral index, αXγ, and log νLγ for the whole sample, and the subclasses (FSRQs, LBLs, and HBLs). The effective spectral index shows that (4) For the spectral index difference, αγXγ(=αXγ – αγ) of the effective spectral index, αXγ, and the γ-ray spectral index, αγ, we obtain (5) Based on the Fermi-detected sources, we can say that the HBLs are different from FSRQs, while the LBLs are similar to FSRQs. [ABSTRACT FROM AUTHOR]