Various samples of sodium cobalt oxyhydrate with relatively large amounts of Na$^{+}$ ions were synthesized by a modified soft-chemical process in which a NaOH aqueous solution was added in the final step of the procedure. From these samples, a superconducting phase diagram was determined for a section of a cobalt valence of $sim$+3.48, which was compared with a previously obtained one of $sim$+3.40. The superconductivity was significantly affected by the isovalent exchanger of Na$^{+}$ and H$_{3}$O$^{+}$, rather than by variation of Co valence, suggesting the presence of multiple kinds of Fermi surface. Furthermore, the high-field magnetic susceptibility measurements for one sample up to 30 T indicated an upper critical field much higher than the Pauli limit supporting the validity of the spin-triplet pairing mechanism.
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