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Approximately seventy resonances were observed for the $^{56}\mathrm{Fe}(p,\ensuremath{\gamma})^{57}\mathrm{Co}$ reaction in the range of proton energy from 1300 to 1800 keV. Gamma-ray energy measurements performed with scintillation and lithium-drift silicon spectrometers yield 6.040\ifmmode\pm\else\textpm\fi{}0.030 MeV for the $Q$ value of the $^{56}\mathrm{Fe}(p,\ensuremath{\gamma})^{57}\mathrm{Co}$ reaction and show the existence of intermediate states in $^{57}\mathrm{Co}$ at 1.381\ifmmode\pm\else\textpm\fi{}0.010, 1.506\ifmmode\pm\else\textpm\fi{}0.020, 1.760\ifmmode\pm\else\textpm\fi{}0.020, 1.921\ifmmode\pm\else\textpm\fi{}0.020, 2.136\ifmmode\pm\else\textpm\fi{}0.020, 2.60\ifmmode\pm\else\textpm\fi{}0.03, 2.74\ifmmode\pm\else\textpm\fi{}0.03, 2.80\ifmmode\pm\else\textpm\fi{}0.03, 2.88\ifmmode\pm\else\textpm\fi{}0.03, 3.11\ifmmode\pm\else\textpm\fi{}0.03, 3.18\ifmmode\pm\else\textpm\fi{}0.03, 3.27\ifmmode\pm\else\textpm\fi{}0.03, and 3.52\ifmmode\pm\else\textpm\fi{}0.03 MeV. Triple-correlation measurements performed with a double sum-coincidence spectrometer permit ${J}^{\ensuremath{\pi}}$ assignments of ${\frac{3}{2}}^{\ensuremath{-}}$, ${\frac{5}{2}}^{\ensuremath{-}}$, ${\frac{5}{2}}^{\ensuremath{-}}$, and ${\frac{5}{2}}^{+}$ to the intermediate states in $^{57}\mathrm{Co}$ at 1.38, 1.76, 1.92, and 2.14 MeV, respectively. These measurements were performed at resonance energies of 1525, 1598, 1622, 1637, 1645, and 1800 keV, where the capturing states at the 1637- and 1800-keV resonances are assigned a ${J}^{\ensuremath{\pi}}$ value of ${\frac{5}{2}}^{+}$, and for the others ${J}^{\ensuremath{\pi}}$ is ${\frac{3}{2}}^{\ensuremath{-}}$. Ambiguities exist for the $\frac{E2}{M1}$ amplitude ratios for all of the mixed secondary gamma-ray transitions observed in these studies. All measurements were in agreement with the established value of ${\frac{7}{2}}^{\ensuremath{-}}$ for the ground state of $^{57}\mathrm{Co}$.