  We study a real scalar septuplet model which has a neutral component regarded
as a dark matter particle candidate. The calculation of its thermal relic
abundance without (with) the Sommerfeld effect suggests that the observed value
corresponds to the mass of the dark matter particle $m_0 \gtrsim
8.8~(25)~\mathrm{TeV}$. Two extra quartic couplings $\lambda_2$ and $\lambda_3$
introduced in this model affect the running of other couplings, and hence the
vacuum stability and the perturbativity up to the Planck scale. Therefore, the
vacuum stability and the perturbativity conditions can constrain these
couplings into a narrow region in the $\lambda_2$-$\lambda_3$ plane. Other
constraints from $h\to\gamma\gamma$, electroweak oblique parameters, and direct
and indirect DM searches are also investigated. Moreover, we survey the vacuum
stability and the perturbativity in the model combining the septuplet with the
Type-I or Type-III seesaw. The running of couplings is further altered when the
energy scale goes above the seesaw scale. In the Type-I case, when the seesaw
scale $\sim 10^{15}~\mathrm{GeV}$, the vacuum stability condition makes the
acceptable region in the $\lambda_2$-$\lambda_3$ plane much narrower. In the
Type-III case, if the seesaw scale is $\lesssim 52~\mathrm{TeV}$ or $\gtrsim
10^{15}~\mathrm{GeV}$, the acceptable region vanishes.
