  We study two models of scalar dark matter from "large" electroweak multiplets
with isospin 5/2 (n=6 members) and 7/2 (n=8), whose scalar potentials preserve
a $Z_2$ symmetry. Because of large annihilation cross sections due to
electroweak interactions, these scalars can constitute all the dark matter only
for masses in the multi-TeV range. For such high masses, Sommerfeld enhancement
and co-annihilations play important roles in the dark matter relic abundance
calculation, reducing the upper bound on the large multiplet's mass by almost a
factor of two. We determine the allowed parameter ranges including both of
these effects and show that these models are as yet unconstrained by dark
matter direct detection experiments, but will be probed by currently-running
and proposed future experiments. We also show that a Landau pole appears in
these models at energy scales below $10^9$ GeV, indicating the presence of
additional new physics below that scale.
