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C1
E (not highly symmetrical or linear, no rotation axis)
Cs
sigma & E (not highly symmetrical or linear, no rotation axis)
Ci
i & E (not highly symmetrical or linear, no rotation axis)
C(infinity-v)
linear (not symmetric)
D(infinity-v)
linear (symmetric)
Cnh
nCn, sigma-h, i, nSn (not high symmetry or linear, rotation axis, no C2 norm to highest-order Cn rot axis, sigma-h)
Cnv
nCn, nsigma-v/d (not high symmetry or linear, rotation axis, no C2 norm to highest-order Cn rot axis, sigma-v)
S2n
(n/2)Sn, (n/2)Cn/2 (not high symmetry or linear, rotation axis, no C2 norm to highest-order Cn rot axis, S2n)
Cn
nCn (not high symmetry or linear, rotation axis, no C2 norm to highest-order Cn rot axis, no sigma-v or -h, no S2n)
Dnh
nCn, nC2, nsigma-v/d, sigma-h, i (not high symmetry or linear, rotation axis, C2 norm to highest-order Cn rot axis, sigma-h)
Dnd
nCn, nC2, nsigma-d, nS2n, i (not high symmetry or linear, rotation axis, C2 norm to highest-order Cn rot axis, sigma-d)
Dn
nCn, nC2 (not high symmetry or linear, rotation axis, C2 norm to highest-order Cn rot axis, no sigma-h or -d)
Td
tetrahedral (high symmetry, has >1 rot axis w/ n>2)
Oh
octahedral (high symmetry, has >1 rot axis w/ n>2)
Ih
icosahedral (high symmetry, has >1 rot axis w/ n>2)