File:Symmetric group S4; lattice of subgroups Hasse diagram; all 30 subgroups.svg

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Description
Subgroups with the same colored cycle graphs bundled together
The lines from C2 to S3 are not drawn straight, to make the pattern more clear

The lattice of subgroups of the Symmetric group S4, represented in a Hasse diagram

These are Sloane's A005432(4) = 30 distinct subgroups. They belong to Sloane's A000638(4) = 11 different types, which are shown on the right.

S4: Symmetric group of order 24
A4: Alternating group of order 12
Dih4: Dihedral group of order 8
S3: Symmetric group of order 6
C22: Klein 4-group
C4: Cyclic group of order 4
C3: 3 element group
C2: 2 element group
C1: Trivial group

The C2 knots are marked with the index numbers (compare this table) of the transpositions (1, 6, 5, 14, 2, 21) and double transpositions (7, 16, 23) each has as its non-identity element.

Edge colors indicate the quotient of the connected groups' orders: red 2, green 3, blue 4
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Watchduck
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