Finite Group Databasebeta

← C₃×H(3,𝔽₃)×C₂₁₁ #30685249

Subgroups — C₃×H(3,𝔽₃)×C₂₁₁ #30685249

Order 17091.

112 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃³×C₂₁₁	C₃	1	4	yes	maximal, complemented
2	H(3,𝔽₃)×C₂₁₁	C₃	1	9	yes	maximal, direct product factor
3	C₃²×C₂₁₁	—	3	12	no	—
4	C₃²×C₂₁₁	C₃²	1	1	yes	characteristic
5	C₃²×C₂₁₁	C₃²	1	12	yes	complemented
6	C₃×C₂₁₁	—	3	12	no	—
7	C₃×C₂₁₁	H(3,𝔽₃)	1	3	yes	direct product factor
8	C₃×C₂₁₁	C₃³	1	1	yes	characteristic
9	C₂₁₁	C₃×H(3,𝔽₃)	1	1	yes	characteristic, direct product factor, Sylow
10	C₃×H(3,𝔽₃)	C₂₁₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
11	C₃³	C₃×C₂₁₁	1	4	yes	complemented
12	H(3,𝔽₃)	C₃×C₂₁₁	1	9	yes	direct product factor
13	C₃²	—	3	12	no	—
14	C₃²	C₃²×C₂₁₁	1	1	yes	characteristic
15	C₃²	C₃²×C₂₁₁	1	12	yes	complemented
16	C₃	—	3	12	no	—
17	C₃	H(3,𝔽₃)×C₂₁₁	1	3	yes	direct product factor
18	C₃	C₃³×C₂₁₁	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₃²×C₂₁₁	C₃²
commutator, Frattini, cosocle	C₃	C₃³×C₂₁₁
LCS	G ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃²×C₂₁₁ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₃³×C₂₁₁	C₃	semidirect
2. H(3,𝔽₃)×C₂₁₁	C₃	direct (N × H)
5. C₃²×C₂₁₁	C₃²	semidirect
7. C₃×C₂₁₁	H(3,𝔽₃)	direct (N × H)
9. C₂₁₁	C₃×H(3,𝔽₃)	direct (N × H)
10. C₃×H(3,𝔽₃)	C₂₁₁	direct (N × H)
11. C₃³	C₃×C₂₁₁	semidirect
12. H(3,𝔽₃)	C₃×C₂₁₁	direct (N × H)
15. C₃²	C₃²×C₂₁₁	semidirect
17. C₃	H(3,𝔽₃)×C₂₁₁	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₃³×C₂₁₁	3
H(3,𝔽₃)×C₂₁₁	3
C₃×H(3,𝔽₃)	211