Finite Group Databasebeta

← C₃×D₁₇×C₂₂₇ #37529165

Subgroups — C₃×D₁₇×C₂₂₇ #37529165

Order 23154.

16 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₁₇×C₂₂₇	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	D₁₇×C₂₂₇	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
3	C₁₇×C₂₂₇	C₂×C₃	1	1	yes	characteristic, complemented, Hall
4	C₂×C₃×C₂₂₇	—	17	1	no	maximal, Hall
5	C₃×C₂₂₇	D₁₇	1	1	yes	characteristic, direct product factor, Hall
6	C₂×C₂₂₇	—	17	1	no	Hall
7	C₂₂₇	C₃×D₁₇	1	1	yes	characteristic, direct product factor, Sylow
8	C₃×D₁₇	C₂₂₇	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₃×C₁₇	C₂×C₂₂₇	1	1	yes	characteristic, complemented, Hall
10	D₁₇	C₃×C₂₂₇	1	1	yes	characteristic, direct product factor, Hall
11	C₁₇	C₂×C₃×C₂₂₇	1	1	yes	characteristic, complemented, Sylow
12	C₂×C₃	—	17	1	no	Hall
13	C₃	D₁₇×C₂₂₇	1	1	yes	characteristic, direct product factor, Sylow
14	C₂	—	17	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₃×C₂₂₇	D₁₇
commutator, cosocle, hypocenter	C₁₇	C₂×C₃×C₂₂₇
Frattini	C₁	G
Fitting, socle	C₃×C₁₇×C₂₂₇	C₂
LCS	G ▷ C₁₇
UCS	C₁ ◁ C₃×C₂₂₇

Complements

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

N	H ≅ G/N	type
1. C₃×C₁₇×C₂₂₇	C₂	semidirect
2. D₁₇×C₂₂₇	C₃	direct (N × H)
3. C₁₇×C₂₂₇	C₂×C₃	semidirect
5. C₃×C₂₂₇	D₁₇	direct (N × H)
7. C₂₂₇	C₃×D₁₇	direct (N × H)
8. C₃×D₁₇	C₂₂₇	direct (N × H)
9. C₃×C₁₇	C₂×C₂₂₇	semidirect
10. D₁₇	C₃×C₂₂₇	direct (N × H)
11. C₁₇	C₂×C₃×C₂₂₇	semidirect
13. C₃	D₁₇×C₂₂₇	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×C₁₇×C₂₂₇	2
D₁₇×C₂₂₇	3
C₂×C₃×C₂₂₇	17
C₃×D₁₇	227