Finite Group Databasebeta

← C₅×D₂₃×C₁₃₇ #44538509

Subgroups — C₅×D₂₃×C₁₃₇ #44538509

Order 31510.

16 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×C₂₃×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	D₂₃×C₁₃₇	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
3	C₂₃×C₁₃₇	C₂×C₅	1	1	yes	characteristic, complemented, Hall
4	C₂×C₅×C₁₃₇	—	23	1	no	maximal, Hall
5	C₅×C₁₃₇	D₂₃	1	1	yes	characteristic, direct product factor, Hall
6	C₂×C₁₃₇	—	23	1	no	Hall
7	C₅×D₂₃	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Hall
8	C₁₃₇	C₅×D₂₃	1	1	yes	characteristic, direct product factor, Sylow
9	C₅×C₂₃	C₂×C₁₃₇	1	1	yes	characteristic, complemented, Hall
10	D₂₃	C₅×C₁₃₇	1	1	yes	characteristic, direct product factor, Hall
11	C₂₃	C₂×C₅×C₁₃₇	1	1	yes	characteristic, complemented, Sylow
12	C₂×C₅	—	23	1	no	Hall
13	C₅	D₂₃×C₁₃₇	1	1	yes	characteristic, direct product factor, Sylow
14	C₂	—	23	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₅×C₁₃₇	D₂₃
commutator, cosocle, hypocenter	C₂₃	C₂×C₅×C₁₃₇
Frattini	C₁	G
Fitting, socle	C₅×C₂₃×C₁₃₇	C₂
LCS	G ▷ C₂₃
UCS	C₁ ◁ C₅×C₁₃₇

Complements

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

N	H ≅ G/N	type
1. C₅×C₂₃×C₁₃₇	C₂	semidirect
2. D₂₃×C₁₃₇	C₅	direct (N × H)
3. C₂₃×C₁₃₇	C₂×C₅	semidirect
5. C₅×C₁₃₇	D₂₃	direct (N × H)
7. C₅×D₂₃	C₁₃₇	direct (N × H)
8. C₁₃₇	C₅×D₂₃	direct (N × H)
9. C₅×C₂₃	C₂×C₁₃₇	semidirect
10. D₂₃	C₅×C₁₃₇	direct (N × H)
11. C₂₃	C₂×C₅×C₁₃₇	semidirect
13. C₅	D₂₃×C₁₃₇	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅×C₂₃×C₁₃₇	2
D₂₃×C₁₃₇	5
C₂×C₅×C₁₃₇	23
C₅×D₂₃	137