Subgroups — D₃×ΓL(1,𝔽₈)×C₁₃₇ #31772129

Order 17262.

36 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×ΓL(1,𝔽₈)×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	D₃×C₇×C₁₃₇	C₃	1	1	yes	maximal, characteristic, complemented
3	C₂×ΓL(1,𝔽₈)×C₁₃₇	—	3	1	no	maximal
4	C₃×C₇×C₁₃₇	C₂×C₃	1	1	yes	characteristic, complemented
5	ΓL(1,𝔽₈)×C₁₃₇	—	2	1	no	—
6	ΓL(1,𝔽₈)×C₁₃₇	D₃	1	1	yes	characteristic, direct product factor
7	C₃×D₃×C₁₃₇	—	7	1	no	maximal, Hall
8	C₂×C₇×C₁₃₇	—	3	1	no	Hall
9	C₃²×C₁₃₇	—	7	1	no	Hall
10	C₇×C₁₃₇	C₃×D₃	1	1	yes	characteristic, complemented, Hall
11	C₂×C₃×C₁₃₇	—	21	1	no	—
12	D₃×C₁₃₇	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
13	C₃×C₁₃₇	—	7	1	no	—
14	C₃×C₁₃₇	—	14	1	no	—
15	C₃×C₁₃₇	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
16	C₂×C₁₃₇	—	3	1	no	Hall
17	C₁₃₇	D₃×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Sylow
18	D₃×ΓL(1,𝔽₈)	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Hall
19	C₃×ΓL(1,𝔽₈)	C₂×C₁₃₇	1	1	yes	characteristic, complemented, Hall
20	D₃×C₇	C₃×C₁₃₇	1	1	yes	characteristic, complemented
21	C₂×ΓL(1,𝔽₈)	—	3	1	no	—
22	C₃×C₇	C₂×C₃×C₁₃₇	1	1	yes	characteristic, complemented
23	ΓL(1,𝔽₈)	—	2	1	no	—
24	ΓL(1,𝔽₈)	D₃×C₁₃₇	1	1	yes	characteristic, direct product factor
25	C₃×D₃	—	7	1	no	Hall
26	C₂×C₇	—	3	1	no	Hall
27	C₃²	—	7	1	no	Sylow
28	C₇	C₃×D₃×C₁₃₇	1	1	yes	characteristic, complemented, Sylow
29	C₂×C₃	—	21	1	no	—
30	D₃	ΓL(1,𝔽₈)×C₁₃₇	1	1	yes	characteristic, direct product factor
31	C₃	—	7	1	no	—
32	C₃	—	14	1	no	—
33	C₃	C₂×ΓL(1,𝔽₈)×C₁₃₇	1	1	yes	characteristic, complemented
34	C₂	—	3	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₁₃₇	D₃×ΓL(1,𝔽₈)
commutator, cosocle, hypocenter	C₃×C₇	C₂×C₃×C₁₃₇
Frattini	C₁	G
Fitting, socle	C₃×C₇×C₁₃₇	C₂×C₃
LCS	G ▷ C₃×C₇
UCS	C₁ ◁ C₁₃₇

Complements

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

N	H ≅ G/N	type
1. C₃×ΓL(1,𝔽₈)×C₁₃₇	C₂	semidirect
2. D₃×C₇×C₁₃₇	C₃	semidirect
4. C₃×C₇×C₁₃₇	C₂×C₃	semidirect
6. ΓL(1,𝔽₈)×C₁₃₇	D₃	direct (N × H)
10. C₇×C₁₃₇	C₃×D₃	semidirect
12. D₃×C₁₃₇	ΓL(1,𝔽₈)	direct (N × H)
15. C₃×C₁₃₇	C₂×ΓL(1,𝔽₈)	semidirect
17. C₁₃₇	D₃×ΓL(1,𝔽₈)	direct (N × H)
18. D₃×ΓL(1,𝔽₈)	C₁₃₇	direct (N × H)
19. C₃×ΓL(1,𝔽₈)	C₂×C₁₃₇	semidirect
20. D₃×C₇	C₃×C₁₃₇	semidirect
22. C₃×C₇	C₂×C₃×C₁₃₇	semidirect
24. ΓL(1,𝔽₈)	D₃×C₁₃₇	direct (N × H)
28. C₇	C₃×D₃×C₁₃₇	semidirect
30. D₃	ΓL(1,𝔽₈)×C₁₃₇	direct (N × H)
33. C₃	C₂×ΓL(1,𝔽₈)×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₃×ΓL(1,𝔽₈)×C₁₃₇	2
D₃×C₇×C₁₃₇	3
C₂×ΓL(1,𝔽₈)×C₁₃₇	3
C₃×D₃×C₁₃₇	7
D₃×ΓL(1,𝔽₈)	137

Subgroups — D3×ΓL(1,𝔽8)×C137 #31772129

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₃×ΓL(1,𝔽₈)×C₁₃₇ #31772129