Subgroups — C₃×ΓL(1,𝔽₈)×D₁₂₇ #30630712

Order 16002.

48 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×ΓL(1,𝔽₈)×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	ΓL(1,𝔽₈)×D₁₂₇	C₃	1	3	yes	maximal, direct product factor
3	C₃×C₇×D₁₂₇	C₃	1	1	yes	maximal, characteristic, complemented
4	C₃×C₇×C₁₂₇	C₂×C₃	1	1	yes	characteristic, complemented
5	ΓL(1,𝔽₈)×C₁₂₇	C₂×C₃	1	3	yes	complemented
6	C₃²×D₁₂₇	—	7	1	no	maximal, Hall
7	C₇×D₁₂₇	C₃²	1	1	yes	characteristic, complemented, Hall
8	C₃²×C₁₂₇	—	7	1	no	Hall
9	C₇×C₁₂₇	C₂×C₃²	1	1	yes	characteristic, complemented, Hall
10	C₃×D₁₂₇	—	7	3	no	—
11	C₃×D₁₂₇	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
12	C₃×C₁₂₇	—	7	3	no	—
13	C₃×C₁₂₇	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
14	D₁₂₇	C₃×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Hall
15	C₁₂₇	C₂×C₃×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Sylow
16	C₂×C₃×ΓL(1,𝔽₈)	—	127	1	no	maximal, Hall
17	C₃×ΓL(1,𝔽₈)	D₁₂₇	1	1	yes	characteristic, direct product factor, Hall
18	C₂×C₃×C₇	—	127	1	no	—
19	C₂×ΓL(1,𝔽₈)	—	127	3	no	—
20	C₃×C₇	C₃×D₁₂₇	1	1	yes	characteristic, complemented
21	ΓL(1,𝔽₈)	C₃×D₁₂₇	1	3	yes	direct product factor
22	C₂×C₃²	—	889	1	no	Hall
23	C₂×C₇	—	127	1	no	Hall
24	C₃²	—	7	1	no	Sylow
25	C₇	C₃²×D₁₂₇	1	1	yes	characteristic, complemented, Sylow
26	C₂×C₃	—	889	3	no	—
27	C₂×C₃	—	127	1	no	—
28	C₃	—	7	3	no	—
29	C₃	ΓL(1,𝔽₈)×D₁₂₇	1	1	yes	characteristic, direct product factor
30	C₂	—	127	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₃	ΓL(1,𝔽₈)×D₁₂₇
commutator, cosocle, hypocenter	C₇×C₁₂₇	C₂×C₃²
Frattini	C₁	G
Fitting, socle	C₃×C₇×C₁₂₇	C₂×C₃
LCS	G ▷ C₇×C₁₂₇
UCS	C₁ ◁ C₃

Complements

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

N	H ≅ G/N	type
1. C₃×ΓL(1,𝔽₈)×C₁₂₇	C₂	semidirect
2. ΓL(1,𝔽₈)×D₁₂₇	C₃	direct (N × H)
3. C₃×C₇×D₁₂₇	C₃	semidirect
4. C₃×C₇×C₁₂₇	C₂×C₃	semidirect
5. ΓL(1,𝔽₈)×C₁₂₇	C₂×C₃	semidirect
7. C₇×D₁₂₇	C₃²	semidirect
9. C₇×C₁₂₇	C₂×C₃²	semidirect
11. C₃×D₁₂₇	ΓL(1,𝔽₈)	direct (N × H)
13. C₃×C₁₂₇	C₂×ΓL(1,𝔽₈)	semidirect
14. D₁₂₇	C₃×ΓL(1,𝔽₈)	direct (N × H)
15. C₁₂₇	C₂×C₃×ΓL(1,𝔽₈)	semidirect
17. C₃×ΓL(1,𝔽₈)	D₁₂₇	direct (N × H)
20. C₃×C₇	C₃×D₁₂₇	semidirect
21. ΓL(1,𝔽₈)	C₃×D₁₂₇	direct (N × H)
25. C₇	C₃²×D₁₂₇	semidirect
29. C₃	ΓL(1,𝔽₈)×D₁₂₇	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×ΓL(1,𝔽₈)×C₁₂₇	2
ΓL(1,𝔽₈)×D₁₂₇	3
C₃×C₇×D₁₂₇	3
C₃²×D₁₂₇	7
C₂×C₃×ΓL(1,𝔽₈)	127

Subgroups — C3×ΓL(1,𝔽8)×D127 #30630712

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×ΓL(1,𝔽₈)×D₁₂₇ #30630712