Subgroups — C₇×#19128624 #49152804

Order 37338.

40 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#14581888	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	C₇²×D₁₂₇	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	C₇²×C₁₂₇	C₂×C₃	1	1	yes	characteristic, complemented, Hall
4	#19128624	C₇	1	1	yes	maximal, characteristic, direct product factor
5	C₇×D₁₂₇⋊C₃	—	7	1	no	maximal
6	#14581888	C₂×C₇	1	1	yes	characteristic, complemented
7	C₇×C₁₂₇⋊C₃	—	7	1	no	—
8	C₇×D₁₂₇	—	3	2	no	—
9	C₇×D₁₂₇	ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
10	C₇×D₁₂₇	C₃×C₇	1	1	yes	characteristic, complemented
11	C₇×C₁₂₇	—	3	2	no	—
12	C₇×C₁₂₇	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
13	C₇×C₁₂₇	C₂×C₃×C₇	1	1	yes	characteristic, complemented
14	D₁₂₇⋊C₃	—	7	1	no	Hall
15	C₁₂₇⋊C₃	—	7	1	no	Hall
16	C₂×C₇×ΓL(1,𝔽₈)	—	127	1	no	maximal, Hall
17	D₁₂₇	C₇×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Hall
18	C₇×ΓL(1,𝔽₈)	—	127	1	no	Hall
19	C₁₂₇	C₂×C₇×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Sylow
20	C₂×C₇²	—	127	1	no	Hall
21	C₇²	D₁₂₇⋊C₃	1	1	yes	characteristic, complemented, Sylow
22	C₂×C₃×C₇	—	889	1	no	—
23	C₂×ΓL(1,𝔽₈)	—	127	1	no	—
24	C₃×C₇	—	889	1	no	—
25	ΓL(1,𝔽₈)	—	127	1	no	—
26	C₂×C₇	—	381	2	no	—
27	C₂×C₇	—	127	1	no	—
28	C₂×C₇	—	127	1	no	—
29	C₇	—	3	2	no	—
30	C₇	C₇×D₁₂₇⋊C₃	1	1	yes	characteristic, complemented
31	C₇	#19128624	1	1	yes	characteristic, direct product factor
32	C₂×C₃	—	889	1	no	Hall
33	C₃	—	889	1	no	Sylow
34	C₂	—	127	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₇	#19128624
commutator, cosocle, hypocenter	C₇×C₁₂₇	C₂×C₃×C₇
Frattini	C₁	G
Fitting, socle	C₇²×C₁₂₇	C₂×C₃
LCS	G ▷ C₇×C₁₂₇
UCS	C₁ ◁ C₇

Complements

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

N	H ≅ G/N	type
1. C₇×#14581888	C₂	semidirect
2. C₇²×D₁₂₇	C₃	semidirect
3. C₇²×C₁₂₇	C₂×C₃	semidirect
4. #19128624	C₇	direct (N × H)
6. #14581888	C₂×C₇	semidirect
9. C₇×D₁₂₇	ΓL(1,𝔽₈)	semidirect
10. C₇×D₁₂₇	C₃×C₇	semidirect
12. C₇×C₁₂₇	C₂×ΓL(1,𝔽₈)	semidirect
13. C₇×C₁₂₇	C₂×C₃×C₇	semidirect
17. D₁₂₇	C₇×ΓL(1,𝔽₈)	semidirect
19. C₁₂₇	C₂×C₇×ΓL(1,𝔽₈)	semidirect
21. C₇²	D₁₂₇⋊C₃	semidirect
30. C₇	C₇×D₁₂₇⋊C₃	semidirect
31. C₇	#19128624	direct (N × H)

Maximal subgroups

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

subgroup	index
C₇×#14581888	2
C₇²×D₁₂₇	3
#19128624	7
C₇×D₁₂₇⋊C₃	7
C₂×C₇×ΓL(1,𝔽₈)	127

Subgroups — C7×#19128624 #49152804

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#19128624 #49152804