Subgroups — C₇×AGL(1,𝔽₆₇) #44526280

Order 30954.

32 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×(C₆₇⋊C₃)⋊C₁₁	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	C₇×D₆₇⋊C₁₁	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	C₇×C₆₇⋊C₁₁	C₂×C₃	1	1	yes	characteristic, complemented, Hall
4	AGL(1,𝔽₆₇)	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₇×D₆₇⋊C₃	C₁₁	1	1	yes	maximal, characteristic, complemented, Hall
6	(C₆₇⋊C₃)⋊C₁₁	C₂×C₇	1	1	yes	characteristic, complemented, Hall
7	D₆₇⋊C₁₁	C₃×C₇	1	1	yes	characteristic, complemented, Hall
8	C₇×C₆₇⋊C₃	C₂×C₁₁	1	1	yes	characteristic, complemented, Hall
9	C₇×D₆₇	C₃×C₁₁	1	1	yes	characteristic, complemented, Hall
10	C₆₇⋊C₁₁	C₂×C₃×C₇	1	1	yes	characteristic, complemented, Hall
11	C₇×C₆₇	C₂×C₃×C₁₁	1	1	yes	characteristic, complemented, Hall
12	C₂×C₃×C₇×C₁₁	—	67	1	no	maximal, Hall
13	D₆₇⋊C₃	C₇×C₁₁	1	1	yes	characteristic, complemented, Hall
14	C₃×C₇×C₁₁	—	67	1	no	Hall
15	C₆₇⋊C₃	C₂×C₇×C₁₁	1	1	yes	characteristic, complemented, Hall
16	C₂×C₇×C₁₁	—	67	1	no	Hall
17	D₆₇	C₃×C₇×C₁₁	1	1	yes	characteristic, complemented, Hall
18	C₇×C₁₁	—	67	1	no	Hall
19	C₆₇	C₂×C₃×C₇×C₁₁	1	1	yes	characteristic, complemented, Sylow
20	C₂×C₃×C₁₁	—	67	1	no	Hall
21	C₂×C₃×C₇	—	67	1	no	Hall
22	C₃×C₁₁	—	67	1	no	Hall
23	C₂×C₁₁	—	67	1	no	Hall
24	C₃×C₇	—	67	1	no	Hall
25	C₂×C₇	—	67	1	no	Hall
26	C₁₁	—	67	1	no	Sylow
27	C₇	AGL(1,𝔽₆₇)	1	1	yes	characteristic, direct product factor, Sylow
28	C₂×C₃	—	67	1	no	Hall
29	C₃	—	67	1	no	Sylow
30	C₂	—	67	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₇	AGL(1,𝔽₆₇)
commutator, cosocle, hypocenter	C₆₇	C₂×C₃×C₇×C₁₁
Frattini	C₁	G
Fitting, socle	C₇×C₆₇	C₂×C₃×C₁₁
LCS	G ▷ C₆₇
UCS	C₁ ◁ C₇

Complements

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

N	H ≅ G/N	type
1. C₇×(C₆₇⋊C₃)⋊C₁₁	C₂	semidirect
2. C₇×D₆₇⋊C₁₁	C₃	semidirect
3. C₇×C₆₇⋊C₁₁	C₂×C₃	semidirect
4. AGL(1,𝔽₆₇)	C₇	direct (N × H)
5. C₇×D₆₇⋊C₃	C₁₁	semidirect
6. (C₆₇⋊C₃)⋊C₁₁	C₂×C₇	semidirect
7. D₆₇⋊C₁₁	C₃×C₇	semidirect
8. C₇×C₆₇⋊C₃	C₂×C₁₁	semidirect
9. C₇×D₆₇	C₃×C₁₁	semidirect
10. C₆₇⋊C₁₁	C₂×C₃×C₇	semidirect
11. C₇×C₆₇	C₂×C₃×C₁₁	semidirect
13. D₆₇⋊C₃	C₇×C₁₁	semidirect
15. C₆₇⋊C₃	C₂×C₇×C₁₁	semidirect
17. D₆₇	C₃×C₇×C₁₁	semidirect
19. C₆₇	C₂×C₃×C₇×C₁₁	semidirect
27. C₇	AGL(1,𝔽₆₇)	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×(C₆₇⋊C₃)⋊C₁₁	2
C₇×D₆₇⋊C₁₁	3
AGL(1,𝔽₆₇)	7
C₇×D₆₇⋊C₃	11
C₂×C₃×C₇×C₁₁	67

Subgroups — C7×AGL(1,𝔽67) #44526280

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×AGL(1,𝔽₆₇) #44526280