Subgroups — AGL(1,𝔽₇)×D₁₆₃ #28331908

Order 13692.

40 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	ΓL(1,𝔽₈)⋊D₁₆₃	C₂	1	1	yes	maximal, characteristic, complemented
2	AGL(1,𝔽₇)×C₁₆₃	C₂	1	1	yes	maximal, characteristic, complemented
3	ΓL(1,𝔽₈)×D₁₆₃	C₂	1	1	yes	maximal, characteristic, complemented
4	D₇×D₁₆₃	C₃	1	1	yes	maximal, characteristic, complemented, Hall
5	ΓL(1,𝔽₈)×C₁₆₃	C₂²	1	1	yes	characteristic, complemented, Hall
6	D₁₁₄₁	C₂×C₃	1	1	yes	characteristic, complemented
7	C₇×D₁₆₃	C₂×C₃	1	1	yes	characteristic, complemented
8	D₇×C₁₆₃	C₂×C₃	1	1	yes	characteristic, complemented
9	C₂×C₃×D₁₆₃	—	7	1	no	maximal, Hall
10	C₇×C₁₆₃	C₂²×C₃	1	1	yes	characteristic, complemented, Hall
11	C₂×C₃×C₁₆₃	—	7	1	no	—
12	C₃×D₁₆₃	—	7	1	no	—
13	C₃×D₁₆₃	—	7	1	no	—
14	C₂×D₁₆₃	—	7	1	no	Hall
15	C₃×C₁₆₃	—	7	1	no	Hall
16	C₂×C₁₆₃	—	7	1	no	—
17	D₁₆₃	—	7	1	no	—
18	D₁₆₃	AGL(1,𝔽₇)	1	1	yes	characteristic, direct product factor
19	C₁₆₃	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic, complemented, Sylow
20	C₂×AGL(1,𝔽₇)	—	163	1	no	maximal, Hall
21	C₂×ΓL(1,𝔽₈)	—	163	1	no	—
22	AGL(1,𝔽₇)	—	163	1	no	—
23	AGL(1,𝔽₇)	D₁₆₃	1	1	yes	characteristic, direct product factor
24	C₂×D₇	—	163	1	no	Hall
25	ΓL(1,𝔽₈)	C₂×D₁₆₃	1	1	yes	characteristic, complemented, Hall
26	C₂×C₇	—	163	1	no	—
27	D₇	—	163	1	no	—
28	D₇	C₃×D₁₆₃	1	1	yes	characteristic, complemented
29	C₂²×C₃	—	1141	1	no	Hall
30	C₇	C₂×C₃×D₁₆₃	1	1	yes	characteristic, complemented, Sylow
31	C₂×C₃	—	7	1	no	—
32	C₂×C₃	—	1141	1	no	—
33	C₂×C₃	—	2282	1	no	—
34	C₂²	—	1141	1	no	Sylow
35	C₃	—	14	1	no	Sylow
36	C₂	—	7	1	no	—
37	C₂	—	1141	1	no	—
38	C₂	—	163	1	no	—

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₇×C₁₆₃	C₂²×C₃
LCS	G ▷ C₇×C₁₆₃
UCS	C₁

Complements

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

N	H ≅ G/N	type
1. ΓL(1,𝔽₈)⋊D₁₆₃	C₂	semidirect
2. AGL(1,𝔽₇)×C₁₆₃	C₂	semidirect
3. ΓL(1,𝔽₈)×D₁₆₃	C₂	semidirect
4. D₇×D₁₆₃	C₃	semidirect
5. ΓL(1,𝔽₈)×C₁₆₃	C₂²	semidirect
6. D₁₁₄₁	C₂×C₃	semidirect
7. C₇×D₁₆₃	C₂×C₃	semidirect
8. D₇×C₁₆₃	C₂×C₃	semidirect
10. C₇×C₁₆₃	C₂²×C₃	semidirect
18. D₁₆₃	AGL(1,𝔽₇)	direct (N × H)
19. C₁₆₃	C₂×AGL(1,𝔽₇)	semidirect
23. AGL(1,𝔽₇)	D₁₆₃	direct (N × H)
25. ΓL(1,𝔽₈)	C₂×D₁₆₃	semidirect
28. D₇	C₃×D₁₆₃	semidirect
30. C₇	C₂×C₃×D₁₆₃	semidirect

Maximal subgroups

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

subgroup	index
ΓL(1,𝔽₈)⋊D₁₆₃	2
AGL(1,𝔽₇)×C₁₆₃	2
ΓL(1,𝔽₈)×D₁₆₃	2
D₇×D₁₆₃	3
C₂×C₃×D₁₆₃	7
C₂×AGL(1,𝔽₇)	163

Subgroups — AGL(1,𝔽7)×D163 #28331908

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — AGL(1,𝔽₇)×D₁₆₃ #28331908