Subgroups — C₂×AGL(1,𝔽₇)×C₂₆₃ #36348317

Order 22092.

40 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	AGL(1,𝔽₇)×C₂₆₃	C₂	1	2	yes	maximal, direct product factor
2	C₂×ΓL(1,𝔽₈)×C₂₆₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×D₇×C₂₆₃	C₃	1	1	yes	maximal, characteristic, complemented, Hall
4	ΓL(1,𝔽₈)×C₂₆₃	C₂²	1	1	yes	characteristic, complemented, Hall
5	C₂×C₇×C₂₆₃	C₂×C₃	1	1	yes	characteristic, complemented
6	D₇×C₂₆₃	C₂×C₃	1	2	yes	complemented
7	C₂²×C₃×C₂₆₃	—	7	1	no	maximal, Hall
8	C₇×C₂₆₃	C₂²×C₃	1	1	yes	characteristic, complemented, Hall
9	C₂×C₃×C₂₆₃	—	7	2	no	—
10	C₂×C₃×C₂₆₃	—	7	1	no	—
11	C₂²×C₂₆₃	—	7	1	no	Hall
12	C₃×C₂₆₃	—	7	1	no	Hall
13	C₂×C₂₆₃	—	7	2	no	—
14	C₂×C₂₆₃	AGL(1,𝔽₇)	1	1	yes	characteristic, direct product factor
15	C₂₆₃	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic, direct product factor, Sylow
16	C₂×AGL(1,𝔽₇)	C₂₆₃	1	1	yes	maximal, characteristic, direct product factor, Hall
17	C₂×ΓL(1,𝔽₈)	C₂×C₂₆₃	1	1	yes	characteristic, complemented
18	AGL(1,𝔽₇)	C₂×C₂₆₃	1	2	yes	direct product factor
19	C₂×D₇	C₃×C₂₆₃	1	1	yes	characteristic, complemented, Hall
20	ΓL(1,𝔽₈)	C₂²×C₂₆₃	1	1	yes	characteristic, complemented, Hall
21	C₂×C₇	C₂×C₃×C₂₆₃	1	1	yes	characteristic, complemented
22	D₇	C₂×C₃×C₂₆₃	1	2	yes	complemented
23	C₂²×C₃	—	7	1	no	Hall
24	C₇	C₂²×C₃×C₂₆₃	1	1	yes	characteristic, complemented, Sylow
25	C₂×C₃	—	7	2	no	—
26	C₂×C₃	—	7	1	no	—
27	C₂²	—	7	1	no	Sylow
28	C₃	—	7	1	no	Sylow
29	C₂	—	7	2	no	—
30	C₂	AGL(1,𝔽₇)×C₂₆₃	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂×C₂₆₃	AGL(1,𝔽₇)
commutator, cosocle, hypocenter	C₇	C₂²×C₃×C₂₆₃
Frattini	C₁	G
Fitting, socle	C₂×C₇×C₂₆₃	C₂×C₃
LCS	G ▷ C₇
UCS	C₁ ◁ C₂×C₂₆₃

Complements

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

N	H ≅ G/N	type
1. AGL(1,𝔽₇)×C₂₆₃	C₂	direct (N × H)
2. C₂×ΓL(1,𝔽₈)×C₂₆₃	C₂	semidirect
3. C₂×D₇×C₂₆₃	C₃	semidirect
4. ΓL(1,𝔽₈)×C₂₆₃	C₂²	semidirect
5. C₂×C₇×C₂₆₃	C₂×C₃	semidirect
6. D₇×C₂₆₃	C₂×C₃	semidirect
8. C₇×C₂₆₃	C₂²×C₃	semidirect
14. C₂×C₂₆₃	AGL(1,𝔽₇)	direct (N × H)
15. C₂₆₃	C₂×AGL(1,𝔽₇)	direct (N × H)
16. C₂×AGL(1,𝔽₇)	C₂₆₃	direct (N × H)
17. C₂×ΓL(1,𝔽₈)	C₂×C₂₆₃	semidirect
18. AGL(1,𝔽₇)	C₂×C₂₆₃	direct (N × H)
19. C₂×D₇	C₃×C₂₆₃	semidirect
20. ΓL(1,𝔽₈)	C₂²×C₂₆₃	semidirect
21. C₂×C₇	C₂×C₃×C₂₆₃	semidirect
22. D₇	C₂×C₃×C₂₆₃	semidirect
24. C₇	C₂²×C₃×C₂₆₃	semidirect
30. C₂	AGL(1,𝔽₇)×C₂₆₃	direct (N × H)

Maximal subgroups

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

subgroup	index
AGL(1,𝔽₇)×C₂₆₃	2
C₂×ΓL(1,𝔽₈)×C₂₆₃	2
C₂×D₇×C₂₆₃	3
C₂²×C₃×C₂₆₃	7
C₂×AGL(1,𝔽₇)	263

Subgroups — C2×AGL(1,𝔽7)×C263 #36348317

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×AGL(1,𝔽₇)×C₂₆₃ #36348317