Subgroups — D₇×ΓL(1,𝔽₈)×C₄₃ #27162436

Order 12642.

34 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×ΓL(1,𝔽₈)×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	C₇×D₇×C₄₃	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	C₇²×C₄₃	C₂×C₃	1	1	yes	characteristic, complemented, Hall
4	C₃×D₇×C₄₃	—	7	1	no	maximal
5	C₂×ΓL(1,𝔽₈)×C₄₃	—	7	1	no	maximal
6	C₃×C₇×C₄₃	—	7	1	no	—
7	ΓL(1,𝔽₈)×C₄₃	D₇	1	1	yes	characteristic, direct product factor
8	C₂×C₇×C₄₃	—	7	1	no	—
9	D₇×C₄₃	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
10	C₇×C₄₃	—	6	1	no	—
11	C₇×C₄₃	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
12	C₇×C₄₃	C₃×D₇	1	1	yes	characteristic, complemented
13	D₇×ΓL(1,𝔽₈)	C₄₃	1	1	yes	maximal, characteristic, direct product factor, Hall
14	C₂×C₃×C₄₃	—	49	1	no	Hall
15	C₇×ΓL(1,𝔽₈)	C₂×C₄₃	1	1	yes	characteristic, complemented, Hall
16	C₃×C₄₃	—	7	1	no	Hall
17	C₇×D₇	C₃×C₄₃	1	1	yes	characteristic, complemented, Hall
18	C₂×C₄₃	—	7	1	no	Hall
19	C₇²	C₂×C₃×C₄₃	1	1	yes	characteristic, complemented, Sylow
20	C₄₃	D₇×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Sylow
21	C₃×D₇	—	7	1	no	—
22	C₂×ΓL(1,𝔽₈)	—	7	1	no	—
23	C₃×C₇	—	7	1	no	—
24	ΓL(1,𝔽₈)	D₇×C₄₃	1	1	yes	characteristic, direct product factor
25	C₂×C₇	—	7	1	no	—
26	D₇	ΓL(1,𝔽₈)×C₄₃	1	1	yes	characteristic, direct product factor
27	C₇	—	6	1	no	—
28	C₇	C₂×ΓL(1,𝔽₈)×C₄₃	1	1	yes	characteristic, complemented
29	C₇	C₃×D₇×C₄₃	1	1	yes	characteristic, complemented
30	C₂×C₃	—	49	1	no	Hall
31	C₃	—	7	1	no	Sylow
32	C₂	—	7	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₄₃	D₇×ΓL(1,𝔽₈)
commutator, cosocle, hypocenter	C₇²	C₂×C₃×C₄₃
Frattini	C₁	G
Fitting, socle	C₇²×C₄₃	C₂×C₃
LCS	G ▷ C₇²
UCS	C₁ ◁ C₄₃

Complements

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

N	H ≅ G/N	type
1. C₇×ΓL(1,𝔽₈)×C₄₃	C₂	semidirect
2. C₇×D₇×C₄₃	C₃	semidirect
3. C₇²×C₄₃	C₂×C₃	semidirect
7. ΓL(1,𝔽₈)×C₄₃	D₇	direct (N × H)
9. D₇×C₄₃	ΓL(1,𝔽₈)	direct (N × H)
11. C₇×C₄₃	C₂×ΓL(1,𝔽₈)	semidirect
12. C₇×C₄₃	C₃×D₇	semidirect
13. D₇×ΓL(1,𝔽₈)	C₄₃	direct (N × H)
15. C₇×ΓL(1,𝔽₈)	C₂×C₄₃	semidirect
17. C₇×D₇	C₃×C₄₃	semidirect
19. C₇²	C₂×C₃×C₄₃	semidirect
20. C₄₃	D₇×ΓL(1,𝔽₈)	direct (N × H)
24. ΓL(1,𝔽₈)	D₇×C₄₃	direct (N × H)
26. D₇	ΓL(1,𝔽₈)×C₄₃	direct (N × H)
28. C₇	C₂×ΓL(1,𝔽₈)×C₄₃	semidirect
29. C₇	C₃×D₇×C₄₃	semidirect

Maximal subgroups

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

subgroup	index
C₇×ΓL(1,𝔽₈)×C₄₃	2
C₇×D₇×C₄₃	3
C₃×D₇×C₄₃	7
C₂×ΓL(1,𝔽₈)×C₄₃	7
D₇×ΓL(1,𝔽₈)	43

Subgroups — D7×ΓL(1,𝔽8)×C43 #27162436

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₇×ΓL(1,𝔽₈)×C₄₃ #27162436