Subgroups — C₁₃×C₂₃×C₂²⋊ΓL(1,𝔽₈) #38737448

Order 25116.

40 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₇×C₁₃×C₂₃	C₃	1	1	yes	maximal, characteristic, complemented, Hall
2	C₁₃×ΓL(1,𝔽₈)×C₂₃	—	4	1	no	maximal, Hall
3	C₂×C₇×C₁₃×C₂₃	—	3	1	no	—
4	A₄×C₁₃×C₂₃	—	7	1	no	maximal, Hall
5	C₇×C₁₃×C₂₃	A₄	1	1	yes	characteristic, complemented, Hall
6	C₂₃×C₂²⋊ΓL(1,𝔽₈)	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂²×C₁₃×C₂₃	ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Hall
8	C₁₃×C₂²⋊ΓL(1,𝔽₈)	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₃×C₁₃×C₂₃	—	28	1	no	Hall
10	C₂²×C₇×C₂₃	C₃×C₁₃	1	1	yes	characteristic, complemented, Hall
11	C₂×C₁₃×C₂₃	—	3	1	no	—
12	ΓL(1,𝔽₈)×C₂₃	—	4	1	no	Hall
13	C₂²×C₇×C₁₃	C₃×C₂₃	1	1	yes	characteristic, complemented, Hall
14	C₂×C₇×C₂₃	—	3	1	no	—
15	C₁₃×C₂₃	C₂²⋊ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Hall
16	A₄×C₂₃	—	7	1	no	Hall
17	C₁₃×ΓL(1,𝔽₈)	—	4	1	no	Hall
18	C₂×C₇×C₁₃	—	3	1	no	—
19	C₇×C₂₃	A₄×C₁₃	1	1	yes	characteristic, complemented, Hall
20	A₄×C₁₃	—	7	1	no	Hall
21	C₂²×C₂₃	C₁₃×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Hall
22	C₇×C₁₃	A₄×C₂₃	1	1	yes	characteristic, complemented, Hall
23	C₂²⋊ΓL(1,𝔽₈)	C₁₃×C₂₃	1	1	yes	characteristic, direct product factor, Hall
24	C₃×C₂₃	—	28	1	no	Hall
25	C₂²×C₁₃	ΓL(1,𝔽₈)×C₂₃	1	1	yes	characteristic, complemented, Hall
26	C₂×C₂₃	—	3	1	no	—
27	C₃×C₁₃	—	28	1	no	Hall
28	C₂²×C₇	C₃×C₁₃×C₂₃	1	1	yes	characteristic, complemented, Hall
29	C₂×C₁₃	—	3	1	no	—
30	C₂₃	C₁₃×C₂²⋊ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Sylow
31	ΓL(1,𝔽₈)	—	4	1	no	Hall
32	C₂×C₇	—	3	1	no	—
33	C₁₃	C₂₃×C₂²⋊ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Sylow
34	A₄	—	7	1	no	Hall
35	C₇	A₄×C₁₃×C₂₃	1	1	yes	characteristic, complemented, Sylow
36	C₂²	C₁₃×ΓL(1,𝔽₈)×C₂₃	1	1	yes	characteristic, complemented, Sylow
37	C₃	—	28	1	no	Sylow
38	C₂	—	3	1	no	—

	subgroup	G/H
center, hypercenter	C₁₃×C₂₃	C₂²⋊ΓL(1,𝔽₈)
commutator, cosocle, hypocenter	C₂²×C₇	C₃×C₁₃×C₂₃
Frattini	C₁	G
Fitting, socle	C₂²×C₇×C₁₃×C₂₃	C₃
LCS	G ▷ C₂²×C₇
UCS	C₁ ◁ C₁₃×C₂₃

Complements

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

N	H ≅ G/N	type
1. C₂²×C₇×C₁₃×C₂₃	C₃	semidirect
5. C₇×C₁₃×C₂₃	A₄	semidirect
6. C₂₃×C₂²⋊ΓL(1,𝔽₈)	C₁₃	direct (N × H)
7. C₂²×C₁₃×C₂₃	ΓL(1,𝔽₈)	semidirect
8. C₁₃×C₂²⋊ΓL(1,𝔽₈)	C₂₃	direct (N × H)
10. C₂²×C₇×C₂₃	C₃×C₁₃	semidirect
13. C₂²×C₇×C₁₃	C₃×C₂₃	semidirect
15. C₁₃×C₂₃	C₂²⋊ΓL(1,𝔽₈)	direct (N × H)
19. C₇×C₂₃	A₄×C₁₃	semidirect
21. C₂²×C₂₃	C₁₃×ΓL(1,𝔽₈)	semidirect
22. C₇×C₁₃	A₄×C₂₃	semidirect
23. C₂²⋊ΓL(1,𝔽₈)	C₁₃×C₂₃	direct (N × H)
25. C₂²×C₁₃	ΓL(1,𝔽₈)×C₂₃	semidirect
28. C₂²×C₇	C₃×C₁₃×C₂₃	semidirect
30. C₂₃	C₁₃×C₂²⋊ΓL(1,𝔽₈)	direct (N × H)
33. C₁₃	C₂₃×C₂²⋊ΓL(1,𝔽₈)	direct (N × H)
35. C₇	A₄×C₁₃×C₂₃	semidirect
36. C₂²	C₁₃×ΓL(1,𝔽₈)×C₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₇×C₁₃×C₂₃	3
C₁₃×ΓL(1,𝔽₈)×C₂₃	4
A₄×C₁₃×C₂₃	7
C₂₃×C₂²⋊ΓL(1,𝔽₈)	13
C₁₃×C₂²⋊ΓL(1,𝔽₈)	23

Subgroups — C13×C23×C22⋊ΓL(1,𝔽8) #38737448

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×C₂₃×C₂²⋊ΓL(1,𝔽₈) #38737448