Subgroups — A₄×ΓL(1,𝔽₈)×C₁₂₇ #44544843

Order 32004.

56 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×ΓL(1,𝔽₈)×C₁₂₇	C₃	1	1	yes	maximal, characteristic, complemented
2	C₇×A₄×C₁₂₇	C₃	1	1	yes	maximal, characteristic, complemented
3	C₂²⋊ΓL(1,𝔽₈)×C₁₂₇	C₃	1	2	yes	maximal, complemented
4	C₃×ΓL(1,𝔽₈)×C₁₂₇	—	4	1	no	maximal, Hall
5	C₂×ΓL(1,𝔽₈)×C₁₂₇	—	3	1	no	—
6	C₃×A₄×C₁₂₇	—	7	1	no	maximal, Hall
7	C₂²×C₇×C₁₂₇	C₃²	1	1	yes	characteristic, complemented, Hall
8	C₃×C₇×C₁₂₇	—	4	1	no	—
9	ΓL(1,𝔽₈)×C₁₂₇	—	4	2	no	—
10	ΓL(1,𝔽₈)×C₁₂₇	A₄	1	1	yes	characteristic, direct product factor
11	C₂×C₇×C₁₂₇	—	3	1	no	—
12	C₂²×C₃×C₁₂₇	—	7	1	no	—
13	A₄×C₁₂₇	—	7	2	no	—
14	A₄×C₁₂₇	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
15	C₃²×C₁₂₇	—	28	1	no	Hall
16	C₇×C₁₂₇	C₃×A₄	1	1	yes	characteristic, complemented, Hall
17	C₂×C₃×C₁₂₇	—	21	1	no	—
18	C₂²×C₁₂₇	C₃×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Hall
19	C₃×C₁₂₇	—	4	1	no	—
20	C₃×C₁₂₇	—	7	1	no	—
21	C₃×C₁₂₇	—	28	2	no	—
22	C₂×C₁₂₇	—	3	1	no	—
23	A₄×ΓL(1,𝔽₈)	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Hall
24	C₁₂₇	A₄×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Sylow
25	C₂²⋊ΓL(1,𝔽₈)	C₃×C₁₂₇	1	2	yes	complemented
26	C₇×A₄	C₃×C₁₂₇	1	1	yes	characteristic, complemented
27	C₂²×ΓL(1,𝔽₈)	C₃×C₁₂₇	1	1	yes	characteristic, complemented
28	C₃×ΓL(1,𝔽₈)	—	4	1	no	Hall
29	C₂×ΓL(1,𝔽₈)	—	3	1	no	—
30	C₃×A₄	—	7	1	no	Hall
31	C₂²×C₇	C₃²×C₁₂₇	1	1	yes	characteristic, complemented, Hall
32	C₃×C₇	—	4	1	no	—
33	ΓL(1,𝔽₈)	—	4	2	no	—
34	ΓL(1,𝔽₈)	A₄×C₁₂₇	1	1	yes	characteristic, direct product factor
35	C₂×C₇	—	3	1	no	—
36	C₂²×C₃	—	7	1	no	—
37	A₄	—	7	2	no	—
38	A₄	ΓL(1,𝔽₈)×C₁₂₇	1	1	yes	characteristic, direct product factor
39	C₃²	—	28	1	no	Sylow
40	C₇	C₃×A₄×C₁₂₇	1	1	yes	characteristic, complemented, Sylow
41	C₂×C₃	—	21	1	no	—
42	C₂²	C₃×ΓL(1,𝔽₈)×C₁₂₇	1	1	yes	characteristic, complemented, Sylow
43	C₃	—	4	1	no	—
44	C₃	—	7	1	no	—
45	C₃	—	28	2	no	—
46	C₂	—	3	1	no	—

	subgroup	G/H
center, hypercenter	C₁₂₇	A₄×ΓL(1,𝔽₈)
commutator, cosocle, hypocenter	C₂²×C₇	C₃²×C₁₂₇
Frattini	C₁	G
Fitting, socle	C₂²×C₇×C₁₂₇	C₃²
LCS	G ▷ C₂²×C₇
UCS	C₁ ◁ C₁₂₇

Complements

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

N	H ≅ G/N	type
1. C₂²×ΓL(1,𝔽₈)×C₁₂₇	C₃	semidirect
2. C₇×A₄×C₁₂₇	C₃	semidirect
3. C₂²⋊ΓL(1,𝔽₈)×C₁₂₇	C₃	semidirect
7. C₂²×C₇×C₁₂₇	C₃²	semidirect
10. ΓL(1,𝔽₈)×C₁₂₇	A₄	direct (N × H)
14. A₄×C₁₂₇	ΓL(1,𝔽₈)	direct (N × H)
16. C₇×C₁₂₇	C₃×A₄	semidirect
18. C₂²×C₁₂₇	C₃×ΓL(1,𝔽₈)	semidirect
23. A₄×ΓL(1,𝔽₈)	C₁₂₇	direct (N × H)
24. C₁₂₇	A₄×ΓL(1,𝔽₈)	direct (N × H)
25. C₂²⋊ΓL(1,𝔽₈)	C₃×C₁₂₇	semidirect
26. C₇×A₄	C₃×C₁₂₇	semidirect
27. C₂²×ΓL(1,𝔽₈)	C₃×C₁₂₇	semidirect
31. C₂²×C₇	C₃²×C₁₂₇	semidirect
34. ΓL(1,𝔽₈)	A₄×C₁₂₇	direct (N × H)
38. A₄	ΓL(1,𝔽₈)×C₁₂₇	direct (N × H)
40. C₇	C₃×A₄×C₁₂₇	semidirect
42. C₂²	C₃×ΓL(1,𝔽₈)×C₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₂²×ΓL(1,𝔽₈)×C₁₂₇	3
C₇×A₄×C₁₂₇	3
C₂²⋊ΓL(1,𝔽₈)×C₁₂₇	3
C₃×ΓL(1,𝔽₈)×C₁₂₇	4
C₃×A₄×C₁₂₇	7
A₄×ΓL(1,𝔽₈)	127

Subgroups — A4×ΓL(1,𝔽8)×C127 #44544843

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — A₄×ΓL(1,𝔽₈)×C₁₂₇ #44544843