Subgroups — A₄×ΓL(1,𝔽₈)×C₇₁ #31822075

Order 17892.

56 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇₁×C₂²⋊ΓL(1,𝔽₈)	C₃	1	2	yes	maximal, complemented
2	C₇×A₄×C₇₁	C₃	1	1	yes	maximal, characteristic, complemented
3	C₂²×ΓL(1,𝔽₈)×C₇₁	C₃	1	1	yes	maximal, characteristic, complemented
4	C₃×ΓL(1,𝔽₈)×C₇₁	—	4	1	no	maximal, Hall
5	C₂×ΓL(1,𝔽₈)×C₇₁	—	3	1	no	—
6	C₃×A₄×C₇₁	—	7	1	no	maximal, Hall
7	C₂²×C₇×C₇₁	C₃²	1	1	yes	characteristic, complemented, Hall
8	C₃×C₇×C₇₁	—	4	1	no	—
9	ΓL(1,𝔽₈)×C₇₁	—	4	2	no	—
10	ΓL(1,𝔽₈)×C₇₁	A₄	1	1	yes	characteristic, direct product factor
11	C₂×C₇×C₇₁	—	3	1	no	—
12	C₂²×C₃×C₇₁	—	7	1	no	—
13	A₄×C₇₁	—	7	2	no	—
14	A₄×C₇₁	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
15	C₃²×C₇₁	—	28	1	no	Hall
16	C₇×C₇₁	C₃×A₄	1	1	yes	characteristic, complemented, Hall
17	C₂×C₃×C₇₁	—	21	1	no	—
18	C₂²×C₇₁	C₃×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Hall
19	A₄×ΓL(1,𝔽₈)	C₇₁	1	1	yes	maximal, characteristic, direct product factor, Hall
20	C₃×C₇₁	—	4	1	no	—
21	C₃×C₇₁	—	7	1	no	—
22	C₃×C₇₁	—	28	2	no	—
23	C₂×C₇₁	—	3	1	no	—
24	C₂²⋊ΓL(1,𝔽₈)	C₃×C₇₁	1	2	yes	complemented
25	C₇×A₄	C₃×C₇₁	1	1	yes	characteristic, complemented
26	C₂²×ΓL(1,𝔽₈)	C₃×C₇₁	1	1	yes	characteristic, complemented
27	C₇₁	A₄×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Sylow
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×C₇₁	1	1	yes	characteristic, complemented, Hall
32	C₃×C₇	—	4	1	no	—
33	ΓL(1,𝔽₈)	—	4	2	no	—
34	ΓL(1,𝔽₈)	A₄×C₇₁	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)×C₇₁	1	1	yes	characteristic, direct product factor
39	C₃²	—	28	1	no	Sylow
40	C₇	C₃×A₄×C₇₁	1	1	yes	characteristic, complemented, Sylow
41	C₂×C₃	—	21	1	no	—
42	C₂²	C₃×ΓL(1,𝔽₈)×C₇₁	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	C₇₁	A₄×ΓL(1,𝔽₈)
commutator, cosocle, hypocenter	C₂²×C₇	C₃²×C₇₁
Frattini	C₁	G
Fitting, socle	C₂²×C₇×C₇₁	C₃²
LCS	G ▷ C₂²×C₇
UCS	C₁ ◁ C₇₁

Complements

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

N	H ≅ G/N	type
1. C₇₁×C₂²⋊ΓL(1,𝔽₈)	C₃	semidirect
2. C₇×A₄×C₇₁	C₃	semidirect
3. C₂²×ΓL(1,𝔽₈)×C₇₁	C₃	semidirect
7. C₂²×C₇×C₇₁	C₃²	semidirect
10. ΓL(1,𝔽₈)×C₇₁	A₄	direct (N × H)
14. A₄×C₇₁	ΓL(1,𝔽₈)	direct (N × H)
16. C₇×C₇₁	C₃×A₄	semidirect
18. C₂²×C₇₁	C₃×ΓL(1,𝔽₈)	semidirect
19. A₄×ΓL(1,𝔽₈)	C₇₁	direct (N × H)
24. C₂²⋊ΓL(1,𝔽₈)	C₃×C₇₁	semidirect
25. C₇×A₄	C₃×C₇₁	semidirect
26. C₂²×ΓL(1,𝔽₈)	C₃×C₇₁	semidirect
27. C₇₁	A₄×ΓL(1,𝔽₈)	direct (N × H)
31. C₂²×C₇	C₃²×C₇₁	semidirect
34. ΓL(1,𝔽₈)	A₄×C₇₁	direct (N × H)
38. A₄	ΓL(1,𝔽₈)×C₇₁	direct (N × H)
40. C₇	C₃×A₄×C₇₁	semidirect
42. C₂²	C₃×ΓL(1,𝔽₈)×C₇₁	semidirect

Maximal subgroups

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

subgroup	index
C₇₁×C₂²⋊ΓL(1,𝔽₈)	3
C₇×A₄×C₇₁	3
C₂²×ΓL(1,𝔽₈)×C₇₁	3
C₃×ΓL(1,𝔽₈)×C₇₁	4
C₃×A₄×C₇₁	7
A₄×ΓL(1,𝔽₈)	71

Subgroups — A4×ΓL(1,𝔽8)×C71 #31822075

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — A₄×ΓL(1,𝔽₈)×C₇₁ #31822075