Subgroups — C₂×C₇₁×C₃⋊AGL(1,𝔽₇) #31822080

Order 17892.

88 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×ΓL(1,𝔽₈)×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₇₁×C₃⋊AGL(1,𝔽₇)	C₂	1	2	yes	maximal, direct product factor
3	C₂×D₂₁×C₇₁	C₃	1	1	yes	maximal, characteristic, complemented
4	C₂×AGL(1,𝔽₇)×C₇₁	—	3	1	no	maximal
5	C₃×ΓL(1,𝔽₈)×C₇₁	C₂²	1	1	yes	characteristic, complemented, Hall
6	AGL(1,𝔽₇)×C₇₁	—	3	2	no	—
7	D₂₁×C₇₁	C₂×C₃	1	2	yes	complemented
8	C₂×ΓL(1,𝔽₈)×C₇₁	—	2	1	no	—
9	C₂×ΓL(1,𝔽₈)×C₇₁	D₃	1	1	yes	characteristic, complemented
10	C₂×C₃×C₇×C₇₁	C₂×C₃	1	1	yes	characteristic, complemented
11	C₂×C₃×D₃×C₇₁	—	7	1	no	maximal, Hall
12	C₂×D₇×C₇₁	—	3	1	no	Hall
13	C₃×C₇×C₇₁	C₂²×C₃	1	1	yes	characteristic, complemented
14	ΓL(1,𝔽₈)×C₇₁	—	2	1	no	—
15	ΓL(1,𝔽₈)×C₇₁	C₂×D₃	1	1	yes	characteristic, complemented
16	C₂×C₃²×C₇₁	—	7	1	no	—
17	C₃×D₃×C₇₁	—	7	2	no	—
18	C₂×C₇×C₇₁	C₃×D₃	1	1	yes	characteristic, complemented
19	D₇×C₇₁	—	3	2	no	—
20	C₂²×C₃×C₇₁	—	21	1	no	—
21	C₂×D₃×C₇₁	—	7	1	no	—
22	C₃²×C₇₁	—	7	1	no	Hall
23	C₇×C₇₁	C₂×C₃×D₃	1	1	yes	characteristic, complemented, Hall
24	C₂×C₃×C₇₁	—	7	1	no	—
25	C₂×C₃×C₇₁	—	14	1	no	—
26	C₂×C₃×C₇₁	—	21	2	no	—
27	C₂×C₃×C₇₁	AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
28	D₃×C₇₁	—	7	2	no	—
29	C₂²×C₇₁	—	21	1	no	Hall
30	C₂×C₃⋊AGL(1,𝔽₇)	C₇₁	1	1	yes	maximal, characteristic, direct product factor, Hall
31	C₃×C₇₁	—	7	1	no	—
32	C₃×C₇₁	—	14	1	no	—
33	C₃×C₇₁	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
34	C₂×C₇₁	—	21	2	no	—
35	C₂×C₇₁	C₃⋊AGL(1,𝔽₇)	1	1	yes	characteristic, direct product factor
36	C₂×C₃×ΓL(1,𝔽₈)	C₂×C₇₁	1	1	yes	characteristic, complemented
37	C₃⋊AGL(1,𝔽₇)	C₂×C₇₁	1	2	yes	direct product factor
38	C₂×D₂₁	C₃×C₇₁	1	1	yes	characteristic, complemented
39	C₂×AGL(1,𝔽₇)	—	3	1	no	—
40	C₇₁	C₂×C₃⋊AGL(1,𝔽₇)	1	1	yes	characteristic, direct product factor, Sylow
41	C₃×ΓL(1,𝔽₈)	C₂²×C₇₁	1	1	yes	characteristic, complemented, Hall
42	C₂×C₃×C₇	C₂×C₃×C₇₁	1	1	yes	characteristic, complemented
43	D₂₁	C₂×C₃×C₇₁	1	2	yes	complemented
44	C₂×ΓL(1,𝔽₈)	—	2	1	no	—
45	C₂×ΓL(1,𝔽₈)	D₃×C₇₁	1	1	yes	characteristic, complemented
46	AGL(1,𝔽₇)	—	3	2	no	—
47	C₂×C₃×D₃	—	7	1	no	Hall
48	C₂×D₇	—	3	1	no	Hall
49	C₃×C₇	C₂²×C₃×C₇₁	1	1	yes	characteristic, complemented
50	ΓL(1,𝔽₈)	—	2	1	no	—
51	ΓL(1,𝔽₈)	C₂×D₃×C₇₁	1	1	yes	characteristic, complemented
52	C₂×C₃²	—	7	1	no	—
53	C₃×D₃	—	7	2	no	—
54	C₂×C₇	C₃×D₃×C₇₁	1	1	yes	characteristic, complemented
55	D₇	—	3	2	no	—
56	C₂²×C₃	—	21	1	no	—
57	C₂×D₃	—	7	1	no	—
58	C₃²	—	7	1	no	Sylow
59	C₇	C₂×C₃×D₃×C₇₁	1	1	yes	characteristic, complemented, Sylow
60	C₂×C₃	—	7	1	no	—
61	C₂×C₃	—	14	1	no	—
62	C₂×C₃	—	21	2	no	—
63	C₂×C₃	AGL(1,𝔽₇)×C₇₁	1	1	yes	characteristic, complemented
64	D₃	—	7	2	no	—
65	C₂²	—	21	1	no	Sylow
66	C₃	—	7	1	no	—
67	C₃	—	14	1	no	—
68	C₃	C₂×AGL(1,𝔽₇)×C₇₁	1	1	yes	characteristic, complemented
69	C₂	—	21	2	no	—
70	C₂	C₇₁×C₃⋊AGL(1,𝔽₇)	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂×C₇₁	C₃⋊AGL(1,𝔽₇)
commutator, cosocle, hypocenter	C₃×C₇	C₂²×C₃×C₇₁
Frattini	C₁	G
Fitting, socle	C₂×C₃×C₇×C₇₁	C₂×C₃
LCS	G ▷ C₃×C₇
UCS	C₁ ◁ C₂×C₇₁

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×ΓL(1,𝔽₈)×C₇₁	C₂	semidirect
2. C₇₁×C₃⋊AGL(1,𝔽₇)	C₂	direct (N × H)
3. C₂×D₂₁×C₇₁	C₃	semidirect
5. C₃×ΓL(1,𝔽₈)×C₇₁	C₂²	semidirect
7. D₂₁×C₇₁	C₂×C₃	semidirect
9. C₂×ΓL(1,𝔽₈)×C₇₁	D₃	semidirect
10. C₂×C₃×C₇×C₇₁	C₂×C₃	semidirect
13. C₃×C₇×C₇₁	C₂²×C₃	semidirect
15. ΓL(1,𝔽₈)×C₇₁	C₂×D₃	semidirect
18. C₂×C₇×C₇₁	C₃×D₃	semidirect
23. C₇×C₇₁	C₂×C₃×D₃	semidirect
27. C₂×C₃×C₇₁	AGL(1,𝔽₇)	semidirect
30. C₂×C₃⋊AGL(1,𝔽₇)	C₇₁	direct (N × H)
33. C₃×C₇₁	C₂×AGL(1,𝔽₇)	semidirect
35. C₂×C₇₁	C₃⋊AGL(1,𝔽₇)	direct (N × H)
36. C₂×C₃×ΓL(1,𝔽₈)	C₂×C₇₁	semidirect
37. C₃⋊AGL(1,𝔽₇)	C₂×C₇₁	direct (N × H)
38. C₂×D₂₁	C₃×C₇₁	semidirect
40. C₇₁	C₂×C₃⋊AGL(1,𝔽₇)	direct (N × H)
41. C₃×ΓL(1,𝔽₈)	C₂²×C₇₁	semidirect
42. C₂×C₃×C₇	C₂×C₃×C₇₁	semidirect
43. D₂₁	C₂×C₃×C₇₁	semidirect
45. C₂×ΓL(1,𝔽₈)	D₃×C₇₁	semidirect
49. C₃×C₇	C₂²×C₃×C₇₁	semidirect
51. ΓL(1,𝔽₈)	C₂×D₃×C₇₁	semidirect
54. C₂×C₇	C₃×D₃×C₇₁	semidirect
59. C₇	C₂×C₃×D₃×C₇₁	semidirect
63. C₂×C₃	AGL(1,𝔽₇)×C₇₁	semidirect
68. C₃	C₂×AGL(1,𝔽₇)×C₇₁	semidirect
70. C₂	C₇₁×C₃⋊AGL(1,𝔽₇)	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃×ΓL(1,𝔽₈)×C₇₁	2
C₇₁×C₃⋊AGL(1,𝔽₇)	2
C₂×D₂₁×C₇₁	3
C₂×AGL(1,𝔽₇)×C₇₁	3
C₂×C₃×D₃×C₇₁	7
C₂×C₃⋊AGL(1,𝔽₇)	71

Subgroups — C2×C71×C3⋊AGL(1,𝔽7) #31822080

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇₁×C₃⋊AGL(1,𝔽₇) #31822080