Subgroups — C₅×C₃₇×C₃⋊AGL(1,𝔽₇) #37552093

Order 23310.

72 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×ΓL(1,𝔽₈)×C₃₇	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	C₅×C₃₇×AGL(1,𝔽₇)	—	3	1	no	maximal
3	C₅×C₃₇×D₂₁	C₃	1	1	yes	maximal, characteristic, complemented
4	C₃₇×C₃⋊AGL(1,𝔽₇)	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₅×ΓL(1,𝔽₈)×C₃₇	—	2	1	no	—
6	C₅×ΓL(1,𝔽₈)×C₃₇	D₃	1	1	yes	characteristic, complemented
7	C₃×C₅×C₇×C₃₇	C₂×C₃	1	1	yes	characteristic, complemented
8	C₃×C₅×D₃×C₃₇	—	7	1	no	maximal, Hall
9	C₅×D₇×C₃₇	—	3	1	no	Hall
10	C₃×ΓL(1,𝔽₈)×C₃₇	C₂×C₅	1	1	yes	characteristic, complemented, Hall
11	C₃²×C₅×C₃₇	—	7	1	no	Hall
12	C₃₇×D₂₁	C₃×C₅	1	1	yes	characteristic, complemented
13	C₃₇×AGL(1,𝔽₇)	—	3	1	no	—
14	C₅×C₇×C₃₇	C₃×D₃	1	1	yes	characteristic, complemented, Hall
15	C₂×C₃×C₅×C₃₇	—	21	1	no	—
16	C₅×D₃×C₃₇	—	7	1	no	—
17	C₃×C₇×C₃₇	C₂×C₃×C₅	1	1	yes	characteristic, complemented
18	ΓL(1,𝔽₈)×C₃₇	—	2	1	no	—
19	ΓL(1,𝔽₈)×C₃₇	C₅×D₃	1	1	yes	characteristic, complemented
20	C₃×D₃×C₃₇	—	7	1	no	Hall
21	C₅×C₃⋊AGL(1,𝔽₇)	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Hall
22	C₃×C₅×C₃₇	—	7	1	no	—
23	C₃×C₅×C₃₇	—	14	1	no	—
24	C₃×C₅×C₃₇	AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
25	D₇×C₃₇	—	3	1	no	Hall
26	C₂×C₅×C₃₇	—	21	1	no	Hall
27	C₃²×C₃₇	—	7	1	no	Hall
28	C₃×C₅×ΓL(1,𝔽₈)	C₂×C₃₇	1	1	yes	characteristic, complemented, Hall
29	C₇×C₃₇	C₃×C₅×D₃	1	1	yes	characteristic, complemented, Hall
30	C₂×C₃×C₃₇	—	21	1	no	—
31	D₃×C₃₇	—	7	1	no	—
32	C₅×D₂₁	C₃×C₃₇	1	1	yes	characteristic, complemented
33	C₅×AGL(1,𝔽₇)	—	3	1	no	—
34	C₅×C₃₇	C₃⋊AGL(1,𝔽₇)	1	1	yes	characteristic, direct product factor, Hall
35	C₃⋊AGL(1,𝔽₇)	C₅×C₃₇	1	1	yes	characteristic, direct product factor, Hall
36	C₃×C₃₇	—	7	1	no	—
37	C₃×C₃₇	—	14	1	no	—
38	C₃×C₃₇	C₅×AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
39	C₃×C₅×C₇	C₂×C₃×C₃₇	1	1	yes	characteristic, complemented
40	C₅×ΓL(1,𝔽₈)	—	2	1	no	—
41	C₅×ΓL(1,𝔽₈)	D₃×C₃₇	1	1	yes	characteristic, complemented
42	C₃×C₅×D₃	—	7	1	no	Hall
43	C₂×C₃₇	—	21	1	no	Hall
44	C₅×D₇	—	3	1	no	Hall
45	C₃×ΓL(1,𝔽₈)	C₂×C₅×C₃₇	1	1	yes	characteristic, complemented, Hall
46	C₃²×C₅	—	7	1	no	Hall
47	D₂₁	C₃×C₅×C₃₇	1	1	yes	characteristic, complemented
48	AGL(1,𝔽₇)	—	3	1	no	—
49	C₃₇	C₅×C₃⋊AGL(1,𝔽₇)	1	1	yes	characteristic, direct product factor, Sylow
50	C₅×C₇	C₃×D₃×C₃₇	1	1	yes	characteristic, complemented, Hall
51	C₂×C₃×C₅	—	21	1	no	—
52	C₅×D₃	—	7	1	no	—
53	C₃×C₇	C₂×C₃×C₅×C₃₇	1	1	yes	characteristic, complemented
54	ΓL(1,𝔽₈)	—	2	1	no	—
55	ΓL(1,𝔽₈)	C₅×D₃×C₃₇	1	1	yes	characteristic, complemented
56	C₃×D₃	—	7	1	no	Hall
57	C₃×C₅	—	7	1	no	—
58	C₃×C₅	—	14	1	no	—
59	C₃×C₅	C₃₇×AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
60	D₇	—	3	1	no	Hall
61	C₂×C₅	—	21	1	no	Hall
62	C₃²	—	7	1	no	Sylow
63	C₇	C₃×C₅×D₃×C₃₇	1	1	yes	characteristic, complemented, Sylow
64	C₂×C₃	—	21	1	no	—
65	D₃	—	7	1	no	—
66	C₅	C₃₇×C₃⋊AGL(1,𝔽₇)	1	1	yes	characteristic, direct product factor, Sylow
67	C₃	—	7	1	no	—
68	C₃	—	14	1	no	—
69	C₃	C₅×C₃₇×AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
70	C₂	—	21	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₅×C₃₇	C₃⋊AGL(1,𝔽₇)
commutator, cosocle, hypocenter	C₃×C₇	C₂×C₃×C₅×C₃₇
Frattini	C₁	G
Fitting, socle	C₃×C₅×C₇×C₃₇	C₂×C₃
LCS	G ▷ C₃×C₇
UCS	C₁ ◁ C₅×C₃₇

Complements

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

N	H ≅ G/N	type
1. C₃×C₅×ΓL(1,𝔽₈)×C₃₇	C₂	semidirect
3. C₅×C₃₇×D₂₁	C₃	semidirect
4. C₃₇×C₃⋊AGL(1,𝔽₇)	C₅	direct (N × H)
6. C₅×ΓL(1,𝔽₈)×C₃₇	D₃	semidirect
7. C₃×C₅×C₇×C₃₇	C₂×C₃	semidirect
10. C₃×ΓL(1,𝔽₈)×C₃₇	C₂×C₅	semidirect
12. C₃₇×D₂₁	C₃×C₅	semidirect
14. C₅×C₇×C₃₇	C₃×D₃	semidirect
17. C₃×C₇×C₃₇	C₂×C₃×C₅	semidirect
19. ΓL(1,𝔽₈)×C₃₇	C₅×D₃	semidirect
21. C₅×C₃⋊AGL(1,𝔽₇)	C₃₇	direct (N × H)
24. C₃×C₅×C₃₇	AGL(1,𝔽₇)	semidirect
28. C₃×C₅×ΓL(1,𝔽₈)	C₂×C₃₇	semidirect
29. C₇×C₃₇	C₃×C₅×D₃	semidirect
32. C₅×D₂₁	C₃×C₃₇	semidirect
34. C₅×C₃₇	C₃⋊AGL(1,𝔽₇)	direct (N × H)
35. C₃⋊AGL(1,𝔽₇)	C₅×C₃₇	direct (N × H)
38. C₃×C₃₇	C₅×AGL(1,𝔽₇)	semidirect
39. C₃×C₅×C₇	C₂×C₃×C₃₇	semidirect
41. C₅×ΓL(1,𝔽₈)	D₃×C₃₇	semidirect
45. C₃×ΓL(1,𝔽₈)	C₂×C₅×C₃₇	semidirect
47. D₂₁	C₃×C₅×C₃₇	semidirect
49. C₃₇	C₅×C₃⋊AGL(1,𝔽₇)	direct (N × H)
50. C₅×C₇	C₃×D₃×C₃₇	semidirect
53. C₃×C₇	C₂×C₃×C₅×C₃₇	semidirect
55. ΓL(1,𝔽₈)	C₅×D₃×C₃₇	semidirect
59. C₃×C₅	C₃₇×AGL(1,𝔽₇)	semidirect
63. C₇	C₃×C₅×D₃×C₃₇	semidirect
66. C₅	C₃₇×C₃⋊AGL(1,𝔽₇)	direct (N × H)
69. C₃	C₅×C₃₇×AGL(1,𝔽₇)	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₅×ΓL(1,𝔽₈)×C₃₇	2
C₅×C₃₇×AGL(1,𝔽₇)	3
C₅×C₃₇×D₂₁	3
C₃₇×C₃⋊AGL(1,𝔽₇)	5
C₃×C₅×D₃×C₃₇	7
C₅×C₃⋊AGL(1,𝔽₇)	37

Subgroups — C5×C37×C3⋊AGL(1,𝔽7) #37552093

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×C₃₇×C₃⋊AGL(1,𝔽₇) #37552093