Subgroups — C₂×C₃×C₅×ΓL(1,𝔽₈)×C₄₇ #44444104

Order 29610.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×ΓL(1,𝔽₈)×C₄₇	C₂	1	1	yes	maximal, characteristic, direct product factor, Hall
2	C₂×C₅×ΓL(1,𝔽₈)×C₄₇	C₃	1	3	yes	maximal, direct product factor
3	C₂×C₃×C₅×C₇×C₄₇	C₃	1	1	yes	maximal, characteristic, complemented
4	C₂×C₃×ΓL(1,𝔽₈)×C₄₇	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₅×ΓL(1,𝔽₈)×C₄₇	C₂×C₃	1	3	yes	direct product factor
6	C₃×C₅×C₇×C₄₇	C₂×C₃	1	1	yes	characteristic, complemented
7	C₂×C₃²×C₅×C₄₇	—	7	1	no	maximal, Hall
8	C₂×C₅×C₇×C₄₇	C₃²	1	1	yes	characteristic, complemented, Hall
9	C₃×ΓL(1,𝔽₈)×C₄₇	C₂×C₅	1	1	yes	characteristic, direct product factor, Hall
10	C₃²×C₅×C₄₇	—	7	1	no	Hall
11	C₂×C₃×C₇×C₄₇	C₃×C₅	1	1	yes	characteristic, complemented
12	C₂×ΓL(1,𝔽₈)×C₄₇	C₃×C₅	1	3	yes	direct product factor
13	C₅×C₇×C₄₇	C₂×C₃²	1	1	yes	characteristic, complemented, Hall
14	C₂×C₃×C₅×C₄₇	—	7	3	no	—
15	C₂×C₃×C₅×C₄₇	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
16	C₃×C₇×C₄₇	C₂×C₃×C₅	1	1	yes	characteristic, complemented
17	ΓL(1,𝔽₈)×C₄₇	C₂×C₃×C₅	1	3	yes	direct product factor
18	C₂×C₃²×C₄₇	—	7	1	no	Hall
19	C₃×C₅×C₄₇	—	7	3	no	—
20	C₃×C₅×C₄₇	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
21	C₂×C₇×C₄₇	C₃²×C₅	1	1	yes	characteristic, complemented, Hall
22	C₂×C₃×C₅×ΓL(1,𝔽₈)	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Hall
23	C₂×C₅×C₄₇	C₃×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Hall
24	C₃²×C₄₇	—	7	1	no	Hall
25	C₇×C₄₇	C₂×C₃²×C₅	1	1	yes	characteristic, complemented, Hall
26	C₃×C₅×ΓL(1,𝔽₈)	C₂×C₄₇	1	1	yes	characteristic, direct product factor, Hall
27	C₂×C₃×C₄₇	—	7	3	no	—
28	C₂×C₃×C₄₇	C₅×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
29	C₅×C₄₇	C₂×C₃×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Hall
30	C₂×C₃×C₅×C₇	C₃×C₄₇	1	1	yes	characteristic, complemented
31	C₂×C₅×ΓL(1,𝔽₈)	C₃×C₄₇	1	3	yes	direct product factor
32	C₃×C₄₇	—	7	3	no	—
33	C₃×C₄₇	C₂×C₅×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
34	C₂×C₃×ΓL(1,𝔽₈)	C₅×C₄₇	1	1	yes	characteristic, direct product factor, Hall
35	C₃×C₅×C₇	C₂×C₃×C₄₇	1	1	yes	characteristic, complemented
36	C₅×ΓL(1,𝔽₈)	C₂×C₃×C₄₇	1	3	yes	direct product factor
37	C₂×C₄₇	C₃×C₅×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Hall
38	C₂×C₃²×C₅	—	7	1	no	Hall
39	C₂×C₅×C₇	C₃²×C₄₇	1	1	yes	characteristic, complemented, Hall
40	C₃×ΓL(1,𝔽₈)	C₂×C₅×C₄₇	1	1	yes	characteristic, direct product factor, Hall
41	C₄₇	C₂×C₃×C₅×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Sylow
42	C₃²×C₅	—	7	1	no	Hall
43	C₂×C₃×C₇	C₃×C₅×C₄₇	1	1	yes	characteristic, complemented
44	C₂×ΓL(1,𝔽₈)	C₃×C₅×C₄₇	1	3	yes	direct product factor
45	C₅×C₇	C₂×C₃²×C₄₇	1	1	yes	characteristic, complemented, Hall
46	C₂×C₃×C₅	—	7	3	no	—
47	C₂×C₃×C₅	ΓL(1,𝔽₈)×C₄₇	1	1	yes	characteristic, direct product factor
48	C₃×C₇	C₂×C₃×C₅×C₄₇	1	1	yes	characteristic, complemented
49	ΓL(1,𝔽₈)	C₂×C₃×C₅×C₄₇	1	3	yes	direct product factor
50	C₂×C₃²	—	7	1	no	Hall
51	C₃×C₅	—	7	3	no	—
52	C₃×C₅	C₂×ΓL(1,𝔽₈)×C₄₇	1	1	yes	characteristic, direct product factor
53	C₂×C₇	C₃²×C₅×C₄₇	1	1	yes	characteristic, complemented, Hall
54	C₂×C₅	C₃×ΓL(1,𝔽₈)×C₄₇	1	1	yes	characteristic, direct product factor, Hall
55	C₃²	—	7	1	no	Sylow
56	C₇	C₂×C₃²×C₅×C₄₇	1	1	yes	characteristic, complemented, Sylow
57	C₂×C₃	—	7	3	no	—
58	C₂×C₃	C₅×ΓL(1,𝔽₈)×C₄₇	1	1	yes	characteristic, direct product factor
59	C₅	C₂×C₃×ΓL(1,𝔽₈)×C₄₇	1	1	yes	characteristic, direct product factor, Sylow
60	C₃	—	7	3	no	—
61	C₃	C₂×C₅×ΓL(1,𝔽₈)×C₄₇	1	1	yes	characteristic, direct product factor
62	C₂	C₃×C₅×ΓL(1,𝔽₈)×C₄₇	1	1	yes	characteristic, direct product factor, Sylow

	subgroup	G/H
center, hypercenter	C₂×C₃×C₅×C₄₇	ΓL(1,𝔽₈)
commutator, cosocle, hypocenter	C₇	C₂×C₃²×C₅×C₄₇
Frattini	C₁	G
Fitting, socle	C₂×C₃×C₅×C₇×C₄₇	C₃
LCS	G ▷ C₇
UCS	C₁ ◁ C₂×C₃×C₅×C₄₇

Complements

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

N	H ≅ G/N	type
1. C₃×C₅×ΓL(1,𝔽₈)×C₄₇	C₂	direct (N × H)
2. C₂×C₅×ΓL(1,𝔽₈)×C₄₇	C₃	direct (N × H)
3. C₂×C₃×C₅×C₇×C₄₇	C₃	semidirect
4. C₂×C₃×ΓL(1,𝔽₈)×C₄₇	C₅	direct (N × H)
5. C₅×ΓL(1,𝔽₈)×C₄₇	C₂×C₃	direct (N × H)
6. C₃×C₅×C₇×C₄₇	C₂×C₃	semidirect
8. C₂×C₅×C₇×C₄₇	C₃²	semidirect
9. C₃×ΓL(1,𝔽₈)×C₄₇	C₂×C₅	direct (N × H)
11. C₂×C₃×C₇×C₄₇	C₃×C₅	semidirect
12. C₂×ΓL(1,𝔽₈)×C₄₇	C₃×C₅	direct (N × H)
13. C₅×C₇×C₄₇	C₂×C₃²	semidirect
15. C₂×C₃×C₅×C₄₇	ΓL(1,𝔽₈)	direct (N × H)
16. C₃×C₇×C₄₇	C₂×C₃×C₅	semidirect
17. ΓL(1,𝔽₈)×C₄₇	C₂×C₃×C₅	direct (N × H)
20. C₃×C₅×C₄₇	C₂×ΓL(1,𝔽₈)	direct (N × H)
21. C₂×C₇×C₄₇	C₃²×C₅	semidirect
22. C₂×C₃×C₅×ΓL(1,𝔽₈)	C₄₇	direct (N × H)
23. C₂×C₅×C₄₇	C₃×ΓL(1,𝔽₈)	direct (N × H)
25. C₇×C₄₇	C₂×C₃²×C₅	semidirect
26. C₃×C₅×ΓL(1,𝔽₈)	C₂×C₄₇	direct (N × H)
28. C₂×C₃×C₄₇	C₅×ΓL(1,𝔽₈)	direct (N × H)
29. C₅×C₄₇	C₂×C₃×ΓL(1,𝔽₈)	direct (N × H)
30. C₂×C₃×C₅×C₇	C₃×C₄₇	semidirect
31. C₂×C₅×ΓL(1,𝔽₈)	C₃×C₄₇	direct (N × H)
33. C₃×C₄₇	C₂×C₅×ΓL(1,𝔽₈)	direct (N × H)
34. C₂×C₃×ΓL(1,𝔽₈)	C₅×C₄₇	direct (N × H)
35. C₃×C₅×C₇	C₂×C₃×C₄₇	semidirect
36. C₅×ΓL(1,𝔽₈)	C₂×C₃×C₄₇	direct (N × H)
37. C₂×C₄₇	C₃×C₅×ΓL(1,𝔽₈)	direct (N × H)
39. C₂×C₅×C₇	C₃²×C₄₇	semidirect
40. C₃×ΓL(1,𝔽₈)	C₂×C₅×C₄₇	direct (N × H)
41. C₄₇	C₂×C₃×C₅×ΓL(1,𝔽₈)	direct (N × H)
43. C₂×C₃×C₇	C₃×C₅×C₄₇	semidirect
44. C₂×ΓL(1,𝔽₈)	C₃×C₅×C₄₇	direct (N × H)
45. C₅×C₇	C₂×C₃²×C₄₇	semidirect
47. C₂×C₃×C₅	ΓL(1,𝔽₈)×C₄₇	direct (N × H)
48. C₃×C₇	C₂×C₃×C₅×C₄₇	semidirect
49. ΓL(1,𝔽₈)	C₂×C₃×C₅×C₄₇	direct (N × H)
52. C₃×C₅	C₂×ΓL(1,𝔽₈)×C₄₇	direct (N × H)
53. C₂×C₇	C₃²×C₅×C₄₇	semidirect
54. C₂×C₅	C₃×ΓL(1,𝔽₈)×C₄₇	direct (N × H)
56. C₇	C₂×C₃²×C₅×C₄₇	semidirect
58. C₂×C₃	C₅×ΓL(1,𝔽₈)×C₄₇	direct (N × H)
59. C₅	C₂×C₃×ΓL(1,𝔽₈)×C₄₇	direct (N × H)
61. C₃	C₂×C₅×ΓL(1,𝔽₈)×C₄₇	direct (N × H)
62. C₂	C₃×C₅×ΓL(1,𝔽₈)×C₄₇	direct (N × H)

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)×C₄₇	2
C₂×C₅×ΓL(1,𝔽₈)×C₄₇	3
C₂×C₃×C₅×C₇×C₄₇	3
C₂×C₃×ΓL(1,𝔽₈)×C₄₇	5
C₂×C₃²×C₅×C₄₇	7
C₂×C₃×C₅×ΓL(1,𝔽₈)	47

Subgroups — C2×C3×C5×ΓL(1,𝔽8)×C47 #44444104

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×C₅×ΓL(1,𝔽₈)×C₄₇ #44444104