Subgroups — C₃×C₅×C₂₉×C₂²⋊ΓL(1,𝔽₈) #49141671

Order 36540.

112 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×C₂₉×C₂²⋊ΓL(1,𝔽₈)	C₃	1	3	yes	maximal, direct product factor
2	C₂²×C₃×C₅×C₇×C₂₉	C₃	1	1	yes	maximal, characteristic, complemented
3	C₃×C₅×ΓL(1,𝔽₈)×C₂₉	—	4	1	no	maximal, Hall
4	C₃×C₂₉×C₂²⋊ΓL(1,𝔽₈)	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₂×C₃×C₅×C₇×C₂₉	—	3	1	no	—
6	C₃×C₅×A₄×C₂₉	—	7	1	no	maximal, Hall
7	C₂²×C₅×C₇×C₂₉	C₃²	1	1	yes	characteristic, complemented, Hall
8	C₅×ΓL(1,𝔽₈)×C₂₉	—	4	3	no	—
9	C₃×C₅×C₇×C₂₉	A₄	1	1	yes	characteristic, complemented
10	C₂₉×C₂²⋊ΓL(1,𝔽₈)	C₃×C₅	1	3	yes	direct product factor
11	C₂²×C₃×C₇×C₂₉	C₃×C₅	1	1	yes	characteristic, complemented
12	C₂×C₅×C₇×C₂₉	—	3	1	no	—
13	C₃×ΓL(1,𝔽₈)×C₂₉	—	4	1	no	Hall
14	C₂²×C₃×C₅×C₂₉	ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
15	C₅×A₄×C₂₉	—	7	3	no	—
16	C₃²×C₅×C₂₉	—	28	1	no	Hall
17	C₃×C₅×C₂²⋊ΓL(1,𝔽₈)	C₂₉	1	1	yes	maximal, characteristic, direct product factor, Hall
18	C₂×C₃×C₇×C₂₉	—	3	1	no	—
19	C₃×A₄×C₂₉	—	7	1	no	Hall
20	C₅×C₇×C₂₉	C₃×A₄	1	1	yes	characteristic, complemented, Hall
21	C₂×C₃×C₅×C₂₉	—	3	1	no	—
22	C₂²×C₇×C₂₉	C₃²×C₅	1	1	yes	characteristic, complemented, Hall
23	C₃×C₇×C₂₉	C₅×A₄	1	1	yes	characteristic, complemented
24	ΓL(1,𝔽₈)×C₂₉	—	4	3	no	—
25	C₂²×C₅×C₂₉	C₃×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Hall
26	C₃×C₅×C₂₉	—	28	3	no	—
27	C₃×C₅×C₂₉	C₂²⋊ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
28	C₂²×C₃×C₅×C₇	C₃×C₂₉	1	1	yes	characteristic, complemented
29	C₅×C₂²⋊ΓL(1,𝔽₈)	C₃×C₂₉	1	3	yes	direct product factor
30	C₂×C₇×C₂₉	—	3	1	no	—
31	C₂²×C₃×C₂₉	C₅×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
32	A₄×C₂₉	—	7	3	no	—
33	C₃×C₅×ΓL(1,𝔽₈)	—	4	1	no	Hall
34	C₂×C₅×C₂₉	—	3	1	no	—
35	C₃²×C₂₉	—	28	1	no	Hall
36	C₃×C₂²⋊ΓL(1,𝔽₈)	C₅×C₂₉	1	1	yes	characteristic, direct product factor, Hall
37	C₂×C₃×C₅×C₇	—	3	1	no	—
38	C₇×C₂₉	C₃×C₅×A₄	1	1	yes	characteristic, complemented, Hall
39	C₃×C₅×A₄	—	7	1	no	Hall
40	C₂×C₃×C₂₉	—	3	1	no	—
41	C₅×C₂₉	C₃×C₂²⋊ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Hall
42	C₂²×C₅×C₇	C₃²×C₂₉	1	1	yes	characteristic, complemented, Hall
43	C₂²×C₂₉	C₃×C₅×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Hall
44	C₃×C₅×C₇	A₄×C₂₉	1	1	yes	characteristic, complemented
45	C₅×ΓL(1,𝔽₈)	—	4	3	no	—
46	C₃×C₂₉	—	28	3	no	—
47	C₃×C₂₉	C₅×C₂²⋊ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
48	C₂²×C₃×C₇	C₃×C₅×C₂₉	1	1	yes	characteristic, complemented
49	C₂²⋊ΓL(1,𝔽₈)	C₃×C₅×C₂₉	1	3	yes	direct product factor
50	C₂×C₅×C₇	—	3	1	no	—
51	C₃×ΓL(1,𝔽₈)	—	4	1	no	Hall
52	C₂²×C₃×C₅	ΓL(1,𝔽₈)×C₂₉	1	1	yes	characteristic, complemented
53	C₅×A₄	—	7	3	no	—
54	C₂×C₂₉	—	3	1	no	—
55	C₃²×C₅	—	28	1	no	Hall
56	C₂×C₃×C₇	—	3	1	no	—
57	C₃×A₄	—	7	1	no	Hall
58	C₅×C₇	C₃×A₄×C₂₉	1	1	yes	characteristic, complemented, Hall
59	C₂×C₃×C₅	—	3	1	no	—
60	C₂₉	C₃×C₅×C₂²⋊ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Sylow
61	C₂²×C₇	C₃²×C₅×C₂₉	1	1	yes	characteristic, complemented, Hall
62	C₃×C₇	C₅×A₄×C₂₉	1	1	yes	characteristic, complemented
63	ΓL(1,𝔽₈)	—	4	3	no	—
64	C₂²×C₅	C₃×ΓL(1,𝔽₈)×C₂₉	1	1	yes	characteristic, complemented, Hall
65	C₃×C₅	—	28	3	no	—
66	C₃×C₅	C₂₉×C₂²⋊ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
67	C₂×C₇	—	3	1	no	—
68	C₂²×C₃	C₅×ΓL(1,𝔽₈)×C₂₉	1	1	yes	characteristic, complemented
69	A₄	—	7	3	no	—
70	C₂×C₅	—	3	1	no	—
71	C₃²	—	28	1	no	Sylow
72	C₇	C₃×C₅×A₄×C₂₉	1	1	yes	characteristic, complemented, Sylow
73	C₂×C₃	—	3	1	no	—
74	C₅	C₃×C₂₉×C₂²⋊ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Sylow
75	C₂²	C₃×C₅×ΓL(1,𝔽₈)×C₂₉	1	1	yes	characteristic, complemented, Sylow
76	C₃	—	28	3	no	—
77	C₃	C₅×C₂₉×C₂²⋊ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
78	C₂	—	3	1	no	—

	subgroup	G/H
center, hypercenter	C₃×C₅×C₂₉	C₂²⋊ΓL(1,𝔽₈)
commutator, cosocle, hypocenter	C₂²×C₇	C₃²×C₅×C₂₉
Frattini	C₁	G
Fitting, socle	C₂²×C₃×C₅×C₇×C₂₉	C₃
LCS	G ▷ C₂²×C₇
UCS	C₁ ◁ C₃×C₅×C₂₉

Complements

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

N	H ≅ G/N	type
1. C₅×C₂₉×C₂²⋊ΓL(1,𝔽₈)	C₃	direct (N × H)
2. C₂²×C₃×C₅×C₇×C₂₉	C₃	semidirect
4. C₃×C₂₉×C₂²⋊ΓL(1,𝔽₈)	C₅	direct (N × H)
7. C₂²×C₅×C₇×C₂₉	C₃²	semidirect
9. C₃×C₅×C₇×C₂₉	A₄	semidirect
10. C₂₉×C₂²⋊ΓL(1,𝔽₈)	C₃×C₅	direct (N × H)
11. C₂²×C₃×C₇×C₂₉	C₃×C₅	semidirect
14. C₂²×C₃×C₅×C₂₉	ΓL(1,𝔽₈)	semidirect
17. C₃×C₅×C₂²⋊ΓL(1,𝔽₈)	C₂₉	direct (N × H)
20. C₅×C₇×C₂₉	C₃×A₄	semidirect
22. C₂²×C₇×C₂₉	C₃²×C₅	semidirect
23. C₃×C₇×C₂₉	C₅×A₄	semidirect
25. C₂²×C₅×C₂₉	C₃×ΓL(1,𝔽₈)	semidirect
27. C₃×C₅×C₂₉	C₂²⋊ΓL(1,𝔽₈)	direct (N × H)
28. C₂²×C₃×C₅×C₇	C₃×C₂₉	semidirect
29. C₅×C₂²⋊ΓL(1,𝔽₈)	C₃×C₂₉	direct (N × H)
31. C₂²×C₃×C₂₉	C₅×ΓL(1,𝔽₈)	semidirect
36. C₃×C₂²⋊ΓL(1,𝔽₈)	C₅×C₂₉	direct (N × H)
38. C₇×C₂₉	C₃×C₅×A₄	semidirect
41. C₅×C₂₉	C₃×C₂²⋊ΓL(1,𝔽₈)	direct (N × H)
42. C₂²×C₅×C₇	C₃²×C₂₉	semidirect
43. C₂²×C₂₉	C₃×C₅×ΓL(1,𝔽₈)	semidirect
44. C₃×C₅×C₇	A₄×C₂₉	semidirect
47. C₃×C₂₉	C₅×C₂²⋊ΓL(1,𝔽₈)	direct (N × H)
48. C₂²×C₃×C₇	C₃×C₅×C₂₉	semidirect
49. C₂²⋊ΓL(1,𝔽₈)	C₃×C₅×C₂₉	direct (N × H)
52. C₂²×C₃×C₅	ΓL(1,𝔽₈)×C₂₉	semidirect
58. C₅×C₇	C₃×A₄×C₂₉	semidirect
60. C₂₉	C₃×C₅×C₂²⋊ΓL(1,𝔽₈)	direct (N × H)
61. C₂²×C₇	C₃²×C₅×C₂₉	semidirect
62. C₃×C₇	C₅×A₄×C₂₉	semidirect
64. C₂²×C₅	C₃×ΓL(1,𝔽₈)×C₂₉	semidirect
66. C₃×C₅	C₂₉×C₂²⋊ΓL(1,𝔽₈)	direct (N × H)
68. C₂²×C₃	C₅×ΓL(1,𝔽₈)×C₂₉	semidirect
72. C₇	C₃×C₅×A₄×C₂₉	semidirect
74. C₅	C₃×C₂₉×C₂²⋊ΓL(1,𝔽₈)	direct (N × H)
75. C₂²	C₃×C₅×ΓL(1,𝔽₈)×C₂₉	semidirect
77. C₃	C₅×C₂₉×C₂²⋊ΓL(1,𝔽₈)	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅×C₂₉×C₂²⋊ΓL(1,𝔽₈)	3
C₂²×C₃×C₅×C₇×C₂₉	3
C₃×C₅×ΓL(1,𝔽₈)×C₂₉	4
C₃×C₂₉×C₂²⋊ΓL(1,𝔽₈)	5
C₃×C₅×A₄×C₂₉	7
C₃×C₅×C₂²⋊ΓL(1,𝔽₈)	29

Subgroups — C3×C5×C29×C22⋊ΓL(1,𝔽8) #49141671

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×C₂₉×C₂²⋊ΓL(1,𝔽₈) #49141671