Subgroups — C₅×A₄×ΓL(1,𝔽₈)×C₂₉ #49141498

Order 36540.

112 conjugacy classes of subgroups.

Conjugacy classes of subgroups

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

	subgroup	G/H
center, hypercenter	C₅×C₂₉	A₄×ΓL(1,𝔽₈)
commutator, cosocle, hypocenter	C₂²×C₇	C₃²×C₅×C₂₉
Frattini	C₁	G
Fitting, socle	C₂²×C₅×C₇×C₂₉	C₃²
LCS	G ▷ C₂²×C₇
UCS	C₁ ◁ C₅×C₂₉

Complements

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

N	H ≅ G/N	type
1. C₂²×C₅×ΓL(1,𝔽₈)×C₂₉	C₃	semidirect
2. C₅×C₇×A₄×C₂₉	C₃	semidirect
3. C₅×C₂₉×C₂²⋊ΓL(1,𝔽₈)	C₃	semidirect
5. A₄×ΓL(1,𝔽₈)×C₂₉	C₅	direct (N × H)
8. C₂²×C₅×C₇×C₂₉	C₃²	semidirect
10. C₅×ΓL(1,𝔽₈)×C₂₉	A₄	direct (N × H)
12. C₂₉×C₂²⋊ΓL(1,𝔽₈)	C₃×C₅	semidirect
13. C₇×A₄×C₂₉	C₃×C₅	semidirect
14. C₂²×ΓL(1,𝔽₈)×C₂₉	C₃×C₅	semidirect
19. C₅×A₄×C₂₉	ΓL(1,𝔽₈)	direct (N × H)
21. C₅×A₄×ΓL(1,𝔽₈)	C₂₉	direct (N × H)
24. C₅×C₇×C₂₉	C₃×A₄	semidirect
26. C₂²×C₇×C₂₉	C₃²×C₅	semidirect
29. ΓL(1,𝔽₈)×C₂₉	C₅×A₄	direct (N × H)
30. C₂²×C₅×C₂₉	C₃×ΓL(1,𝔽₈)	semidirect
34. C₅×C₂²⋊ΓL(1,𝔽₈)	C₃×C₂₉	semidirect
35. C₅×C₇×A₄	C₃×C₂₉	semidirect
36. C₂²×C₅×ΓL(1,𝔽₈)	C₃×C₂₉	semidirect
40. A₄×C₂₉	C₅×ΓL(1,𝔽₈)	direct (N × H)
44. A₄×ΓL(1,𝔽₈)	C₅×C₂₉	direct (N × H)
46. C₇×C₂₉	C₃×C₅×A₄	semidirect
49. C₅×C₂₉	A₄×ΓL(1,𝔽₈)	direct (N × H)
50. C₂²×C₅×C₇	C₃²×C₂₉	semidirect
51. C₂²×C₂₉	C₃×C₅×ΓL(1,𝔽₈)	semidirect
54. C₅×ΓL(1,𝔽₈)	A₄×C₂₉	direct (N × H)
58. C₂²⋊ΓL(1,𝔽₈)	C₃×C₅×C₂₉	semidirect
59. C₇×A₄	C₃×C₅×C₂₉	semidirect
60. C₂²×ΓL(1,𝔽₈)	C₃×C₅×C₂₉	semidirect
65. C₅×A₄	ΓL(1,𝔽₈)×C₂₉	direct (N × H)
70. C₅×C₇	C₃×A₄×C₂₉	semidirect
72. C₂₉	C₅×A₄×ΓL(1,𝔽₈)	direct (N × H)
73. C₂²×C₇	C₃²×C₅×C₂₉	semidirect
76. ΓL(1,𝔽₈)	C₅×A₄×C₂₉	direct (N × H)
77. C₂²×C₅	C₃×ΓL(1,𝔽₈)×C₂₉	semidirect
84. A₄	C₅×ΓL(1,𝔽₈)×C₂₉	direct (N × H)
87. C₇	C₃×C₅×A₄×C₂₉	semidirect
89. C₅	A₄×ΓL(1,𝔽₈)×C₂₉	direct (N × H)
90. C₂²	C₃×C₅×ΓL(1,𝔽₈)×C₂₉	semidirect

Maximal subgroups

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

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

Subgroups — C5×A4×ΓL(1,𝔽8)×C29 #49141498

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×A₄×ΓL(1,𝔽₈)×C₂₉ #49141498