Subgroups — C₅²×D₃×#3861 #36348025

Order 22050.

720 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅²×#3861	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	C₅²×D₃×C₇²	C₃	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅²×#3861	—	3	1	no	maximal
4	C₅×D₃×#3861	C₅	1	6	yes	maximal, direct product factor
5	C₅²×#3861	—	2	1	no	—
6	C₅²×#3861	D₃	1	1	yes	characteristic, direct product factor
7	C₃×C₅²×C₇²	C₂×C₃	1	1	yes	characteristic, complemented
8	C₅²×D₃×ΓL(1,𝔽₈)	—	7	8	no	maximal
9	C₂×C₅²×C₇²	—	3	1	no	Hall
10	C₃×C₅×#3861	C₂×C₅	1	6	yes	complemented
11	C₃×C₅²×ΓL(1,𝔽₈)	—	7	8	no	—
12	C₅×D₃×C₇²	C₃×C₅	1	6	yes	complemented
13	C₂×C₅×#3861	—	3	6	no	—
14	C₅²×C₇²	C₃×D₃	1	1	yes	characteristic, complemented, Hall
15	C₅²×D₃×C₇	ΓL(1,𝔽₈)	1	8	yes	complemented
16	C₂×C₅²×ΓL(1,𝔽₈)	—	21	8	no	—
17	D₃×#3861	C₅²	1	1	yes	characteristic, direct product factor, Hall
18	C₃×C₅×C₇²	C₂×C₃×C₅	1	6	yes	complemented
19	C₅×#3861	—	2	6	no	—
20	C₅×#3861	C₅×D₃	1	6	yes	direct product factor
21	C₅×D₃×ΓL(1,𝔽₈)	—	7	48	no	—
22	C₃×C₅²×C₇	C₂×ΓL(1,𝔽₈)	1	8	yes	complemented
23	C₅²×ΓL(1,𝔽₈)	—	7	8	no	—
24	C₅²×ΓL(1,𝔽₈)	—	14	8	no	—
25	C₂×C₅×C₇²	—	3	6	no	—
26	C₃×C₅²×D₃	—	49	1	no	Hall
27	C₃×#3861	C₂×C₅²	1	1	yes	characteristic, complemented, Hall
28	C₂×C₅²×C₇	—	3	8	no	—
29	C₃×C₅×ΓL(1,𝔽₈)	—	7	48	no	—
30	D₃×C₇²	C₃×C₅²	1	1	yes	characteristic, complemented
31	C₂×#3861	—	3	1	no	—
32	C₅×C₇²	C₃×C₅×D₃	1	6	yes	complemented
33	C₃²×C₅²	—	49	1	no	Hall
34	C₅×D₃×C₇	C₅×ΓL(1,𝔽₈)	1	48	yes	complemented
35	C₂×C₅×ΓL(1,𝔽₈)	—	21	48	no	—
36	C₅²×C₇	D₃×ΓL(1,𝔽₈)	1	8	yes	complemented
37	C₂×C₃×C₅²	—	147	1	no	—
38	C₅²×D₃	#3861	1	1	yes	characteristic, direct product factor
39	C₃×C₇²	C₂×C₃×C₅²	1	1	yes	characteristic, complemented
40	#3861	—	2	1	no	—
41	#3861	C₅²×D₃	1	1	yes	characteristic, direct product factor
42	D₃×ΓL(1,𝔽₈)	—	7	8	no	—
43	C₃×C₅×C₇	C₂×C₅×ΓL(1,𝔽₈)	1	48	yes	complemented
44	C₅×ΓL(1,𝔽₈)	—	7	48	no	—
45	C₅×ΓL(1,𝔽₈)	—	14	48	no	—
46	C₂×C₇²	—	3	1	no	Hall
47	C₃×C₅×D₃	—	49	6	no	—
48	C₃×C₅²	—	49	1	no	—
49	C₃×C₅²	—	98	1	no	—
50	C₃×C₅²	C₂×#3861	1	1	yes	characteristic, complemented
51	C₂×C₅×C₇	—	3	48	no	—
52	C₃×ΓL(1,𝔽₈)	—	7	8	no	—
53	C₂×C₅²	—	3	1	no	Hall
54	C₇²	C₃×C₅²×D₃	1	1	yes	characteristic, complemented, Sylow
55	C₃²×C₅	—	49	6	no	—
56	D₃×C₇	C₅²×ΓL(1,𝔽₈)	1	8	yes	complemented
57	C₂×ΓL(1,𝔽₈)	—	21	8	no	—
58	C₅×C₇	C₅×D₃×ΓL(1,𝔽₈)	1	48	yes	complemented
59	C₂×C₃×C₅	—	147	6	no	—
60	C₅×D₃	C₅×#3861	1	6	yes	direct product factor
61	C₅²	D₃×#3861	1	1	yes	characteristic, direct product factor, Sylow
62	C₃×C₇	C₂×C₅²×ΓL(1,𝔽₈)	1	8	yes	complemented
63	ΓL(1,𝔽₈)	—	7	8	no	—
64	ΓL(1,𝔽₈)	—	14	8	no	—
65	C₃×D₃	—	49	1	no	Hall
66	C₃×C₅	—	49	6	no	—
67	C₃×C₅	—	98	6	no	—
68	C₃×C₅	C₂×C₅×#3861	1	6	yes	complemented
69	C₂×C₇	—	3	8	no	—
70	C₂×C₅	—	3	6	no	—
71	C₃²	—	49	1	no	Sylow
72	C₇	C₅²×D₃×ΓL(1,𝔽₈)	1	8	yes	complemented
73	C₂×C₃	—	147	1	no	—
74	D₃	C₅²×#3861	1	1	yes	characteristic, direct product factor
75	C₅	C₅×D₃×#3861	1	6	yes	direct product factor
76	C₃	—	49	1	no	—
77	C₃	—	98	1	no	—
78	C₃	C₂×C₅²×#3861	1	1	yes	characteristic, complemented
79	C₂	—	3	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₅²	D₃×#3861
commutator, cosocle, hypocenter	C₃×C₇²	C₂×C₃×C₅²
Frattini	C₁	G
Fitting, socle	C₃×C₅²×C₇²	C₂×C₃
LCS	G ▷ C₃×C₇²
UCS	C₁ ◁ C₅²

Complements

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

N	H ≅ G/N	type
1. C₃×C₅²×#3861	C₂	semidirect
2. C₅²×D₃×C₇²	C₃	semidirect
4. C₅×D₃×#3861	C₅	direct (N × H)
6. C₅²×#3861	D₃	direct (N × H)
7. C₃×C₅²×C₇²	C₂×C₃	semidirect
10. C₃×C₅×#3861	C₂×C₅	semidirect
12. C₅×D₃×C₇²	C₃×C₅	semidirect
14. C₅²×C₇²	C₃×D₃	semidirect
15. C₅²×D₃×C₇	ΓL(1,𝔽₈)	semidirect
17. D₃×#3861	C₅²	direct (N × H)
18. C₃×C₅×C₇²	C₂×C₃×C₅	semidirect
20. C₅×#3861	C₅×D₃	direct (N × H)
22. C₃×C₅²×C₇	C₂×ΓL(1,𝔽₈)	semidirect
27. C₃×#3861	C₂×C₅²	semidirect
30. D₃×C₇²	C₃×C₅²	semidirect
32. C₅×C₇²	C₃×C₅×D₃	semidirect
34. C₅×D₃×C₇	C₅×ΓL(1,𝔽₈)	semidirect
36. C₅²×C₇	D₃×ΓL(1,𝔽₈)	semidirect
38. C₅²×D₃	#3861	direct (N × H)
39. C₃×C₇²	C₂×C₃×C₅²	semidirect
41. #3861	C₅²×D₃	direct (N × H)
43. C₃×C₅×C₇	C₂×C₅×ΓL(1,𝔽₈)	semidirect
50. C₃×C₅²	C₂×#3861	semidirect
54. C₇²	C₃×C₅²×D₃	semidirect
56. D₃×C₇	C₅²×ΓL(1,𝔽₈)	semidirect
58. C₅×C₇	C₅×D₃×ΓL(1,𝔽₈)	semidirect
60. C₅×D₃	C₅×#3861	direct (N × H)
61. C₅²	D₃×#3861	direct (N × H)
62. C₃×C₇	C₂×C₅²×ΓL(1,𝔽₈)	semidirect
68. C₃×C₅	C₂×C₅×#3861	semidirect
72. C₇	C₅²×D₃×ΓL(1,𝔽₈)	semidirect
74. D₃	C₅²×#3861	direct (N × H)
75. C₅	C₅×D₃×#3861	direct (N × H)
78. C₃	C₂×C₅²×#3861	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₅²×#3861	2
C₅²×D₃×C₇²	3
C₂×C₅²×#3861	3
C₅×D₃×#3861	5
C₅²×D₃×ΓL(1,𝔽₈)	7

Subgroups — C52×D3×#3861 #36348025

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅²×D₃×#3861 #36348025