Subgroups — C₃×C₅²×D₃×C₇² #36348068

Order 22050.

720 conjugacy classes of subgroups.

Conjugacy classes of subgroups

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

	subgroup	G/H
center, hypercenter	C₃×C₅²×C₇²	D₃
commutator, cosocle, hypocenter	C₃	C₂×C₃×C₅²×C₇²
Frattini	C₁	G
Fitting, socle	C₃²×C₅²×C₇²	C₂
LCS	G ▷ C₃
UCS	C₁ ◁ C₃×C₅²×C₇²

Complements

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

N	H ≅ G/N	type
1. C₃²×C₅²×C₇²	C₂	semidirect
2. C₅²×D₃×C₇²	C₃	direct (N × H)
4. C₃×C₅×D₃×C₇²	C₅	direct (N × H)
6. C₃×C₅²×C₇²	D₃	direct (N × H)
7. C₃×C₅²×C₇²	C₂×C₃	semidirect
8. C₃×C₅²×D₃×C₇	C₇	direct (N × H)
10. C₃²×C₅×C₇²	C₂×C₅	semidirect
11. C₃²×C₅²×C₇	C₂×C₇	semidirect
13. C₅×D₃×C₇²	C₃×C₅	direct (N × H)
14. C₅²×C₇²	C₃×D₃	direct (N × H)
16. C₅²×D₃×C₇	C₃×C₇	direct (N × H)
17. C₃×D₃×C₇²	C₅²	direct (N × H)
19. C₃×C₅×C₇²	C₅×D₃	direct (N × H)
20. C₃×C₅×C₇²	C₂×C₃×C₅	semidirect
21. C₃×C₅×D₃×C₇	C₅×C₇	direct (N × H)
23. C₃×C₅²×C₇	D₃×C₇	direct (N × H)
24. C₃×C₅²×C₇	C₂×C₃×C₇	semidirect
26. C₃×C₅²×D₃	C₇²	direct (N × H)
27. C₃²×C₇²	C₂×C₅²	semidirect
29. C₃²×C₅×C₇	C₂×C₅×C₇	semidirect
31. D₃×C₇²	C₃×C₅²	direct (N × H)
32. C₅×C₇²	C₃×C₅×D₃	direct (N × H)
33. C₃²×C₅²	C₂×C₇²	semidirect
35. C₅×D₃×C₇	C₃×C₅×C₇	direct (N × H)
36. C₅²×C₇	C₃×D₃×C₇	direct (N × H)
38. C₅²×D₃	C₃×C₇²	direct (N × H)
40. C₃×C₇²	C₅²×D₃	direct (N × H)
41. C₃×C₇²	C₂×C₃×C₅²	semidirect
42. C₃×D₃×C₇	C₅²×C₇	direct (N × H)
44. C₃×C₅×C₇	C₅×D₃×C₇	direct (N × H)
45. C₃×C₅×C₇	C₂×C₃×C₅×C₇	semidirect
47. C₃×C₅×D₃	C₅×C₇²	direct (N × H)
49. C₃×C₅²	D₃×C₇²	direct (N × H)
50. C₃×C₅²	C₂×C₃×C₇²	semidirect
52. C₃²×C₇	C₂×C₅²×C₇	semidirect
54. C₇²	C₃×C₅²×D₃	direct (N × H)
55. C₃²×C₅	C₂×C₅×C₇²	semidirect
57. D₃×C₇	C₃×C₅²×C₇	direct (N × H)
58. C₅×C₇	C₃×C₅×D₃×C₇	direct (N × H)
60. C₅×D₃	C₃×C₅×C₇²	direct (N × H)
61. C₅²	C₃×D₃×C₇²	direct (N × H)
63. C₃×C₇	C₅²×D₃×C₇	direct (N × H)
64. C₃×C₇	C₂×C₃×C₅²×C₇	semidirect
65. C₃×D₃	C₅²×C₇²	direct (N × H)
67. C₃×C₅	C₅×D₃×C₇²	direct (N × H)
68. C₃×C₅	C₂×C₃×C₅×C₇²	semidirect
71. C₃²	C₂×C₅²×C₇²	semidirect
72. C₇	C₃×C₅²×D₃×C₇	direct (N × H)
74. D₃	C₃×C₅²×C₇²	direct (N × H)
75. C₅	C₃×C₅×D₃×C₇²	direct (N × H)
77. C₃	C₂×C₃×C₅²×C₇²	semidirect
78. C₃	C₅²×D₃×C₇²	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃²×C₅²×C₇²	2
C₅²×D₃×C₇²	3
C₂×C₃×C₅²×C₇²	3
C₃×C₅×D₃×C₇²	5
C₃×C₅²×D₃×C₇	7

Subgroups — C3×C52×D3×C72 #36348068

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅²×D₃×C₇² #36348068