Subgroups — C₂×C₅²×C₇×D₅₁ #49088052

Order 35700.

320 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅²×C₇×D₅₁	C₂	1	2	yes	maximal, direct product factor
2	C₂×C₃×C₅²×C₇×C₁₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅²×C₇×D₁₇	—	3	1	no	maximal, Hall
4	C₃×C₅²×C₇×C₁₇	C₂²	1	1	yes	characteristic, complemented, Hall
5	C₂×C₅×C₇×D₅₁	C₅	1	6	yes	maximal, direct product factor
6	C₅²×C₇×D₁₇	—	3	2	no	—
7	C₂×C₅²×C₇×C₁₇	D₃	1	1	yes	characteristic, complemented
8	C₂×C₅²×D₅₁	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₅×C₇×D₅₁	C₂×C₅	1	12	yes	direct product factor
10	C₂×C₃×C₅×C₇×C₁₇	C₂×C₅	1	6	yes	complemented
11	C₅²×C₇×C₁₇	C₂×D₃	1	1	yes	characteristic, complemented, Hall
12	C₅²×D₅₁	C₂×C₇	1	2	yes	direct product factor
13	C₂×C₃×C₅²×C₁₇	C₂×C₇	1	1	yes	characteristic, complemented
14	C₂×C₅×C₇×D₁₇	—	3	6	no	—
15	C₂×C₅²×D₃×C₇	—	17	1	no	maximal, Hall
16	C₃×C₅×C₇×C₁₇	C₂²×C₅	1	6	yes	complemented
17	C₂×C₅²×D₁₇	—	3	1	no	Hall
18	C₂×C₇×D₅₁	C₅²	1	1	yes	characteristic, direct product factor, Hall
19	C₃×C₅²×C₁₇	C₂²×C₇	1	1	yes	characteristic, complemented, Hall
20	C₂×C₅×C₇×C₁₇	C₅×D₃	1	6	yes	complemented
21	C₅×C₇×D₁₇	—	3	12	no	—
22	C₂×C₃×C₅²×C₇	D₁₇	1	1	yes	characteristic, complemented
23	C₅²×D₃×C₇	—	17	2	no	—
24	C₂×C₅×D₅₁	C₅×C₇	1	6	yes	direct product factor
25	C₂×C₅²×C₁₇	D₃×C₇	1	1	yes	characteristic, complemented
26	C₅²×D₁₇	—	3	2	no	—
27	C₂×C₃×C₇×C₁₇	C₂×C₅²	1	1	yes	characteristic, complemented
28	C₇×D₅₁	C₂×C₅²	1	2	yes	direct product factor
29	C₂²×C₅²×C₇	—	51	1	no	Hall
30	C₅×C₇×C₁₇	C₂×C₅×D₃	1	6	yes	complemented
31	C₃×C₅²×C₇	C₂×D₁₇	1	1	yes	characteristic, complemented, Hall
32	C₂×C₃×C₅×C₁₇	C₂×C₅×C₇	1	6	yes	complemented
33	C₅×D₅₁	C₂×C₅×C₇	1	12	yes	direct product factor
34	C₂×C₇×D₁₇	—	3	1	no	Hall
35	C₅²×C₁₇	C₂×D₃×C₇	1	1	yes	characteristic, complemented, Hall
36	C₂×C₅×D₃×C₇	—	17	6	no	—
37	C₃×C₇×C₁₇	C₂²×C₅²	1	1	yes	characteristic, complemented, Hall
38	C₂×C₅²×C₇	—	51	2	no	—
39	C₂×C₅²×C₇	D₅₁	1	1	yes	characteristic, direct product factor
40	C₂×C₅×D₁₇	—	3	6	no	—
41	C₂×C₅²×D₃	—	17	1	no	Hall
42	C₃×C₅×C₁₇	C₂²×C₅×C₇	1	6	yes	complemented
43	C₂×C₇×C₁₇	C₅²×D₃	1	1	yes	characteristic, complemented
44	C₇×D₁₇	—	3	2	no	—
45	C₂×C₃×C₅×C₇	C₅×D₁₇	1	6	yes	complemented
46	C₅×D₃×C₇	—	17	12	no	—
47	C₂×D₅₁	C₅²×C₇	1	1	yes	characteristic, direct product factor, Hall
48	C₅²×C₇	C₂×D₅₁	1	1	yes	characteristic, direct product factor, Hall
49	C₂×C₅×C₁₇	C₅×D₃×C₇	1	6	yes	complemented
50	C₅×D₁₇	—	3	12	no	—
51	C₂×C₃×C₅²	C₇×D₁₇	1	1	yes	characteristic, complemented
52	C₅²×D₃	—	17	2	no	—
53	C₂²×C₅×C₇	—	51	6	no	—
54	C₇×C₁₇	C₂×C₅²×D₃	1	1	yes	characteristic, complemented, Hall
55	C₃×C₅×C₇	C₂×C₅×D₁₇	1	6	yes	complemented
56	C₂×C₃×C₁₇	C₂×C₅²×C₇	1	1	yes	characteristic, complemented
57	D₅₁	C₂×C₅²×C₇	1	2	yes	direct product factor
58	C₂²×C₅²	—	51	1	no	Hall
59	C₅×C₁₇	C₂×C₅×D₃×C₇	1	6	yes	complemented
60	C₂×D₃×C₇	—	17	1	no	Hall
61	C₃×C₅²	C₂×C₇×D₁₇	1	1	yes	characteristic, complemented, Hall
62	C₂×C₅×C₇	—	51	12	no	—
63	C₂×C₅×C₇	C₅×D₅₁	1	6	yes	direct product factor
64	C₂×D₁₇	—	3	1	no	Hall
65	C₂×C₅×D₃	—	17	6	no	—
66	C₃×C₁₇	C₂²×C₅²×C₇	1	1	yes	characteristic, complemented, Hall
67	C₂×C₅²	—	51	2	no	—
68	C₂×C₅²	C₇×D₅₁	1	1	yes	characteristic, direct product factor
69	C₂×C₃×C₇	C₅²×D₁₇	1	1	yes	characteristic, complemented
70	D₃×C₇	—	17	2	no	—
71	C₅×C₇	C₂×C₅×D₅₁	1	6	yes	direct product factor
72	C₂×C₁₇	C₅²×D₃×C₇	1	1	yes	characteristic, complemented
73	D₁₇	—	3	2	no	—
74	C₂×C₃×C₅	C₅×C₇×D₁₇	1	6	yes	complemented
75	C₅×D₃	—	17	12	no	—
76	C₂²×C₇	—	51	1	no	Hall
77	C₅²	C₂×C₇×D₅₁	1	1	yes	characteristic, direct product factor, Sylow
78	C₃×C₇	C₂×C₅²×D₁₇	1	1	yes	characteristic, complemented, Hall
79	C₂²×C₅	—	51	6	no	—
80	C₁₇	C₂×C₅²×D₃×C₇	1	1	yes	characteristic, complemented, Sylow
81	C₃×C₅	C₂×C₅×C₇×D₁₇	1	6	yes	complemented
82	C₂×C₇	—	51	2	no	—
83	C₂×C₇	C₅²×D₅₁	1	1	yes	characteristic, direct product factor
84	C₂×D₃	—	17	1	no	Hall
85	C₂×C₅	—	51	12	no	—
86	C₂×C₅	C₅×C₇×D₅₁	1	6	yes	direct product factor
87	C₇	C₂×C₅²×D₅₁	1	1	yes	characteristic, direct product factor, Sylow
88	C₂×C₃	C₅²×C₇×D₁₇	1	1	yes	characteristic, complemented
89	D₃	—	17	2	no	—
90	C₅	C₂×C₅×C₇×D₅₁	1	6	yes	direct product factor
91	C₂²	—	51	1	no	Sylow
92	C₃	C₂×C₅²×C₇×D₁₇	1	1	yes	characteristic, complemented, Sylow
93	C₂	—	51	2	no	—
94	C₂	C₅²×C₇×D₅₁	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂×C₅²×C₇	D₅₁
commutator, cosocle, hypocenter	C₃×C₁₇	C₂²×C₅²×C₇
Frattini	C₁	G
Fitting, socle	C₂×C₃×C₅²×C₇×C₁₇	C₂
LCS	G ▷ C₃×C₁₇
UCS	C₁ ◁ C₂×C₅²×C₇

Complements

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

N	H ≅ G/N	type
1. C₅²×C₇×D₅₁	C₂	direct (N × H)
2. C₂×C₃×C₅²×C₇×C₁₇	C₂	semidirect
4. C₃×C₅²×C₇×C₁₇	C₂²	semidirect
5. C₂×C₅×C₇×D₅₁	C₅	direct (N × H)
7. C₂×C₅²×C₇×C₁₇	D₃	semidirect
8. C₂×C₅²×D₅₁	C₇	direct (N × H)
9. C₅×C₇×D₅₁	C₂×C₅	direct (N × H)
10. C₂×C₃×C₅×C₇×C₁₇	C₂×C₅	semidirect
11. C₅²×C₇×C₁₇	C₂×D₃	semidirect
12. C₅²×D₅₁	C₂×C₇	direct (N × H)
13. C₂×C₃×C₅²×C₁₇	C₂×C₇	semidirect
16. C₃×C₅×C₇×C₁₇	C₂²×C₅	semidirect
18. C₂×C₇×D₅₁	C₅²	direct (N × H)
19. C₃×C₅²×C₁₇	C₂²×C₇	semidirect
20. C₂×C₅×C₇×C₁₇	C₅×D₃	semidirect
22. C₂×C₃×C₅²×C₇	D₁₇	semidirect
24. C₂×C₅×D₅₁	C₅×C₇	direct (N × H)
25. C₂×C₅²×C₁₇	D₃×C₇	semidirect
27. C₂×C₃×C₇×C₁₇	C₂×C₅²	semidirect
28. C₇×D₅₁	C₂×C₅²	direct (N × H)
30. C₅×C₇×C₁₇	C₂×C₅×D₃	semidirect
31. C₃×C₅²×C₇	C₂×D₁₇	semidirect
32. C₂×C₃×C₅×C₁₇	C₂×C₅×C₇	semidirect
33. C₅×D₅₁	C₂×C₅×C₇	direct (N × H)
35. C₅²×C₁₇	C₂×D₃×C₇	semidirect
37. C₃×C₇×C₁₇	C₂²×C₅²	semidirect
39. C₂×C₅²×C₇	D₅₁	direct (N × H)
42. C₃×C₅×C₁₇	C₂²×C₅×C₇	semidirect
43. C₂×C₇×C₁₇	C₅²×D₃	semidirect
45. C₂×C₃×C₅×C₇	C₅×D₁₇	semidirect
47. C₂×D₅₁	C₅²×C₇	direct (N × H)
48. C₅²×C₇	C₂×D₅₁	direct (N × H)
49. C₂×C₅×C₁₇	C₅×D₃×C₇	semidirect
51. C₂×C₃×C₅²	C₇×D₁₇	semidirect
54. C₇×C₁₇	C₂×C₅²×D₃	semidirect
55. C₃×C₅×C₇	C₂×C₅×D₁₇	semidirect
56. C₂×C₃×C₁₇	C₂×C₅²×C₇	semidirect
57. D₅₁	C₂×C₅²×C₇	direct (N × H)
59. C₅×C₁₇	C₂×C₅×D₃×C₇	semidirect
61. C₃×C₅²	C₂×C₇×D₁₇	semidirect
63. C₂×C₅×C₇	C₅×D₅₁	direct (N × H)
66. C₃×C₁₇	C₂²×C₅²×C₇	semidirect
68. C₂×C₅²	C₇×D₅₁	direct (N × H)
69. C₂×C₃×C₇	C₅²×D₁₇	semidirect
71. C₅×C₇	C₂×C₅×D₅₁	direct (N × H)
72. C₂×C₁₇	C₅²×D₃×C₇	semidirect
74. C₂×C₃×C₅	C₅×C₇×D₁₇	semidirect
77. C₅²	C₂×C₇×D₅₁	direct (N × H)
78. C₃×C₇	C₂×C₅²×D₁₇	semidirect
80. C₁₇	C₂×C₅²×D₃×C₇	semidirect
81. C₃×C₅	C₂×C₅×C₇×D₁₇	semidirect
83. C₂×C₇	C₅²×D₅₁	direct (N × H)
86. C₂×C₅	C₅×C₇×D₅₁	direct (N × H)
87. C₇	C₂×C₅²×D₅₁	direct (N × H)
88. C₂×C₃	C₅²×C₇×D₁₇	semidirect
90. C₅	C₂×C₅×C₇×D₅₁	direct (N × H)
92. C₃	C₂×C₅²×C₇×D₁₇	semidirect
94. C₂	C₅²×C₇×D₅₁	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅²×C₇×D₅₁	2
C₂×C₃×C₅²×C₇×C₁₇	2
C₂×C₅²×C₇×D₁₇	3
C₂×C₅×C₇×D₅₁	5
C₂×C₅²×D₅₁	7
C₂×C₅²×D₃×C₇	17

Subgroups — C2×C52×C7×D51 #49088052

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅²×C₇×D₅₁ #49088052