Subgroups — C₂×C₃×C₅²×C₇×D₁₇ #49088040

Order 35700.

320 conjugacy classes of subgroups.

Conjugacy classes of subgroups

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

	subgroup	G/H
center, hypercenter	C₂×C₃×C₅²×C₇	D₁₇
commutator, cosocle, hypocenter	C₁₇	C₂²×C₃×C₅²×C₇
Frattini	C₁	G
Fitting, socle	C₂×C₃×C₅²×C₇×C₁₇	C₂
LCS	G ▷ C₁₇
UCS	C₁ ◁ C₂×C₃×C₅²×C₇

Complements

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

N	H ≅ G/N	type
1. C₃×C₅²×C₇×D₁₇	C₂	direct (N × H)
2. C₂×C₃×C₅²×C₇×C₁₇	C₂	semidirect
3. C₂×C₅²×C₇×D₁₇	C₃	direct (N × H)
4. C₃×C₅²×C₇×C₁₇	C₂²	semidirect
5. C₂×C₃×C₅×C₇×D₁₇	C₅	direct (N × H)
6. C₅²×C₇×D₁₇	C₂×C₃	direct (N × H)
7. C₂×C₅²×C₇×C₁₇	C₂×C₃	semidirect
8. C₂×C₃×C₅²×D₁₇	C₇	direct (N × H)
9. C₃×C₅×C₇×D₁₇	C₂×C₅	direct (N × H)
10. C₂×C₃×C₅×C₇×C₁₇	C₂×C₅	semidirect
11. C₅²×C₇×C₁₇	C₂²×C₃	semidirect
12. C₃×C₅²×D₁₇	C₂×C₇	direct (N × H)
13. C₂×C₃×C₅²×C₁₇	C₂×C₇	semidirect
14. C₂×C₅×C₇×D₁₇	C₃×C₅	direct (N × H)
16. C₃×C₅×C₇×C₁₇	C₂²×C₅	semidirect
17. C₂×C₅²×D₁₇	C₃×C₇	direct (N × H)
18. C₂×C₃×C₇×D₁₇	C₅²	direct (N × H)
19. C₃×C₅²×C₁₇	C₂²×C₇	semidirect
20. C₂×C₅×C₇×C₁₇	C₂×C₃×C₅	semidirect
21. C₅×C₇×D₁₇	C₂×C₃×C₅	direct (N × H)
23. C₂×C₃×C₅²×C₇	D₁₇	direct (N × H)
24. C₂×C₃×C₅×D₁₇	C₅×C₇	direct (N × H)
25. C₂×C₅²×C₁₇	C₂×C₃×C₇	semidirect
26. C₅²×D₁₇	C₂×C₃×C₇	direct (N × H)
27. C₂×C₃×C₇×C₁₇	C₂×C₅²	semidirect
28. C₃×C₇×D₁₇	C₂×C₅²	direct (N × H)
30. C₅×C₇×C₁₇	C₂²×C₃×C₅	semidirect
31. C₃×C₅²×C₇	C₂×D₁₇	direct (N × H)
32. C₂×C₃×C₅×C₁₇	C₂×C₅×C₇	semidirect
33. C₃×C₅×D₁₇	C₂×C₅×C₇	direct (N × H)
34. C₂×C₇×D₁₇	C₃×C₅²	direct (N × H)
35. C₅²×C₁₇	C₂²×C₃×C₇	semidirect
37. C₃×C₇×C₁₇	C₂²×C₅²	semidirect
39. C₂×C₅²×C₇	C₃×D₁₇	direct (N × H)
40. C₂×C₅×D₁₇	C₃×C₅×C₇	direct (N × H)
42. C₃×C₅×C₁₇	C₂²×C₅×C₇	semidirect
43. C₂×C₇×C₁₇	C₂×C₃×C₅²	semidirect
44. C₇×D₁₇	C₂×C₃×C₅²	direct (N × H)
46. C₂×C₃×C₅×C₇	C₅×D₁₇	direct (N × H)
47. C₂×C₃×D₁₇	C₅²×C₇	direct (N × H)
48. C₅²×C₇	C₂×C₃×D₁₇	direct (N × H)
49. C₂×C₅×C₁₇	C₂×C₃×C₅×C₇	semidirect
50. C₅×D₁₇	C₂×C₃×C₅×C₇	direct (N × H)
52. C₂×C₃×C₅²	C₇×D₁₇	direct (N × H)
54. C₇×C₁₇	C₂²×C₃×C₅²	semidirect
55. C₃×C₅×C₇	C₂×C₅×D₁₇	direct (N × H)
56. C₂×C₃×C₁₇	C₂×C₅²×C₇	semidirect
57. C₃×D₁₇	C₂×C₅²×C₇	direct (N × H)
59. C₅×C₁₇	C₂²×C₃×C₅×C₇	semidirect
61. C₃×C₅²	C₂×C₇×D₁₇	direct (N × H)
63. C₂×C₅×C₇	C₃×C₅×D₁₇	direct (N × H)
64. C₂×D₁₇	C₃×C₅²×C₇	direct (N × H)
66. C₃×C₁₇	C₂²×C₅²×C₇	semidirect
68. C₂×C₅²	C₃×C₇×D₁₇	direct (N × H)
70. C₂×C₃×C₇	C₅²×D₁₇	direct (N × H)
71. C₅×C₇	C₂×C₃×C₅×D₁₇	direct (N × H)
72. C₂×C₁₇	C₂×C₃×C₅²×C₇	semidirect
73. D₁₇	C₂×C₃×C₅²×C₇	direct (N × H)
75. C₂×C₃×C₅	C₅×C₇×D₁₇	direct (N × H)
77. C₅²	C₂×C₃×C₇×D₁₇	direct (N × H)
78. C₃×C₇	C₂×C₅²×D₁₇	direct (N × H)
80. C₁₇	C₂²×C₃×C₅²×C₇	semidirect
81. C₃×C₅	C₂×C₅×C₇×D₁₇	direct (N × H)
83. C₂×C₇	C₃×C₅²×D₁₇	direct (N × H)
86. C₂×C₅	C₃×C₅×C₇×D₁₇	direct (N × H)
87. C₇	C₂×C₃×C₅²×D₁₇	direct (N × H)
89. C₂×C₃	C₅²×C₇×D₁₇	direct (N × H)
90. C₅	C₂×C₃×C₅×C₇×D₁₇	direct (N × H)
92. C₃	C₂×C₅²×C₇×D₁₇	direct (N × H)
94. C₂	C₃×C₅²×C₇×D₁₇	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×C₅²×C₇×D₁₇	2
C₂×C₃×C₅²×C₇×C₁₇	2
C₂×C₅²×C₇×D₁₇	3
C₂×C₃×C₅×C₇×D₁₇	5
C₂×C₃×C₅²×D₁₇	7
C₂²×C₃×C₅²×C₇	17

Subgroups — C2×C3×C52×C7×D17 #49088040

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×C₅²×C₇×D₁₇ #49088040