Subgroups — C₂×C₇×C₁₇×C₃⋊(C₅⋊D₅) #49088059

Order 35700.

320 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×C₁₇×C₃⋊(C₅⋊D₅)	C₂	1	2	yes	maximal, direct product factor
2	C₂×C₃×C₅²×C₇×C₁₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₇×C₁₇×C₅⋊D₅	—	3	1	no	maximal, Hall
4	C₃×C₅²×C₇×C₁₇	C₂²	1	1	yes	characteristic, complemented, Hall
5	C₂×C₇×C₁₇×D₁₅	—	5	6	no	maximal
6	C₇×C₁₇×C₅⋊D₅	—	3	2	no	—
7	C₂×C₅²×C₇×C₁₇	D₃	1	1	yes	characteristic, complemented
8	C₂×C₁₇×C₃⋊(C₅⋊D₅)	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₇×C₁₇×D₁₅	—	5	12	no	—
10	C₂×C₃×C₅×C₇×C₁₇	D₅	1	6	yes	complemented
11	C₅²×C₇×C₁₇	C₂×D₃	1	1	yes	characteristic, complemented, Hall
12	C₁₇×C₃⋊(C₅⋊D₅)	C₂×C₇	1	2	yes	direct product factor
13	C₂×C₃×C₅²×C₁₇	C₂×C₇	1	1	yes	characteristic, complemented
14	C₂×C₇×D₅×C₁₇	—	15	6	no	—
15	C₂×C₇×C₃⋊(C₅⋊D₅)	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Hall
16	C₃×C₅×C₇×C₁₇	C₂×D₅	1	6	yes	complemented
17	C₂×C₁₇×C₅⋊D₅	—	3	1	no	Hall
18	C₂×D₃×C₇×C₁₇	—	25	1	no	Hall
19	C₃×C₅²×C₁₇	C₂²×C₇	1	1	yes	characteristic, complemented, Hall
20	C₂×C₅×C₇×C₁₇	D₁₅	1	6	yes	complemented
21	C₇×D₅×C₁₇	—	15	12	no	—
22	C₂×C₃×C₅²×C₇	C₂×C₁₇	1	1	yes	characteristic, complemented
23	C₇×C₃⋊(C₅⋊D₅)	C₂×C₁₇	1	2	yes	direct product factor
24	C₂×C₁₇×D₁₅	—	5	6	no	—
25	C₂×C₅²×C₁₇	D₃×C₇	1	1	yes	characteristic, complemented
26	C₁₇×C₅⋊D₅	—	3	2	no	—
27	C₂×C₃×C₇×C₁₇	C₅⋊D₅	1	1	yes	characteristic, complemented
28	D₃×C₇×C₁₇	—	25	2	no	—
29	C₂×C₇×C₅⋊D₅	—	3	1	no	Hall
30	C₅×C₇×C₁₇	C₂×D₁₅	1	6	yes	complemented
31	C₃×C₅²×C₇	C₂²×C₁₇	1	1	yes	characteristic, complemented, Hall
32	C₂×C₃×C₅×C₁₇	C₇×D₅	1	6	yes	complemented
33	C₁₇×D₁₅	—	5	12	no	—
34	C₂²×C₇×C₁₇	—	75	1	no	Hall
35	C₅²×C₁₇	C₂×D₃×C₇	1	1	yes	characteristic, complemented, Hall
36	C₂×C₇×D₁₅	—	5	6	no	—
37	C₃×C₇×C₁₇	C₂×C₅⋊D₅	1	1	yes	characteristic, complemented, Hall
38	C₂×C₅²×C₇	D₃×C₁₇	1	1	yes	characteristic, complemented
39	C₇×C₅⋊D₅	—	3	2	no	—
40	C₂×D₅×C₁₇	—	15	6	no	—
41	C₂×C₃⋊(C₅⋊D₅)	C₇×C₁₇	1	1	yes	characteristic, direct product factor, Hall
42	C₃×C₅×C₁₇	C₂×C₇×D₅	1	6	yes	complemented
43	C₂×C₇×C₁₇	—	75	2	no	—
44	C₂×C₇×C₁₇	C₃⋊(C₅⋊D₅)	1	1	yes	characteristic, direct product factor
45	C₂×C₃×C₅×C₇	D₅×C₁₇	1	6	yes	complemented
46	C₇×D₁₅	—	5	12	no	—
47	C₂×D₃×C₁₇	—	25	1	no	Hall
48	C₅²×C₇	C₂×D₃×C₁₇	1	1	yes	characteristic, complemented, Hall
49	C₂×C₅×C₁₇	C₇×D₁₅	1	6	yes	complemented
50	D₅×C₁₇	—	15	12	no	—
51	C₂×C₃×C₅²	C₂×C₇×C₁₇	1	1	yes	characteristic, complemented
52	C₃⋊(C₅⋊D₅)	C₂×C₇×C₁₇	1	2	yes	direct product factor
53	C₂×C₇×D₅	—	15	6	no	—
54	C₇×C₁₇	C₂×C₃⋊(C₅⋊D₅)	1	1	yes	characteristic, direct product factor, Hall
55	C₃×C₅×C₇	C₂×D₅×C₁₇	1	6	yes	complemented
56	C₂×C₃×C₁₇	C₇×C₅⋊D₅	1	1	yes	characteristic, complemented
57	D₃×C₁₇	—	25	2	no	—
58	C₂×C₅⋊D₅	—	3	1	no	Hall
59	C₅×C₁₇	C₂×C₇×D₁₅	1	6	yes	complemented
60	C₂×D₃×C₇	—	25	1	no	Hall
61	C₃×C₅²	C₂²×C₇×C₁₇	1	1	yes	characteristic, complemented, Hall
62	C₂×C₅×C₇	C₁₇×D₁₅	1	6	yes	complemented
63	C₇×D₅	—	15	12	no	—
64	C₂²×C₁₇	—	75	1	no	Hall
65	C₂×D₁₅	—	5	6	no	—
66	C₃×C₁₇	C₂×C₇×C₅⋊D₅	1	1	yes	characteristic, complemented, Hall
67	C₂×C₅²	D₃×C₇×C₁₇	1	1	yes	characteristic, complemented
68	C₅⋊D₅	—	3	2	no	—
69	C₂×C₃×C₇	C₁₇×C₅⋊D₅	1	1	yes	characteristic, complemented
70	D₃×C₇	—	25	2	no	—
71	C₅×C₇	C₂×C₁₇×D₁₅	1	6	yes	complemented
72	C₂×C₁₇	—	75	2	no	—
73	C₂×C₁₇	C₇×C₃⋊(C₅⋊D₅)	1	1	yes	characteristic, direct product factor
74	C₂×C₃×C₅	C₇×D₅×C₁₇	1	6	yes	complemented
75	D₁₅	—	5	12	no	—
76	C₂²×C₇	—	75	1	no	Hall
77	C₅²	C₂×D₃×C₇×C₁₇	1	1	yes	characteristic, complemented, Sylow
78	C₃×C₇	C₂×C₁₇×C₅⋊D₅	1	1	yes	characteristic, complemented, Hall
79	C₂×D₅	—	15	6	no	—
80	C₁₇	C₂×C₇×C₃⋊(C₅⋊D₅)	1	1	yes	characteristic, direct product factor, Sylow
81	C₃×C₅	C₂×C₇×D₅×C₁₇	1	6	yes	complemented
82	C₂×C₇	—	75	2	no	—
83	C₂×C₇	C₁₇×C₃⋊(C₅⋊D₅)	1	1	yes	characteristic, direct product factor
84	C₂×D₃	—	25	1	no	Hall
85	C₂×C₅	C₇×C₁₇×D₁₅	1	6	yes	complemented
86	D₅	—	15	12	no	—
87	C₇	C₂×C₁₇×C₃⋊(C₅⋊D₅)	1	1	yes	characteristic, direct product factor, Sylow
88	C₂×C₃	C₇×C₁₇×C₅⋊D₅	1	1	yes	characteristic, complemented
89	D₃	—	25	2	no	—
90	C₅	C₂×C₇×C₁₇×D₁₅	1	6	yes	complemented
91	C₂²	—	75	1	no	Sylow
92	C₃	C₂×C₇×C₁₇×C₅⋊D₅	1	1	yes	characteristic, complemented, Sylow
93	C₂	—	75	2	no	—
94	C₂	C₇×C₁₇×C₃⋊(C₅⋊D₅)	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂×C₇×C₁₇	C₃⋊(C₅⋊D₅)
commutator, cosocle, hypocenter	C₃×C₅²	C₂²×C₇×C₁₇
Frattini	C₁	G
Fitting, socle	C₂×C₃×C₅²×C₇×C₁₇	C₂
LCS	G ▷ C₃×C₅²
UCS	C₁ ◁ C₂×C₇×C₁₇

Complements

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

N	H ≅ G/N	type
1. C₇×C₁₇×C₃⋊(C₅⋊D₅)	C₂	direct (N × H)
2. C₂×C₃×C₅²×C₇×C₁₇	C₂	semidirect
4. C₃×C₅²×C₇×C₁₇	C₂²	semidirect
7. C₂×C₅²×C₇×C₁₇	D₃	semidirect
8. C₂×C₁₇×C₃⋊(C₅⋊D₅)	C₇	direct (N × H)
10. C₂×C₃×C₅×C₇×C₁₇	D₅	semidirect
11. C₅²×C₇×C₁₇	C₂×D₃	semidirect
12. C₁₇×C₃⋊(C₅⋊D₅)	C₂×C₇	direct (N × H)
13. C₂×C₃×C₅²×C₁₇	C₂×C₇	semidirect
15. C₂×C₇×C₃⋊(C₅⋊D₅)	C₁₇	direct (N × H)
16. C₃×C₅×C₇×C₁₇	C₂×D₅	semidirect
19. C₃×C₅²×C₁₇	C₂²×C₇	semidirect
20. C₂×C₅×C₇×C₁₇	D₁₅	semidirect
22. C₂×C₃×C₅²×C₇	C₂×C₁₇	semidirect
23. C₇×C₃⋊(C₅⋊D₅)	C₂×C₁₇	direct (N × H)
25. C₂×C₅²×C₁₇	D₃×C₇	semidirect
27. C₂×C₃×C₇×C₁₇	C₅⋊D₅	semidirect
30. C₅×C₇×C₁₇	C₂×D₁₅	semidirect
31. C₃×C₅²×C₇	C₂²×C₁₇	semidirect
32. C₂×C₃×C₅×C₁₇	C₇×D₅	semidirect
35. C₅²×C₁₇	C₂×D₃×C₇	semidirect
37. C₃×C₇×C₁₇	C₂×C₅⋊D₅	semidirect
38. C₂×C₅²×C₇	D₃×C₁₇	semidirect
41. C₂×C₃⋊(C₅⋊D₅)	C₇×C₁₇	direct (N × H)
42. C₃×C₅×C₁₇	C₂×C₇×D₅	semidirect
44. C₂×C₇×C₁₇	C₃⋊(C₅⋊D₅)	direct (N × H)
45. C₂×C₃×C₅×C₇	D₅×C₁₇	semidirect
48. C₅²×C₇	C₂×D₃×C₁₇	semidirect
49. C₂×C₅×C₁₇	C₇×D₁₅	semidirect
51. C₂×C₃×C₅²	C₂×C₇×C₁₇	semidirect
52. C₃⋊(C₅⋊D₅)	C₂×C₇×C₁₇	direct (N × H)
54. C₇×C₁₇	C₂×C₃⋊(C₅⋊D₅)	direct (N × H)
55. C₃×C₅×C₇	C₂×D₅×C₁₇	semidirect
56. C₂×C₃×C₁₇	C₇×C₅⋊D₅	semidirect
59. C₅×C₁₇	C₂×C₇×D₁₅	semidirect
61. C₃×C₅²	C₂²×C₇×C₁₇	semidirect
62. C₂×C₅×C₇	C₁₇×D₁₅	semidirect
66. C₃×C₁₇	C₂×C₇×C₅⋊D₅	semidirect
67. C₂×C₅²	D₃×C₇×C₁₇	semidirect
69. C₂×C₃×C₇	C₁₇×C₅⋊D₅	semidirect
71. C₅×C₇	C₂×C₁₇×D₁₅	semidirect
73. C₂×C₁₇	C₇×C₃⋊(C₅⋊D₅)	direct (N × H)
74. C₂×C₃×C₅	C₇×D₅×C₁₇	semidirect
77. C₅²	C₂×D₃×C₇×C₁₇	semidirect
78. C₃×C₇	C₂×C₁₇×C₅⋊D₅	semidirect
80. C₁₇	C₂×C₇×C₃⋊(C₅⋊D₅)	direct (N × H)
81. C₃×C₅	C₂×C₇×D₅×C₁₇	semidirect
83. C₂×C₇	C₁₇×C₃⋊(C₅⋊D₅)	direct (N × H)
85. C₂×C₅	C₇×C₁₇×D₁₅	semidirect
87. C₇	C₂×C₁₇×C₃⋊(C₅⋊D₅)	direct (N × H)
88. C₂×C₃	C₇×C₁₇×C₅⋊D₅	semidirect
90. C₅	C₂×C₇×C₁₇×D₁₅	semidirect
92. C₃	C₂×C₇×C₁₇×C₅⋊D₅	semidirect
94. C₂	C₇×C₁₇×C₃⋊(C₅⋊D₅)	direct (N × H)

Maximal subgroups

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

subgroup	index
C₇×C₁₇×C₃⋊(C₅⋊D₅)	2
C₂×C₃×C₅²×C₇×C₁₇	2
C₂×C₇×C₁₇×C₅⋊D₅	3
C₂×C₇×C₁₇×D₁₅	5
C₂×C₁₇×C₃⋊(C₅⋊D₅)	7
C₂×C₇×C₃⋊(C₅⋊D₅)	17

Subgroups — C2×C7×C17×C3⋊(C5⋊D5) #49088059

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇×C₁₇×C₃⋊(C₅⋊D₅) #49088059