Subgroups — C₁₇×((C₅⋊D₅)⋊C₃)⋊D₇ #49087771

Order 35700.

120 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×C₁₇×(C₅⋊D₅)⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₁₇×C₇⋊#3874	C₂	1	2	yes	maximal, complemented
3	C₁₇×(C₅⋊D₅)⋊D₇	—	3	1	no	maximal, Hall
4	C₇×C₁₇×C₅²⋊C₃	C₂²	1	1	yes	characteristic, complemented, Hall
5	C₇×C₁₇×C₅⋊D₅	D₃	1	1	yes	characteristic, complemented
6	C₅×C₁₇×D₃₅	—	3	2	no	—
7	C₁₇×#3874⋊C₂	—	7	1	no	maximal, Hall
8	C₅²×C₇×C₁₇	C₂×D₃	1	1	yes	characteristic, complemented, Hall
9	C₁₇×(C₅⋊D₅)⋊C₃	D₇	1	1	yes	characteristic, complemented
10	C₁₇×#3874	—	7	2	no	—
11	D₅×D₇×C₁₇	—	15	2	no	—
12	((C₅⋊D₅)⋊C₃)⋊D₇	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Hall
13	D₅²×C₁₇	—	21	1	no	Hall
14	C₂×C₁₇×D₂₁	—	25	1	no	maximal, Hall
15	C₁₇×C₅²⋊C₃	C₂×D₇	1	1	yes	characteristic, complemented, Hall
16	C₁₇×D₃₅	—	3	2	no	—
17	C₇×D₅×C₁₇	—	15	2	no	—
18	C₅×D₇×C₁₇	—	15	2	no	—
19	C₇×(C₅⋊D₅)⋊C₃	C₂×C₁₇	1	1	yes	characteristic, complemented
20	C₇⋊#3874	C₂×C₁₇	1	2	yes	complemented
21	C₁₇×C₅⋊D₅	D₂₁	1	1	yes	characteristic, complemented
22	C₅×D₅×C₁₇	—	21	2	no	—
23	C₂×C₃×C₇×C₁₇	—	25	1	no	—
24	C₁₇×D₂₁	—	25	2	no	—
25	(C₅⋊D₅)⋊D₇	—	3	1	no	Hall
26	C₅×C₇×C₁₇	—	3	2	no	—
27	C₇×C₅²⋊C₃	C₂²×C₁₇	1	1	yes	characteristic, complemented, Hall
28	C₂×D₇×C₁₇	—	75	1	no	Hall
29	C₅²×C₁₇	C₂×D₂₁	1	1	yes	characteristic, complemented, Hall
30	C₃×C₇×C₁₇	—	25	1	no	Hall
31	C₇×C₅⋊D₅	D₃×C₁₇	1	1	yes	characteristic, complemented
32	C₅×D₃₅	—	3	2	no	—
33	C₂×D₅×C₁₇	—	105	2	no	—
34	#3874⋊C₂	—	7	1	no	Hall
35	C₂×C₇×C₁₇	—	25	1	no	—
36	D₇×C₁₇	—	15	2	no	—
37	C₂×D₃×C₁₇	—	175	1	no	Hall
38	C₅²×C₇	C₂×D₃×C₁₇	1	1	yes	characteristic, complemented, Hall
39	C₂×C₅×C₁₇	—	105	2	no	—
40	D₅×C₁₇	—	15	2	no	—
41	D₅×C₁₇	—	21	2	no	—
42	(C₅⋊D₅)⋊C₃	D₇×C₁₇	1	1	yes	characteristic, complemented
43	#3874	—	7	2	no	—
44	D₅×D₇	—	15	2	no	—
45	C₇×C₁₇	#3874⋊C₂	1	1	yes	characteristic, complemented, Hall
46	C₂×C₃×C₁₇	—	25	1	no	—
47	D₃×C₁₇	—	175	2	no	—
48	D₅²	—	21	1	no	Hall
49	C₅×C₁₇	—	3	2	no	—
50	C₂×D₂₁	—	25	1	no	Hall
51	C₅²⋊C₃	C₂×D₇×C₁₇	1	1	yes	characteristic, complemented, Hall
52	D₃₅	—	3	2	no	—
53	C₅×D₇	—	15	2	no	—
54	C₇×D₅	—	15	2	no	—
55	C₂²×C₁₇	—	525	1	no	Hall
56	C₃×C₁₇	—	25	1	no	Hall
57	C₅⋊D₅	C₁₇×D₂₁	1	1	yes	characteristic, complemented
58	C₅×D₅	—	21	2	no	—
59	C₂×C₃×C₇	—	25	1	no	—
60	D₂₁	—	25	2	no	—
61	C₅×C₇	—	3	2	no	—
62	C₂×C₁₇	—	25	1	no	—
63	C₂×C₁₇	—	105	2	no	—
64	C₂×D₇	—	75	1	no	Hall
65	C₅²	C₂×C₁₇×D₂₁	1	1	yes	characteristic, complemented, Sylow
66	C₃×C₇	—	25	1	no	Hall
67	C₂×D₅	—	105	2	no	—
68	C₁₇	((C₅⋊D₅)⋊C₃)⋊D₇	1	1	yes	characteristic, direct product factor, Sylow
69	C₂×C₇	—	25	1	no	—
70	D₇	—	15	2	no	—
71	C₂×D₃	—	175	1	no	Hall
72	C₂×C₅	—	105	2	no	—
73	D₅	—	15	2	no	—
74	D₅	—	21	2	no	—
75	C₇	C₁₇×#3874⋊C₂	1	1	yes	characteristic, complemented, Sylow
76	C₂×C₃	—	25	1	no	—
77	D₃	—	175	2	no	—
78	C₅	—	3	2	no	—
79	C₂²	—	525	1	no	Sylow
80	C₃	—	25	1	no	Sylow
81	C₂	—	25	1	no	—
82	C₂	—	105	2	no	—

	subgroup	G/H
center, hypercenter	C₁₇	((C₅⋊D₅)⋊C₃)⋊D₇
commutator, cosocle, hypocenter	C₇×C₅²⋊C₃	C₂²×C₁₇
Frattini	C₁	G
Fitting, socle	C₅²×C₇×C₁₇	C₂×D₃
LCS	G ▷ C₇×C₅²⋊C₃
UCS	C₁ ◁ C₁₇

Complements

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

N	H ≅ G/N	type
1. C₇×C₁₇×(C₅⋊D₅)⋊C₃	C₂	semidirect
2. C₁₇×C₇⋊#3874	C₂	semidirect
4. C₇×C₁₇×C₅²⋊C₃	C₂²	semidirect
5. C₇×C₁₇×C₅⋊D₅	D₃	semidirect
8. C₅²×C₇×C₁₇	C₂×D₃	semidirect
9. C₁₇×(C₅⋊D₅)⋊C₃	D₇	semidirect
12. ((C₅⋊D₅)⋊C₃)⋊D₇	C₁₇	direct (N × H)
15. C₁₇×C₅²⋊C₃	C₂×D₇	semidirect
19. C₇×(C₅⋊D₅)⋊C₃	C₂×C₁₇	semidirect
20. C₇⋊#3874	C₂×C₁₇	semidirect
21. C₁₇×C₅⋊D₅	D₂₁	semidirect
27. C₇×C₅²⋊C₃	C₂²×C₁₇	semidirect
29. C₅²×C₁₇	C₂×D₂₁	semidirect
31. C₇×C₅⋊D₅	D₃×C₁₇	semidirect
38. C₅²×C₇	C₂×D₃×C₁₇	semidirect
42. (C₅⋊D₅)⋊C₃	D₇×C₁₇	semidirect
45. C₇×C₁₇	#3874⋊C₂	semidirect
51. C₅²⋊C₃	C₂×D₇×C₁₇	semidirect
57. C₅⋊D₅	C₁₇×D₂₁	semidirect
65. C₅²	C₂×C₁₇×D₂₁	semidirect
68. C₁₇	((C₅⋊D₅)⋊C₃)⋊D₇	direct (N × H)
75. C₇	C₁₇×#3874⋊C₂	semidirect

Maximal subgroups

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

subgroup	index
C₇×C₁₇×(C₅⋊D₅)⋊C₃	2
C₁₇×C₇⋊#3874	2
C₁₇×(C₅⋊D₅)⋊D₇	3
C₁₇×#3874⋊C₂	7
((C₅⋊D₅)⋊C₃)⋊D₇	17
C₂×C₁₇×D₂₁	25

Subgroups — C17×((C5⋊D5)⋊C3)⋊D7 #49087771

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₇×((C₅⋊D₅)⋊C₃)⋊D₇ #49087771