Subgroups — C₁₇×(C₃⋊D₃)⋊(C₅⋊D₅) #29506034

Order 15300.

352 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅²×C₁₇×C₃⋊D₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×C₁₇×C₃⋊(C₅⋊D₅)	C₂	1	2	yes	maximal, complemented
3	D₃×C₁₇×C₅⋊D₅	—	3	2	no	maximal
4	C₃²×C₅²×C₁₇	C₂²	1	1	yes	characteristic, complemented, Hall
5	C₁₇×(C₃⋊D₃)⋊D₅	—	5	6	no	maximal
6	C₁₇×C₃⋊(C₅⋊D₅)	D₃	1	2	yes	complemented
7	C₅²×D₃×C₁₇	—	3	2	no	—
8	C₅²×D₃×C₁₇	—	6	1	no	—
9	C₃×C₁₇×C₅⋊D₅	—	3	2	no	—
10	C₂×C₁₇×C₅⋊D₅	—	9	1	no	Hall
11	C₅×C₁₇×C₃⋊D₃	D₅	1	6	yes	complemented
12	C₃×C₁₇×D₁₅	—	5	12	no	—
13	C₃×C₅²×C₁₇	—	2	1	no	—
14	C₃×C₅²×C₁₇	C₂×D₃	1	2	yes	complemented
15	D₃×D₅×C₁₇	—	15	12	no	—
16	(C₃⋊D₃)⋊(C₅⋊D₅)	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Hall
17	C₂×C₅²×C₁₇	—	9	1	no	—
18	C₁₇×C₅⋊D₅	—	3	2	no	—
19	C₃²×C₅×C₁₇	C₂×D₅	1	6	yes	complemented
20	D₃²×C₁₇	—	25	1	no	Hall
21	C₁₇×D₁₅	—	5	12	no	—
22	C₅×D₃×C₁₇	—	3	12	no	—
23	C₅×D₃×C₁₇	—	6	6	no	—
24	C₃×D₅×C₁₇	—	15	12	no	—
25	C₅²×C₃⋊D₃	C₂×C₁₇	1	1	yes	characteristic, complemented
26	C₃×C₃⋊(C₅⋊D₅)	C₂×C₁₇	1	2	yes	complemented
27	C₅²×C₁₇	D₃²	1	1	yes	characteristic, complemented, Hall
28	C₂×D₅×C₁₇	—	45	6	no	—
29	C₁₇×C₃⋊D₃	C₅⋊D₅	1	1	yes	characteristic, complemented
30	C₃×D₃×C₁₇	—	25	2	no	—
31	D₃×C₅⋊D₅	—	3	2	no	—
32	C₃×C₅×C₁₇	—	2	6	no	—
33	C₃×C₅×C₁₇	D₃×D₅	1	12	yes	complemented
34	C₃²×C₅²	C₂²×C₁₇	1	1	yes	characteristic, complemented, Hall
35	C₂×D₃×C₁₇	—	75	2	no	—
36	(C₃⋊D₃)⋊D₅	—	5	6	no	—
37	C₂×C₅×C₁₇	—	9	6	no	—
38	D₅×C₁₇	—	15	12	no	—
39	C₃²×C₁₇	C₂×C₅⋊D₅	1	1	yes	characteristic, complemented, Hall
40	C₃⋊(C₅⋊D₅)	D₃×C₁₇	1	2	yes	complemented
41	C₅²×D₃	—	3	2	no	—
42	C₅²×D₃	—	6	1	no	—
43	C₃×C₅⋊D₅	—	3	2	no	—
44	C₂×C₃×C₁₇	—	75	2	no	—
45	D₃×C₁₇	—	3	2	no	—
46	D₃×C₁₇	—	6	1	no	—
47	D₃×C₁₇	—	25	2	no	—
48	C₂×C₅⋊D₅	—	9	1	no	Hall
49	C₅×C₃⋊D₃	D₅×C₁₇	1	6	yes	complemented
50	C₃×D₁₅	—	5	12	no	—
51	C₅×C₁₇	(C₃⋊D₃)⋊D₅	1	6	yes	complemented
52	C₃×C₅²	—	2	1	no	—
53	C₃×C₅²	C₂×D₃×C₁₇	1	2	yes	complemented
54	C₂²×C₁₇	—	225	1	no	Hall
55	D₃×D₅	—	15	12	no	—
56	C₃×C₁₇	—	2	1	no	—
57	C₃×C₁₇	D₃×C₅⋊D₅	1	2	yes	complemented
58	C₂×C₅²	—	9	1	no	—
59	C₅⋊D₅	—	3	2	no	—
60	C₃²×C₅	C₂×D₅×C₁₇	1	6	yes	complemented
61	D₃²	—	25	1	no	Hall
62	C₂×C₁₇	—	9	1	no	—
63	C₂×C₁₇	—	75	2	no	—
64	D₁₅	—	5	12	no	—
65	C₃×D₅	—	15	12	no	—
66	C₅×D₃	—	3	12	no	—
67	C₅×D₃	—	6	6	no	—
68	C₅²	D₃²×C₁₇	1	1	yes	characteristic, complemented, Sylow
69	C₂×D₅	—	45	6	no	—
70	C₃⋊D₃	C₁₇×C₅⋊D₅	1	1	yes	characteristic, complemented
71	C₃×D₃	—	25	2	no	—
72	C₁₇	(C₃⋊D₃)⋊(C₅⋊D₅)	1	1	yes	characteristic, direct product factor, Sylow
73	C₃×C₅	—	2	6	no	—
74	C₃×C₅	D₃×D₅×C₁₇	1	12	yes	complemented
75	C₂×D₃	—	75	2	no	—
76	C₂×C₅	—	9	6	no	—
77	D₅	—	15	12	no	—
78	C₃²	C₂×C₁₇×C₅⋊D₅	1	1	yes	characteristic, complemented, Sylow
79	C₂×C₃	—	75	2	no	—
80	D₃	—	3	2	no	—
81	D₃	—	6	1	no	—
82	D₃	—	25	2	no	—
83	C₅	C₁₇×(C₃⋊D₃)⋊D₅	1	6	yes	complemented
84	C₂²	—	225	1	no	Sylow
85	C₃	—	2	1	no	—
86	C₃	D₃×C₁₇×C₅⋊D₅	1	2	yes	complemented
87	C₂	—	9	1	no	—
88	C₂	—	75	2	no	—

	subgroup	G/H
center, hypercenter	C₁₇	(C₃⋊D₃)⋊(C₅⋊D₅)
commutator, cosocle, hypocenter	C₃²×C₅²	C₂²×C₁₇
Frattini	C₁	G
Fitting, socle	C₃²×C₅²×C₁₇	C₂²
LCS	G ▷ C₃²×C₅²
UCS	C₁ ◁ C₁₇

Complements

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

N	H ≅ G/N	type
1. C₅²×C₁₇×C₃⋊D₃	C₂	semidirect
2. C₃×C₁₇×C₃⋊(C₅⋊D₅)	C₂	semidirect
4. C₃²×C₅²×C₁₇	C₂²	semidirect
6. C₁₇×C₃⋊(C₅⋊D₅)	D₃	semidirect
11. C₅×C₁₇×C₃⋊D₃	D₅	semidirect
14. C₃×C₅²×C₁₇	C₂×D₃	semidirect
16. (C₃⋊D₃)⋊(C₅⋊D₅)	C₁₇	direct (N × H)
19. C₃²×C₅×C₁₇	C₂×D₅	semidirect
25. C₅²×C₃⋊D₃	C₂×C₁₇	semidirect
26. C₃×C₃⋊(C₅⋊D₅)	C₂×C₁₇	semidirect
27. C₅²×C₁₇	D₃²	semidirect
29. C₁₇×C₃⋊D₃	C₅⋊D₅	semidirect
33. C₃×C₅×C₁₇	D₃×D₅	semidirect
34. C₃²×C₅²	C₂²×C₁₇	semidirect
39. C₃²×C₁₇	C₂×C₅⋊D₅	semidirect
40. C₃⋊(C₅⋊D₅)	D₃×C₁₇	semidirect
49. C₅×C₃⋊D₃	D₅×C₁₇	semidirect
51. C₅×C₁₇	(C₃⋊D₃)⋊D₅	semidirect
53. C₃×C₅²	C₂×D₃×C₁₇	semidirect
57. C₃×C₁₇	D₃×C₅⋊D₅	semidirect
60. C₃²×C₅	C₂×D₅×C₁₇	semidirect
68. C₅²	D₃²×C₁₇	semidirect
70. C₃⋊D₃	C₁₇×C₅⋊D₅	semidirect
72. C₁₇	(C₃⋊D₃)⋊(C₅⋊D₅)	direct (N × H)
74. C₃×C₅	D₃×D₅×C₁₇	semidirect
78. C₃²	C₂×C₁₇×C₅⋊D₅	semidirect
83. C₅	C₁₇×(C₃⋊D₃)⋊D₅	semidirect
86. C₃	D₃×C₁₇×C₅⋊D₅	semidirect

Maximal subgroups

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

subgroup	index
C₅²×C₁₇×C₃⋊D₃	2
C₃×C₁₇×C₃⋊(C₅⋊D₅)	2
D₃×C₁₇×C₅⋊D₅	3
C₁₇×(C₃⋊D₃)⋊D₅	5
(C₃⋊D₃)⋊(C₅⋊D₅)	17

Subgroups — C17×(C3⋊D3)⋊(C5⋊D5) #29506034

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₇×(C₃⋊D₃)⋊(C₅⋊D₅) #29506034