Subgroups — C₅²×(C₃⋊D₃)⋊D₁₇ #29506022

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×C₅²×D₅₁	C₂	1	2	yes	maximal, complemented
3	C₅²×D₃×D₁₇	—	3	2	no	maximal
4	C₃²×C₅²×C₁₇	C₂²	1	1	yes	characteristic, complemented, Hall
5	C₅×(C₃⋊D₃)⋊D₁₇	C₅	1	6	yes	maximal, direct product factor
6	C₅²×D₅₁	D₃	1	2	yes	complemented
7	C₅²×D₃×C₁₇	—	3	2	no	—
8	C₅²×D₃×C₁₇	—	6	1	no	—
9	C₃×C₅²×D₁₇	—	3	2	no	—
10	C₂×C₅²×D₁₇	—	9	1	no	Hall
11	C₅×C₁₇×C₃⋊D₃	C₂×C₅	1	6	yes	complemented
12	C₃×C₅×D₅₁	C₂×C₅	1	12	yes	complemented
13	C₃×C₅²×C₁₇	—	2	1	no	—
14	C₃×C₅²×C₁₇	C₂×D₃	1	2	yes	complemented
15	C₅×D₃×D₁₇	—	3	12	no	—
16	C₅²×D₃²	—	17	1	no	maximal, Hall
17	C₂×C₅²×C₁₇	—	9	1	no	—
18	C₅²×D₁₇	—	3	2	no	—
19	C₃²×C₅×C₁₇	C₂²×C₅	1	6	yes	complemented
20	(C₃⋊D₃)⋊D₁₇	C₅²	1	1	yes	characteristic, direct product factor, Hall
21	C₅×D₅₁	C₅×D₃	1	12	yes	complemented
22	C₅×D₃×C₁₇	—	3	12	no	—
23	C₅×D₃×C₁₇	—	6	6	no	—
24	C₃×C₅×D₁₇	—	3	12	no	—
25	C₅²×C₃⋊D₃	D₁₇	1	1	yes	characteristic, complemented
26	C₃×C₅²×D₃	—	17	2	no	—
27	C₅²×C₁₇	D₃²	1	1	yes	characteristic, complemented, Hall
28	C₂×C₅×D₁₇	—	9	6	no	—
29	C₁₇×C₃⋊D₃	C₂×C₅²	1	1	yes	characteristic, complemented
30	C₃×D₅₁	C₂×C₅²	1	2	yes	complemented
31	C₂×C₅²×D₃	—	51	2	no	—
32	C₃×C₅×C₁₇	—	2	6	no	—
33	C₃×C₅×C₁₇	C₂×C₅×D₃	1	12	yes	complemented
34	C₃²×C₅²	C₂×D₁₇	1	1	yes	characteristic, complemented, Hall
35	D₃×D₁₇	—	3	2	no	—
36	C₅×D₃²	—	17	6	no	—
37	C₂×C₅×C₁₇	—	9	6	no	—
38	C₅×D₁₇	—	3	12	no	—
39	C₃²×C₁₇	C₂²×C₅²	1	1	yes	characteristic, complemented, Hall
40	C₂×C₃×C₅²	—	51	2	no	—
41	C₅²×D₃	—	3	2	no	—
42	C₅²×D₃	—	6	1	no	—
43	C₅²×D₃	—	17	2	no	—
44	D₅₁	C₅²×D₃	1	2	yes	complemented
45	C₃×D₁₇	—	3	2	no	—
46	D₃×C₁₇	—	3	2	no	—
47	D₃×C₁₇	—	6	1	no	—
48	C₂²×C₅²	—	153	1	no	Hall
49	C₅×C₃⋊D₃	C₅×D₁₇	1	6	yes	complemented
50	C₃×C₅×D₃	—	17	12	no	—
51	C₅×C₁₇	C₅×D₃²	1	6	yes	complemented
52	C₃×C₅²	—	2	1	no	—
53	C₃×C₅²	D₃×D₁₇	1	2	yes	complemented
54	C₂×D₁₇	—	9	1	no	Hall
55	C₂×C₅×D₃	—	51	12	no	—
56	C₃×C₁₇	—	2	1	no	—
57	C₃×C₁₇	C₂×C₅²×D₃	1	2	yes	complemented
58	C₂×C₅²	—	9	1	no	—
59	C₂×C₅²	—	51	2	no	—
60	C₃²×C₅	C₂×C₅×D₁₇	1	6	yes	complemented
61	D₃²	—	17	1	no	Hall
62	C₂×C₁₇	—	9	1	no	—
63	D₁₇	—	3	2	no	—
64	C₂×C₃×C₅	—	51	12	no	—
65	C₅×D₃	—	3	12	no	—
66	C₅×D₃	—	6	6	no	—
67	C₅×D₃	—	17	12	no	—
68	C₅²	(C₃⋊D₃)⋊D₁₇	1	1	yes	characteristic, direct product factor, Sylow
69	C₂²×C₅	—	153	6	no	—
70	C₃⋊D₃	C₅²×D₁₇	1	1	yes	characteristic, complemented
71	C₃×D₃	—	17	2	no	—
72	C₁₇	C₅²×D₃²	1	1	yes	characteristic, complemented, Sylow
73	C₃×C₅	—	2	6	no	—
74	C₃×C₅	C₅×D₃×D₁₇	1	12	yes	complemented
75	C₂×D₃	—	51	2	no	—
76	C₂×C₅	—	9	6	no	—
77	C₂×C₅	—	51	12	no	—
78	C₃²	C₂×C₅²×D₁₇	1	1	yes	characteristic, complemented, Sylow
79	C₂×C₃	—	51	2	no	—
80	D₃	—	3	2	no	—
81	D₃	—	6	1	no	—
82	D₃	—	17	2	no	—
83	C₅	C₅×(C₃⋊D₃)⋊D₁₇	1	6	yes	direct product factor
84	C₂²	—	153	1	no	Sylow
85	C₃	—	2	1	no	—
86	C₃	C₅²×D₃×D₁₇	1	2	yes	complemented
87	C₂	—	9	1	no	—
88	C₂	—	51	2	no	—

	subgroup	G/H
center, hypercenter	C₅²	(C₃⋊D₃)⋊D₁₇
commutator, cosocle, hypocenter	C₃²×C₁₇	C₂²×C₅²
Frattini	C₁	G
Fitting, socle	C₃²×C₅²×C₁₇	C₂²
LCS	G ▷ C₃²×C₁₇
UCS	C₁ ◁ C₅²

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×C₅²×D₅₁	C₂	semidirect
4. C₃²×C₅²×C₁₇	C₂²	semidirect
5. C₅×(C₃⋊D₃)⋊D₁₇	C₅	direct (N × H)
6. C₅²×D₅₁	D₃	semidirect
11. C₅×C₁₇×C₃⋊D₃	C₂×C₅	semidirect
12. C₃×C₅×D₅₁	C₂×C₅	semidirect
14. C₃×C₅²×C₁₇	C₂×D₃	semidirect
19. C₃²×C₅×C₁₇	C₂²×C₅	semidirect
20. (C₃⋊D₃)⋊D₁₇	C₅²	direct (N × H)
21. C₅×D₅₁	C₅×D₃	semidirect
25. C₅²×C₃⋊D₃	D₁₇	semidirect
27. C₅²×C₁₇	D₃²	semidirect
29. C₁₇×C₃⋊D₃	C₂×C₅²	semidirect
30. C₃×D₅₁	C₂×C₅²	semidirect
33. C₃×C₅×C₁₇	C₂×C₅×D₃	semidirect
34. C₃²×C₅²	C₂×D₁₇	semidirect
39. C₃²×C₁₇	C₂²×C₅²	semidirect
44. D₅₁	C₅²×D₃	semidirect
49. C₅×C₃⋊D₃	C₅×D₁₇	semidirect
51. C₅×C₁₇	C₅×D₃²	semidirect
53. C₃×C₅²	D₃×D₁₇	semidirect
57. C₃×C₁₇	C₂×C₅²×D₃	semidirect
60. C₃²×C₅	C₂×C₅×D₁₇	semidirect
68. C₅²	(C₃⋊D₃)⋊D₁₇	direct (N × H)
70. C₃⋊D₃	C₅²×D₁₇	semidirect
72. C₁₇	C₅²×D₃²	semidirect
74. C₃×C₅	C₅×D₃×D₁₇	semidirect
78. C₃²	C₂×C₅²×D₁₇	semidirect
83. C₅	C₅×(C₃⋊D₃)⋊D₁₇	direct (N × H)
86. C₃	C₅²×D₃×D₁₇	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×C₅²×D₅₁	2
C₅²×D₃×D₁₇	3
C₅×(C₃⋊D₃)⋊D₁₇	5
C₅²×D₃²	17

Subgroups — C52×(C3⋊D3)⋊D17 #29506022

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅²×(C₃⋊D₃)⋊D₁₇ #29506022