Subgroups — D₅₁⋊D₁₅₉ #44570013

Order 32436.

88 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅₃×D₅₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×D₂₇₀₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₇×C₃⋊D₁₅₉	C₂	1	1	yes	maximal, characteristic, complemented
4	D₅₁⋊D₅₃	—	3	1	no	maximal
5	D₁₇×D₁₅₉	—	3	1	no	maximal
6	C₃²×C₁₇×C₅₃	C₂²	1	1	yes	characteristic, complemented, Hall
7	D₂₇₀₃	D₃	1	1	yes	characteristic, complemented
8	C₅₃×D₅₁	D₃	1	1	yes	characteristic, complemented
9	C₁₇×D₁₅₉	—	3	1	no	—
10	C₁₇×D₁₅₉	—	3	1	no	—
11	C₁₇×D₁₅₉	—	6	1	no	—
12	C₃×D₉₀₁	—	3	1	no	—
13	C₃×D₁₇×C₅₃	—	3	1	no	—
14	D₁₇×D₅₃	—	9	1	no	Hall
15	C₃×C₁₇×C₅₃	—	2	1	no	—
16	C₃×C₁₇×C₅₃	C₂×D₃	1	1	yes	characteristic, complemented
17	C₃×C₁₇×C₅₃	C₂×D₃	1	1	yes	characteristic, complemented
18	D₃×D₁₅₉	—	17	1	no	maximal, Hall
19	D₉₀₁	—	3	1	no	—
20	C₁₇×D₅₃	—	9	1	no	—
21	D₁₇×C₅₃	—	3	1	no	—
22	C₃⋊D₁₅₉	D₁₇	1	1	yes	characteristic, complemented
23	C₃×D₁₅₉	—	17	1	no	—
24	C₃×D₃×C₅₃	—	17	1	no	—
25	C₁₇×C₅₃	D₃²	1	1	yes	characteristic, complemented, Hall
26	C₂×D₁₅₉	—	51	1	no	—
27	D₃×D₅₃	—	51	1	no	—
28	(C₃⋊D₃)⋊D₁₇	—	53	1	no	maximal, Hall
29	C₃²×C₅₃	C₂×D₁₇	1	1	yes	characteristic, complemented, Hall
30	C₂×C₃×C₅₃	—	51	1	no	—
31	D₁₅₉	—	3	1	no	—
32	D₁₅₉	—	3	1	no	—
33	D₁₅₉	—	6	1	no	—
34	D₁₅₉	—	17	1	no	—
35	C₃×D₅₃	—	51	1	no	—
36	D₃×C₅₃	—	17	1	no	—
37	C₁₇×C₃⋊D₃	—	53	1	no	—
38	C₃×D₅₁	—	53	1	no	—
39	C₃×D₅₁	D₅₃	1	1	yes	characteristic, complemented
40	C₂×D₅₃	—	153	1	no	Hall
41	D₃×D₁₇	—	159	1	no	—
42	D₃×D₁₇	—	159	1	no	—
43	C₃×C₅₃	—	2	1	no	—
44	C₃×C₅₃	D₃×D₁₇	1	1	yes	characteristic, complemented
45	C₃×C₅₃	D₃×D₁₇	1	1	yes	characteristic, complemented
46	C₃²×C₁₇	C₂×D₅₃	1	1	yes	characteristic, complemented, Hall
47	C₂×C₅₃	—	51	1	no	—
48	D₅₃	—	9	1	no	—
49	D₅₃	—	51	1	no	—
50	D₅₁	—	53	1	no	—
51	D₅₁	D₁₅₉	1	1	yes	characteristic, complemented
52	C₃×D₁₇	—	3	1	no	—
53	C₃×D₁₇	—	159	1	no	—
54	D₃×C₁₇	—	318	1	no	—
55	D₃×C₁₇	—	159	1	no	—
56	D₃×C₁₇	—	159	1	no	—
57	C₂×D₁₇	—	477	1	no	Hall
58	C₅₃	(C₃⋊D₃)⋊D₁₇	1	1	yes	characteristic, complemented, Sylow
59	C₃×C₁₇	—	2	1	no	—
60	C₃×C₁₇	D₃×D₅₃	1	1	yes	characteristic, complemented
61	C₃×C₁₇	C₂×D₁₅₉	1	1	yes	characteristic, complemented
62	D₃²	—	901	1	no	Hall
63	C₂×C₁₇	—	477	1	no	—
64	D₁₇	—	3	1	no	—
65	D₁₇	—	159	1	no	—
66	C₃⋊D₃	—	53	1	no	—
67	C₃×D₃	—	17	1	no	—
68	C₃×D₃	—	901	1	no	—
69	C₁₇	D₃×D₁₅₉	1	1	yes	characteristic, complemented, Sylow
70	C₂×D₃	—	2703	1	no	—
71	C₂×D₃	—	2703	1	no	—
72	C₃²	D₁₇×D₅₃	1	1	yes	characteristic, complemented, Sylow
73	C₂×C₃	—	51	1	no	—
74	C₂×C₃	—	2703	1	no	—
75	D₃	—	17	1	no	—
76	D₃	—	318	1	no	—
77	D₃	—	901	1	no	—
78	D₃	—	159	1	no	—
79	D₃	—	159	1	no	—
80	C₂²	—	8109	1	no	Sylow
81	C₃	—	2	1	no	—
82	C₃	D₁₇×D₁₅₉	1	1	yes	characteristic, complemented
83	C₃	D₅₁⋊D₅₃	1	1	yes	characteristic, complemented
84	C₂	—	51	1	no	—
85	C₂	—	2703	1	no	—
86	C₂	—	477	1	no	—

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃²×C₁₇×C₅₃	C₂²
LCS	G ▷ C₃²×C₁₇×C₅₃
UCS	C₁

Complements

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

N	H ≅ G/N	type
1. C₃×C₅₃×D₅₁	C₂	semidirect
2. C₃×D₂₇₀₃	C₂	semidirect
3. C₁₇×C₃⋊D₁₅₉	C₂	semidirect
6. C₃²×C₁₇×C₅₃	C₂²	semidirect
7. D₂₇₀₃	D₃	semidirect
8. C₅₃×D₅₁	D₃	semidirect
16. C₃×C₁₇×C₅₃	C₂×D₃	semidirect
17. C₃×C₁₇×C₅₃	C₂×D₃	semidirect
22. C₃⋊D₁₅₉	D₁₇	semidirect
25. C₁₇×C₅₃	D₃²	semidirect
29. C₃²×C₅₃	C₂×D₁₇	semidirect
39. C₃×D₅₁	D₅₃	semidirect
44. C₃×C₅₃	D₃×D₁₇	semidirect
45. C₃×C₅₃	D₃×D₁₇	semidirect
46. C₃²×C₁₇	C₂×D₅₃	semidirect
51. D₅₁	D₁₅₉	semidirect
58. C₅₃	(C₃⋊D₃)⋊D₁₇	semidirect
60. C₃×C₁₇	D₃×D₅₃	semidirect
61. C₃×C₁₇	C₂×D₁₅₉	semidirect
69. C₁₇	D₃×D₁₅₉	semidirect
72. C₃²	D₁₇×D₅₃	semidirect
82. C₃	D₁₇×D₁₅₉	semidirect
83. C₃	D₅₁⋊D₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₅₃×D₅₁	2
C₃×D₂₇₀₃	2
C₁₇×C₃⋊D₁₅₉	2
D₅₁⋊D₅₃	3
D₁₇×D₁₅₉	3
D₃×D₁₅₉	17
(C₃⋊D₃)⋊D₁₇	53

Subgroups — D51⋊D159 #44570013

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₅₁⋊D₁₅₉ #44570013