Group #49178411 — subgroups

Subgroups — D₄₅⋊D₂₀₉ #49178411

Order 37620.

120 conjugacy classes of subgroups; only the 34 normal subgroups are shown.

normal subgroups only

This group has 120 conjugacy classes of subgroups — too many to draw. Only its 34 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₁×C₁₉×D₄₅	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×D₁₈₈₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₉×D₁₀₄₅	C₂	1	1	yes	maximal, characteristic, complemented
4	D₁₅⋊D₂₀₉	—	3	1	no	maximal
5	C₅×C₉×C₁₁×C₁₉	C₂²	1	1	yes	characteristic, complemented, Hall
6	D₉×D₂₀₉	—	5	1	no	maximal, Hall
7	C₁₁×C₁₉×D₁₅	—	3	1	no	—
8	C₅×D₆₂₇	—	3	1	no	—
9	C₃×D₁₀₄₅	D₃	1	1	yes	characteristic
10	D₅×D₂₀₉	—	9	1	no	Hall
11	D₁₈₈₁	D₅	1	1	yes	characteristic, complemented
12	C₁₁×D₉×C₁₉	—	5	1	no	—
13	C₉×D₂₀₉	—	5	1	no	—
14	D₄₅⋊D₁₉	—	11	1	no	maximal, Hall
15	C₃×C₅×C₁₁×C₁₉	C₂×D₃	1	1	yes	characteristic
16	D₃×D₂₀₉	—	15	1	no	—
17	D₁₀₄₅	D₉	1	1	yes	characteristic, complemented
18	D₅×C₁₁×C₁₉	—	9	1	no	—
19	C₅×D₂₀₉	—	9	1	no	—
20	D₄₅⋊D₁₁	—	19	1	no	maximal, Hall
21	C₉×C₁₁×C₁₉	C₂×D₅	1	1	yes	characteristic, complemented, Hall
22	C₁₉×D₄₅	D₁₁	1	1	yes	characteristic, complemented
23	C₅×D₁₇₁	—	11	1	no	—
24	C₉×D₉₅	—	11	1	no	—
25	D₆₂₇	—	3	1	no	—
26	D₃×C₁₁×C₁₉	—	15	1	no	—
27	C₃×D₂₀₉	—	5	1	no	—
28	D₁₅⋊D₁₉	—	33	1	no	—
29	C₅×C₁₁×C₁₉	C₂×D₉	1	1	yes	characteristic, complemented, Hall
30	C₁₁×D₄₅	D₁₉	1	1	yes	characteristic, complemented
31	C₅×D₉₉	—	19	1	no	—
32	C₉×D₅₅	—	19	1	no	—
33	C₅×C₉×C₁₉	C₂×D₁₁	1	1	yes	characteristic, complemented, Hall
34	C₂×D₂₀₉	—	45	1	no	Hall
35	D₉×D₁₉	—	55	1	no	Hall
36	D₁₅⋊D₁₁	—	57	1	no	—
37	C₃×C₁₁×C₁₉	D₃×D₅	1	1	yes	characteristic
38	C₁₉×D₁₅	—	3	1	no	—
39	C₅×D₅₇	—	33	1	no	—
40	C₃×D₉₅	—	11	1	no	—
41	C₅×C₉×C₁₁	C₂×D₁₉	1	1	yes	characteristic, complemented, Hall
42	C₂×C₁₁×C₁₉	—	45	1	no	—
43	D₂₀₉	—	5	1	no	—
44	D₂₀₉	—	9	1	no	—
45	D₉×D₁₁	—	95	1	no	Hall
46	D₅×D₁₉	—	99	1	no	Hall
47	D₁₇₁	—	11	1	no	—
48	C₉×D₁₉	—	55	1	no	—
49	D₉×C₁₉	—	5	1	no	—
50	C₁₁×D₁₅	—	3	1	no	—
51	C₅×D₃₃	—	57	1	no	—
52	C₃×D₅₅	—	19	1	no	—
53	C₃×C₅×C₁₉	D₃×D₁₁	1	1	yes	characteristic
54	D₃×D₁₉	—	165	1	no	—
55	D₅×D₁₁	—	171	1	no	Hall
56	C₁₁×C₁₉	D₅×D₉	1	1	yes	characteristic, complemented, Hall
57	D₉₉	—	19	1	no	—
58	C₉×D₁₁	—	95	1	no	—
59	C₁₁×D₉	—	5	1	no	—
60	D₉₅	—	11	1	no	—
61	C₅×D₁₉	—	99	1	no	—
62	D₅×C₁₉	—	9	1	no	—
63	D₅×D₉	—	209	1	no	Hall
64	C₉×C₁₉	D₅×D₁₁	1	1	yes	characteristic, complemented, Hall
65	C₃×C₅×C₁₁	D₃×D₁₉	1	1	yes	characteristic
66	D₃×D₁₁	—	285	1	no	—
67	D₅₇	—	33	1	no	—
68	C₃×D₁₉	—	55	1	no	—
69	D₃×C₁₉	—	15	1	no	—
70	D₅₅	—	19	1	no	—
71	C₅×D₁₁	—	171	1	no	—
72	D₅×C₁₁	—	9	1	no	—
73	C₉×C₁₁	D₅×D₁₉	1	1	yes	characteristic, complemented, Hall
74	C₅×C₁₉	D₉×D₁₁	1	1	yes	characteristic, complemented, Hall
75	D₄₅	D₂₀₉	1	1	yes	characteristic, complemented
76	C₉×D₅	—	209	1	no	—
77	C₅×D₉	—	209	1	no	—
78	C₂×D₁₉	—	495	1	no	Hall
79	D₃₃	—	57	1	no	—
80	C₃×D₁₁	—	95	1	no	—
81	D₃×C₁₁	—	15	1	no	—
82	D₃×D₅	—	627	1	no	—
83	C₃×C₁₉	D₁₅⋊D₁₁	1	1	yes	characteristic
84	C₅×C₁₁	D₉×D₁₉	1	1	yes	characteristic, complemented, Hall
85	C₅×C₉	C₂×D₂₀₉	1	1	yes	characteristic, complemented, Hall
86	C₂×D₁₁	—	855	1	no	Hall
87	C₂×C₁₉	—	45	1	no	—
88	D₁₉	—	55	1	no	—
89	D₁₉	—	99	1	no	—
90	C₂×D₉	—	1045	1	no	Hall
91	C₃×C₁₁	D₁₅⋊D₁₉	1	1	yes	characteristic
92	D₁₅	—	3	1	no	—
93	C₃×D₅	—	209	1	no	—
94	C₅×D₃	—	627	1	no	—
95	C₂×C₁₁	—	45	1	no	—
96	D₁₁	—	95	1	no	—
97	D₁₁	—	171	1	no	—
98	C₂×D₅	—	1881	1	no	Hall
99	C₁₉	D₄₅⋊D₁₁	1	1	yes	characteristic, complemented, Sylow
100	C₂×C₉	—	1045	1	no	—
101	D₉	—	5	1	no	—
102	D₉	—	209	1	no	—
103	C₃×C₅	D₃×D₂₀₉	1	1	yes	characteristic
104	C₂×D₃	—	3135	1	no	—
105	C₁₁	D₄₅⋊D₁₉	1	1	yes	characteristic, complemented, Sylow
106	C₂×C₅	—	1881	1	no	—
107	D₅	—	9	1	no	—
108	D₅	—	209	1	no	—
109	C₉	D₅×D₂₀₉	1	1	yes	characteristic, complemented, Sylow
110	C₂×C₃	—	1045	1	no	—
111	D₃	—	15	1	no	—
112	D₃	—	627	1	no	—
113	C₅	D₉×D₂₀₉	1	1	yes	characteristic, complemented, Sylow
114	C₂²	—	9405	1	no	Sylow
115	C₃	D₁₅⋊D₂₀₉	1	1	yes	characteristic
116	C₂	—	1045	1	no	—
117	C₂	—	45	1	no	—
118	C₂	—	1881	1	no	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₁×C₁₉×D₄₅

C₂

yes

maximal, characteristic, complemented

C₅×D₁₈₈₁

C₂

yes

maximal, characteristic, complemented

C₉×D₁₀₄₅

C₂

yes

maximal, characteristic, complemented

D₁₅⋊D₂₀₉

—

maximal

C₅×C₉×C₁₁×C₁₉

C₂²

yes

characteristic, complemented, Hall

D₉×D₂₀₉

—

maximal, Hall

C₁₁×C₁₉×D₁₅

—

C₅×D₆₂₇

—

C₃×D₁₀₄₅

D₃

yes

characteristic

D₅×D₂₀₉

—

Hall

D₁₈₈₁

D₅

yes

characteristic, complemented

C₁₁×D₉×C₁₉

—

C₉×D₂₀₉

—

D₄₅⋊D₁₉

—

maximal, Hall

C₃×C₅×C₁₁×C₁₉

C₂×D₃

yes

characteristic

D₃×D₂₀₉

—

D₁₀₄₅

D₉

yes

characteristic, complemented

D₅×C₁₁×C₁₉

—

C₅×D₂₀₉

—

D₄₅⋊D₁₁

—

maximal, Hall

C₉×C₁₁×C₁₉

C₂×D₅

yes

characteristic, complemented, Hall

C₁₉×D₄₅

D₁₁

yes

characteristic, complemented

C₅×D₁₇₁

—

C₉×D₉₅

—

D₆₂₇

—

D₃×C₁₁×C₁₉

—

C₃×D₂₀₉

—

D₁₅⋊D₁₉

—

C₅×C₁₁×C₁₉

C₂×D₉

yes

characteristic, complemented, Hall

C₁₁×D₄₅

D₁₉

yes

characteristic, complemented

C₅×D₉₉

—

C₉×D₅₅

—

C₅×C₉×C₁₉

C₂×D₁₁

yes

characteristic, complemented, Hall

C₂×D₂₀₉

—

Hall

D₉×D₁₉

—

Hall

D₁₅⋊D₁₁

—

C₃×C₁₁×C₁₉

D₃×D₅

yes

characteristic

C₁₉×D₁₅

—

C₅×D₅₇

—

C₃×D₉₅

—

C₅×C₉×C₁₁

C₂×D₁₉

yes

characteristic, complemented, Hall

C₂×C₁₁×C₁₉

—

D₂₀₉

—

D₂₀₉

—

D₉×D₁₁

—

Hall

D₅×D₁₉

—

Hall

D₁₇₁

—

C₉×D₁₉

—

D₉×C₁₉

—

C₁₁×D₁₅

—

C₅×D₃₃

—

C₃×D₅₅

—

C₃×C₅×C₁₉

D₃×D₁₁

yes

characteristic

D₃×D₁₉

—

165

—

D₅×D₁₁

—

171

Hall

C₁₁×C₁₉

D₅×D₉

yes

characteristic, complemented, Hall

D₉₉

—

C₉×D₁₁

—

C₁₁×D₉

—

D₉₅

—

C₅×D₁₉

—

D₅×C₁₉

—

D₅×D₉

—

209

Hall

C₉×C₁₉

D₅×D₁₁

yes

characteristic, complemented, Hall

C₃×C₅×C₁₁

D₃×D₁₉

yes

characteristic

D₃×D₁₁

—

285

—

D₅₇

—

C₃×D₁₉

—

D₃×C₁₉

—

D₅₅

—

C₅×D₁₁

—

171

—

D₅×C₁₁

—

C₉×C₁₁

D₅×D₁₉

yes

characteristic, complemented, Hall

C₅×C₁₉

D₉×D₁₁

yes

characteristic, complemented, Hall

D₄₅

D₂₀₉

yes

characteristic, complemented

C₉×D₅

—

209

—

C₅×D₉

—

209

—

C₂×D₁₉

—

495

Hall

D₃₃

—

C₃×D₁₁

—

D₃×C₁₁

—

D₃×D₅

—

627

—

C₃×C₁₉

D₁₅⋊D₁₁

yes

characteristic

C₅×C₁₁

D₉×D₁₉

yes

characteristic, complemented, Hall

C₅×C₉

C₂×D₂₀₉

yes

characteristic, complemented, Hall

C₂×D₁₁

—

855

Hall

C₂×C₁₉

—

D₁₉

—

D₁₉

—

C₂×D₉

—

1045

Hall

C₃×C₁₁

D₁₅⋊D₁₉

yes

characteristic

D₁₅

—

C₃×D₅

—

209

—

C₅×D₃

—

627

—

C₂×C₁₁

—

D₁₁

—

D₁₁

—

171

—

C₂×D₅

—

1881

Hall

C₁₉

D₄₅⋊D₁₁

yes

characteristic, complemented, Sylow

100

C₂×C₉

—

1045

—

101

D₉

—

102

D₉

—

209

—

103

C₃×C₅

D₃×D₂₀₉

yes

characteristic

104

C₂×D₃

—

3135

—

105

C₁₁

D₄₅⋊D₁₉

yes

characteristic, complemented, Sylow

106

C₂×C₅

—

1881

—

107

D₅

—

108

D₅

—

209

—

109

C₉

D₅×D₂₀₉

yes

characteristic, complemented, Sylow

110

C₂×C₃

—

1045

—

111

D₃

—

112

D₃

—

627

—

113

C₅

D₉×D₂₀₉

yes

characteristic, complemented, Sylow

114

C₂²

—

9405

Sylow

115

C₃

D₁₅⋊D₂₀₉

yes

characteristic

116

C₂

—

1045

—

117

C₂

—

118

C₂

—

1881

—

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₅×C₉×C₁₁×C₁₉	C₂²
Frattini	C₃	D₁₅⋊D₂₀₉
socle	C₃×C₅×C₁₁×C₁₉	C₂×D₃
LCS	G ▷ C₅×C₉×C₁₁×C₁₉
UCS	C₁

subgroup

G/H

center, hypercenter

C₁

commutator, Fitting, cosocle, hypocenter

C₅×C₉×C₁₁×C₁₉

C₂²

Frattini

C₃

D₁₅⋊D₂₀₉

socle

C₃×C₅×C₁₁×C₁₉

C₂×D₃

LCS

G ▷ C₅×C₉×C₁₁×C₁₉

UCS

C₁

N	H ≅ G/N	type
1. C₁₁×C₁₉×D₄₅	C₂	semidirect
2. C₅×D₁₈₈₁	C₂	semidirect
3. C₉×D₁₀₄₅	C₂	semidirect
5. C₅×C₉×C₁₁×C₁₉	C₂²	semidirect
11. D₁₈₈₁	D₅	semidirect
17. D₁₀₄₅	D₉	semidirect
21. C₉×C₁₁×C₁₉	C₂×D₅	semidirect
22. C₁₉×D₄₅	D₁₁	semidirect
29. C₅×C₁₁×C₁₉	C₂×D₉	semidirect
30. C₁₁×D₄₅	D₁₉	semidirect
33. C₅×C₉×C₁₉	C₂×D₁₁	semidirect
41. C₅×C₉×C₁₁	C₂×D₁₉	semidirect
56. C₁₁×C₁₉	D₅×D₉	semidirect
64. C₉×C₁₉	D₅×D₁₁	semidirect
73. C₉×C₁₁	D₅×D₁₉	semidirect
74. C₅×C₁₉	D₉×D₁₁	semidirect
75. D₄₅	D₂₀₉	semidirect
84. C₅×C₁₁	D₉×D₁₉	semidirect
85. C₅×C₉	C₂×D₂₀₉	semidirect
99. C₁₉	D₄₅⋊D₁₁	semidirect
105. C₁₁	D₄₅⋊D₁₉	semidirect
109. C₉	D₅×D₂₀₉	semidirect
113. C₅	D₉×D₂₀₉	semidirect

H ≅ G/N

type

1. C₁₁×C₁₉×D₄₅

C₂

semidirect

2. C₅×D₁₈₈₁

C₂

semidirect

3. C₉×D₁₀₄₅

C₂

semidirect

5. C₅×C₉×C₁₁×C₁₉

C₂²

semidirect

11. D₁₈₈₁

D₅

semidirect

17. D₁₀₄₅

D₉

semidirect

21. C₉×C₁₁×C₁₉

C₂×D₅

semidirect

22. C₁₉×D₄₅

D₁₁

semidirect

29. C₅×C₁₁×C₁₉

C₂×D₉

semidirect

30. C₁₁×D₄₅

D₁₉

semidirect

33. C₅×C₉×C₁₉

C₂×D₁₁

semidirect

41. C₅×C₉×C₁₁

C₂×D₁₉

semidirect

56. C₁₁×C₁₉

D₅×D₉

semidirect

64. C₉×C₁₉

D₅×D₁₁

semidirect

73. C₉×C₁₁

D₅×D₁₉

semidirect

74. C₅×C₁₉

D₉×D₁₁

semidirect

75. D₄₅

D₂₀₉

semidirect

84. C₅×C₁₁

D₉×D₁₉

semidirect

85. C₅×C₉

C₂×D₂₀₉

semidirect

99. C₁₉

D₄₅⋊D₁₁

semidirect

105. C₁₁

D₄₅⋊D₁₉

semidirect

109. C₉

D₅×D₂₀₉

semidirect

113. C₅

D₉×D₂₀₉

semidirect

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

subgroup	index
C₁₁×C₁₉×D₄₅	2
C₅×D₁₈₈₁	2
C₉×D₁₀₄₅	2
D₁₅⋊D₂₀₉	3
D₉×D₂₀₉	5
D₄₅⋊D₁₉	11
D₄₅⋊D₁₁	19

subgroup

index

C₁₁×C₁₉×D₄₅

C₅×D₁₈₈₁

C₉×D₁₀₄₅

D₁₅⋊D₂₀₉

D₉×D₂₀₉

D₄₅⋊D₁₉

D₄₅⋊D₁₁

Subgroups — D₄₅⋊D₂₀₉ #49178411

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D45⋊D209 #49178411

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₄₅⋊D₂₀₉ #49178411