Group #11766971 — subgroups

Subgroups — C₉×#2250 #11766971

Order 1152.

297 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 297 conjugacy classes of subgroups — too many to draw. Only its 108 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	C₂×C₉×SD₃₂	C₂	1	2	yes	maximal, complemented
2	C₂×C₉×D₁₆	C₂	1	2	yes	maximal, complemented
3	C₉×#495	C₂	1	1	yes	maximal, characteristic, complemented
4	C₉×#493	C₂	1	1	yes	maximal, characteristic, complemented
5	C₉×#346	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#2250	C₃	1	1	yes	maximal, characteristic
7	C₂×C₉×Q₁₆	C₂²	1	1	yes	characteristic
8	C₂×C₉×SD₁₆	—	2	1	no	—
9	C₂×C₉×D₈	—	2	1	no	—
10	C₂×C₉×D₈	C₂²	1	2	yes	—
11	C₂×C₉×D₈	C₂²	1	1	yes	characteristic
12	C₉×#127	—	2	1	no	—
13	C₉×#124	—	2	1	no	—
14	C₉×SD₃₂	—	2	4	no	—
15	C₉×D₁₆	—	2	4	no	—
16	C₂×C₉×C₁₆	C₂²	1	2	yes	—
17	C₄×C₈×C₉	C₂²	1	1	yes	characteristic
18	C₂×C₃×SD₃₂	C₂×C₃	1	2	yes	—
19	C₂×C₃×D₁₆	C₂×C₃	1	2	yes	—
20	C₃×#495	C₂×C₃	1	1	yes	characteristic
21	C₃×#493	C₂×C₃	1	1	yes	characteristic
22	C₃×#346	C₂×C₃	1	1	yes	characteristic
23	C₂×Q₈×C₉	—	2	1	no	—
24	C₂×D₄×C₉	—	2	1	no	—
25	C₂×D₄×C₉	—	2	2	no	—
26	C₂×D₄×C₉	—	4	2	no	—
27	C₉×Q₁₆	—	2	2	no	—
28	C₉×SD₁₆	—	4	2	no	—
29	C₉×D₈	—	2	2	no	—
30	C₉×D₈	—	2	2	no	—
31	C₉×D₈	—	2	2	no	—
32	C₉×D₈	—	4	2	no	—
33	C₂×C₈×C₉	D₄	1	1	yes	characteristic
34	C₂×C₈×C₉	C₂³	1	1	yes	characteristic
35	C₉×C₂²⋊C₄	—	4	2	no	—
36	C₄²×C₉	D₄	1	1	yes	characteristic
37	C₉×C₁₆	D₄	1	4	yes	complemented
38	#2250	C₉	1	1	yes	characteristic, direct product factor, Sylow
39	C₂×C₃×Q₁₆	C₂²×C₃	1	1	yes	characteristic
40	C₂×C₃×SD₁₆	—	2	1	no	—
41	C₂×C₃×D₈	—	2	1	no	—
42	C₂×C₃×D₈	C₂²×C₃	1	1	yes	characteristic
43	C₂×C₃×D₈	C₂²×C₃	1	2	yes	—
44	C₃×#127	—	2	1	no	—
45	C₃×#124	—	2	1	no	—
46	C₃×SD₃₂	—	2	4	no	—
47	C₃×D₁₆	—	2	4	no	—
48	C₂×C₃×C₁₆	C₂²×C₃	1	2	yes	—
49	C₃×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
50	C₂³×C₉	—	4	1	no	—
51	C₂³×C₉	—	4	2	no	—
52	Q₈×C₉	—	4	1	no	—
53	Q₈×C₉	—	4	1	no	—
54	D₄×C₉	—	4	1	no	—
55	D₄×C₉	—	4	2	no	—
56	D₄×C₉	—	4	2	no	—
57	D₄×C₉	—	4	1	no	—
58	D₄×C₉	—	8	4	no	—
59	C₂×C₄×C₉	—	4	1	no	—
60	C₂×C₄×C₉	D₈	1	2	yes	—
61	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
62	C₈×C₉	D₈	1	2	yes	—
63	C₈×C₉	C₂×D₄	1	2	yes	—
64	C₂×SD₃₂	C₂×C₉	1	2	yes	complemented
65	C₂×D₁₆	C₂×C₉	1	2	yes	complemented
66	#495	C₂×C₉	1	1	yes	characteristic, complemented
67	#493	C₂×C₉	1	1	yes	characteristic, complemented
68	#346	C₂×C₉	1	1	yes	characteristic, complemented
69	C₂×C₃×Q₈	—	2	1	no	—
70	C₂×C₃×D₄	—	2	1	no	—
71	C₂×C₃×D₄	—	2	2	no	—
72	C₂×C₃×D₄	—	4	2	no	—
73	C₃×Q₁₆	—	2	2	no	—
74	C₃×SD₁₆	—	4	2	no	—
75	C₃×D₈	—	2	2	no	—
76	C₃×D₈	—	2	2	no	—
77	C₃×D₈	—	2	2	no	—
78	C₃×D₈	—	4	2	no	—
79	C₂×C₃×C₈	C₃×D₄	1	1	yes	characteristic
80	C₂×C₃×C₈	C₂³×C₃	1	1	yes	characteristic
81	C₃×C₂²⋊C₄	—	4	2	no	—
82	C₃×C₄²	C₃×D₄	1	1	yes	characteristic
83	C₃×C₁₆	C₃×D₄	1	4	yes	—
84	C₂²×C₉	—	8	2	no	—
85	C₂²×C₉	—	8	1	no	—
86	C₂²×C₉	—	8	4	no	—
87	C₂²×C₉	—	8	2	no	—
88	C₂²×C₉	C₂×D₈	1	1	yes	characteristic
89	C₄×C₉	—	2	2	no	—
90	C₄×C₉	—	8	1	no	—
91	C₄×C₉	#127	1	1	yes	characteristic
92	C₄×C₉	C₂×D₈	1	1	yes	characteristic
93	C₂×Q₁₆	C₂²×C₉	1	1	yes	characteristic
94	C₂×SD₁₆	—	2	1	no	—
95	C₂×D₈	—	2	1	no	—
96	C₂×D₈	C₂²×C₉	1	2	yes	—
97	C₂×D₈	C₂²×C₉	1	1	yes	characteristic
98	#127	—	2	1	no	—
99	#124	—	2	1	no	—
100	SD₃₂	—	2	4	no	—
101	D₁₆	—	2	4	no	—
102	C₂×C₁₆	C₂²×C₉	1	2	yes	—
103	C₄×C₈	C₂²×C₉	1	1	yes	characteristic
104	C₂³×C₃	—	4	1	no	—
105	C₂³×C₃	—	4	2	no	—
106	C₃×Q₈	—	4	1	no	—
107	C₃×Q₈	—	4	1	no	—
108	C₃×D₄	—	4	1	no	—
109	C₃×D₄	—	4	2	no	—
110	C₃×D₄	—	4	2	no	—
111	C₃×D₄	—	4	1	no	—
112	C₃×D₄	—	8	4	no	—
113	C₂×C₃×C₄	—	4	1	no	—
114	C₂×C₃×C₄	C₃×D₈	1	2	yes	—
115	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
116	C₃×C₈	C₃×D₈	1	2	yes	—
117	C₃×C₈	C₂×C₃×D₄	1	2	yes	—
118	C₂×C₉	—	16	2	no	—
119	C₂×C₉	—	16	1	no	—
120	C₂×C₉	#493	1	1	yes	characteristic
121	C₂×C₉	#509	1	2	yes	—
122	C₂×Q₈	—	2	1	no	—
123	C₂×D₄	—	2	1	no	—
124	C₂×D₄	—	2	2	no	—
125	C₂×D₄	—	4	2	no	—
126	Q₁₆	—	2	2	no	—
127	SD₁₆	—	4	2	no	—
128	D₈	—	2	2	no	—
129	D₈	—	2	2	no	—
130	D₈	—	2	2	no	—
131	D₈	—	4	2	no	—
132	C₂×C₈	D₄×C₉	1	1	yes	characteristic
133	C₂×C₈	C₂³×C₉	1	1	yes	characteristic
134	C₂²⋊C₄	—	4	2	no	—
135	C₄²	D₄×C₉	1	1	yes	characteristic
136	C₁₆	D₄×C₉	1	4	yes	complemented
137	C₂²×C₃	—	8	4	no	—
138	C₂²×C₃	—	8	2	no	—
139	C₂²×C₃	—	8	2	no	—
140	C₂²×C₃	—	8	1	no	—
141	C₂²×C₃	C₂×C₃×D₈	1	1	yes	characteristic
142	C₃×C₄	—	2	2	no	—
143	C₃×C₄	—	8	1	no	—
144	C₃×C₄	C₃×#127	1	1	yes	characteristic
145	C₃×C₄	C₂×C₃×D₈	1	1	yes	characteristic
146	C₉	#2250	1	1	yes	characteristic, direct product factor, Sylow
147	C₂³	—	4	1	no	—
148	C₂³	—	4	2	no	—
149	Q₈	—	4	1	no	—
150	Q₈	—	4	1	no	—
151	D₄	—	4	1	no	—
152	D₄	—	4	2	no	—
153	D₄	—	4	2	no	—
154	D₄	—	4	1	no	—
155	D₄	—	8	4	no	—
156	C₂×C₄	—	4	1	no	—
157	C₂×C₄	C₉×D₈	1	2	yes	—
158	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
159	C₈	C₉×D₈	1	2	yes	—
160	C₈	C₂×D₄×C₉	1	2	yes	—
161	C₂×C₃	—	16	2	no	—
162	C₂×C₃	—	16	1	no	—
163	C₂×C₃	C₃×#493	1	1	yes	characteristic
164	C₂×C₃	C₃×#509	1	2	yes	—
165	C₂²	—	8	2	no	—
166	C₂²	—	8	1	no	—
167	C₂²	—	8	4	no	—
168	C₂²	—	8	2	no	—
169	C₂²	C₂×C₉×D₈	1	1	yes	characteristic
170	C₄	—	2	2	no	—
171	C₄	—	8	1	no	—
172	C₄	C₉×#127	1	1	yes	characteristic
173	C₄	C₂×C₉×D₈	1	1	yes	characteristic
174	C₃	C₃×#2250	1	1	yes	characteristic
175	C₂	—	16	2	no	—
176	C₂	—	16	1	no	—
177	C₂	C₉×#493	1	1	yes	characteristic
178	C₂	C₉×#509	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₉×SD₃₂

C₂

yes

maximal, complemented

C₂×C₉×D₁₆

C₂

yes

maximal, complemented

C₉×#495

C₂

yes

maximal, characteristic, complemented

C₉×#493

C₂

yes

maximal, characteristic, complemented

C₉×#346

C₂

yes

maximal, characteristic, complemented

C₃×#2250

C₃

yes

maximal, characteristic

C₂×C₉×Q₁₆

C₂²

yes

characteristic

C₂×C₉×SD₁₆

—

C₂×C₉×D₈

—

C₂×C₉×D₈

C₂²

yes

—

C₂×C₉×D₈

C₂²

yes

characteristic

C₉×#127

—

C₉×#124

—

C₉×SD₃₂

—

C₉×D₁₆

—

C₂×C₉×C₁₆

C₂²

yes

—

C₄×C₈×C₉

C₂²

yes

characteristic

C₂×C₃×SD₃₂

C₂×C₃

yes

—

C₂×C₃×D₁₆

C₂×C₃

yes

—

C₃×#495

C₂×C₃

yes

characteristic

C₃×#493

C₂×C₃

yes

characteristic

C₃×#346

C₂×C₃

yes

characteristic

C₂×Q₈×C₉

—

C₂×D₄×C₉

—

C₂×D₄×C₉

—

C₂×D₄×C₉

—

C₉×Q₁₆

—

C₉×SD₁₆

—

C₉×D₈

—

C₉×D₈

—

C₉×D₈

—

C₉×D₈

—

C₂×C₈×C₉

D₄

yes

characteristic

C₂×C₈×C₉

C₂³

yes

characteristic

C₉×C₂²⋊C₄

—

C₄²×C₉

D₄

yes

characteristic

C₉×C₁₆

D₄

yes

complemented

#2250

C₉

yes

characteristic, direct product factor, Sylow

C₂×C₃×Q₁₆

C₂²×C₃

yes

characteristic

C₂×C₃×SD₁₆

—

C₂×C₃×D₈

—

C₂×C₃×D₈

C₂²×C₃

yes

characteristic

C₂×C₃×D₈

C₂²×C₃

yes

—

C₃×#127

—

C₃×#124

—

C₃×SD₃₂

—

C₃×D₁₆

—

C₂×C₃×C₁₆

C₂²×C₃

yes

—

C₃×C₄×C₈

C₂²×C₃

yes

characteristic

C₂³×C₉

—

C₂³×C₉

—

Q₈×C₉

—

Q₈×C₉

—

D₄×C₉

—

D₄×C₉

—

D₄×C₉

—

D₄×C₉

—

D₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

D₈

yes

—

C₂×C₄×C₉

C₂×D₄

yes

characteristic

C₈×C₉

D₈

yes

—

C₈×C₉

C₂×D₄

yes

—

C₂×SD₃₂

C₂×C₉

yes

complemented

C₂×D₁₆

C₂×C₉

yes

complemented

#495

C₂×C₉

yes

characteristic, complemented

#493

C₂×C₉

yes

characteristic, complemented

#346

C₂×C₉

yes

characteristic, complemented

C₂×C₃×Q₈

—

C₂×C₃×D₄

—

C₂×C₃×D₄

—

C₂×C₃×D₄

—

C₃×Q₁₆

—

C₃×SD₁₆

—

C₃×D₈

—

C₃×D₈

—

C₃×D₈

—

C₃×D₈

—

C₂×C₃×C₈

C₃×D₄

yes

characteristic

C₂×C₃×C₈

C₂³×C₃

yes

characteristic

C₃×C₂²⋊C₄

—

C₃×C₄²

C₃×D₄

yes

characteristic

C₃×C₁₆

C₃×D₄

yes

—

C₂²×C₉

—

C₂²×C₉

—

C₂²×C₉

—

C₂²×C₉

—

C₂²×C₉

C₂×D₈

yes

characteristic

C₄×C₉

—

C₄×C₉

—

C₄×C₉

#127

yes

characteristic

C₄×C₉

C₂×D₈

yes

characteristic

C₂×Q₁₆

C₂²×C₉

yes

characteristic

C₂×SD₁₆

—

C₂×D₈

—

C₂×D₈

C₂²×C₉

yes

—

C₂×D₈

C₂²×C₉

yes

characteristic

#127

—

#124

—

100

SD₃₂

—

101

D₁₆

—

102

C₂×C₁₆

C₂²×C₉

yes

—

103

C₄×C₈

C₂²×C₉

yes

characteristic

104

C₂³×C₃

—

105

C₂³×C₃

—

106

C₃×Q₈

—

107

C₃×Q₈

—

108

C₃×D₄

—

109

C₃×D₄

—

110

C₃×D₄

—

111

C₃×D₄

—

112

C₃×D₄

—

113

C₂×C₃×C₄

—

114

C₂×C₃×C₄

C₃×D₈

yes

—

115

C₂×C₃×C₄

C₂×C₃×D₄

yes

characteristic

116

C₃×C₈

C₃×D₈

yes

—

117

C₃×C₈

C₂×C₃×D₄

yes

—

118

C₂×C₉

—

119

C₂×C₉

—

120

C₂×C₉

#493

yes

characteristic

121

C₂×C₉

#509

yes

—

122

C₂×Q₈

—

123

C₂×D₄

—

124

C₂×D₄

—

125

C₂×D₄

—

126

Q₁₆

—

127

SD₁₆

—

128

D₈

—

129

D₈

—

130

D₈

—

131

D₈

—

132

C₂×C₈

D₄×C₉

yes

characteristic

133

C₂×C₈

C₂³×C₉

yes

characteristic

134

C₂²⋊C₄

—

135

C₄²

D₄×C₉

yes

characteristic

136

C₁₆

D₄×C₉

yes

complemented

137

C₂²×C₃

—

138

C₂²×C₃

—

139

C₂²×C₃

—

140

C₂²×C₃

—

141

C₂²×C₃

C₂×C₃×D₈

yes

characteristic

142

C₃×C₄

—

143

C₃×C₄

—

144

C₃×C₄

C₃×#127

yes

characteristic

145

C₃×C₄

C₂×C₃×D₈

yes

characteristic

146

C₉

#2250

yes

characteristic, direct product factor, Sylow

147

C₂³

—

148

C₂³

—

149

Q₈

—

150

Q₈

—

151

D₄

—

152

D₄

—

153

D₄

—

154

D₄

—

155

D₄

—

156

C₂×C₄

—

157

C₂×C₄

C₉×D₈

yes

—

158

C₂×C₄

C₂×D₄×C₉

yes

characteristic

159

C₈

C₉×D₈

yes

—

160

C₈

C₂×D₄×C₉

yes

—

161

C₂×C₃

—

162

C₂×C₃

—

163

C₂×C₃

C₃×#493

yes

characteristic

164

C₂×C₃

C₃×#509

yes

—

165

C₂²

—

166

C₂²

—

167

C₂²

—

168

C₂²

—

169

C₂²

C₂×C₉×D₈

yes

characteristic

170

C₄

—

171

C₄

—

172

C₄

C₉×#127

yes

characteristic

173

C₄

C₂×C₉×D₈

yes

characteristic

174

C₃

C₃×#2250

yes

characteristic

175

C₂

—

176

C₂

—

177

C₂

C₉×#493

yes

characteristic

178

C₂

C₉×#509

yes

—

	subgroup	G/H
center	C₂²×C₉	C₂×D₈
commutator	C₂×C₈	C₂³×C₉
Frattini, cosocle	C₂×C₃×C₈	C₂³×C₃
socle	C₂²×C₃	C₂×C₃×D₈
prime radical	C₃×#2250	C₃
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₉ ◁ C₄²×C₉ ◁ C₄×C₈×C₉ ◁ G

subgroup

G/H

center

C₂²×C₉

C₂×D₈

commutator

C₂×C₈

C₂³×C₉

Frattini, cosocle

C₂×C₃×C₈

C₂³×C₃

socle

C₂²×C₃

C₂×C₃×D₈

prime radical

C₃×#2250

C₃

LCS

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₉ ◁ C₄²×C₉ ◁ C₄×C₈×C₉ ◁ G

N	H ≅ G/N	type
1. C₂×C₉×SD₃₂	C₂	semidirect
2. C₂×C₉×D₁₆	C₂	semidirect
3. C₉×#495	C₂	semidirect
4. C₉×#493	C₂	semidirect
5. C₉×#346	C₂	semidirect
37. C₉×C₁₆	D₄	semidirect
38. #2250	C₉	direct (N × H)
64. C₂×SD₃₂	C₂×C₉	semidirect
65. C₂×D₁₆	C₂×C₉	semidirect
66. #495	C₂×C₉	semidirect
67. #493	C₂×C₉	semidirect
68. #346	C₂×C₉	semidirect
136. C₁₆	D₄×C₉	semidirect
146. C₉	#2250	direct (N × H)

H ≅ G/N

type

1. C₂×C₉×SD₃₂

C₂

semidirect

2. C₂×C₉×D₁₆

C₂

semidirect

3. C₉×#495

C₂

semidirect

4. C₉×#493

C₂

semidirect

5. C₉×#346

C₂

semidirect

37. C₉×C₁₆

D₄

semidirect

38. #2250

C₉

direct (N × H)

64. C₂×SD₃₂

C₂×C₉

semidirect

65. C₂×D₁₆

C₂×C₉

semidirect

66. #495

C₂×C₉

semidirect

67. #493

C₂×C₉

semidirect

68. #346

C₂×C₉

semidirect

136. C₁₆

D₄×C₉

semidirect

146. C₉

#2250

direct (N × H)

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

subgroup	index
C₂×C₉×SD₃₂	2
C₂×C₉×D₁₆	2
C₉×#495	2
C₉×#493	2
C₉×#346	2
C₃×#2250	3

subgroup

index

C₂×C₉×SD₃₂

C₂×C₉×D₁₆

Subgroups — C₉×#2250 #11766971

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C9×#2250 #11766971

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#2250 #11766971