Group #11782469 — subgroups

Subgroups — #11782469

Order 1152.

270 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 86 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	C₉×#487	C₂	1	1	yes	maximal, characteristic
2	#10588765	C₂	1	1	yes	maximal, characteristic
3	#10588760	C₂	1	1	yes	maximal, characteristic
4	#10588760	C₂	1	1	yes	maximal, characteristic
5	#10588758	C₂	1	1	yes	maximal, characteristic
6	#10588361	C₂	1	1	yes	maximal, characteristic
7	C₈×C₉⋊C₈	C₂	1	1	yes	maximal, characteristic
8	#71212	—	3	1	no	maximal
9	C₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic
10	C₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic
11	C₂×#3674	—	2	2	no	—
12	#63431	C₂²	1	1	yes	characteristic
13	#63431	C₂²	1	1	yes	characteristic
14	C₉×#103	—	2	2	no	—
15	C₄×C₈×C₉	C₂²	1	1	yes	characteristic
16	#63376	—	2	2	no	—
17	#63367	—	2	2	no	—
18	#63367	—	2	2	no	—
19	#63366	—	2	2	no	—
20	C₄×C₉⋊C₈	C₂²	1	1	yes	characteristic
21	C₄×C₉⋊C₈	C₂²	1	1	yes	characteristic
22	C₃×#487	D₃	1	1	yes	characteristic
23	#5092	—	3	1	no	—
24	#5080	—	3	1	no	—
25	#5080	—	3	1	no	—
26	#5078	—	3	1	no	—
27	#4681	—	3	1	no	—
28	C₈×C₃⋊C₈	—	3	1	no	—
29	C₂×Q₈×C₉	—	2	1	no	—
30	C₂×Q₈×C₉	—	2	1	no	—
31	C₂×Dic₁₈	—	2	1	no	—
32	C₂×Dic₁₈	—	2	1	no	—
33	C₂×C₈×C₉	D₄	1	2	yes	—
34	C₉×C₄⋊C₄	—	2	1	no	—
35	C₉×C₄⋊C₄	—	2	1	no	—
36	C₉×C₄⋊C₄	—	2	1	no	—
37	C₉×C₄⋊C₄	—	2	1	no	—
38	C₄²×C₉	C₂³	1	1	yes	characteristic
39	#3674	—	4	4	no	—
40	C₄⋊Dic₉	—	2	1	no	—
41	C₄⋊Dic₉	—	2	1	no	—
42	C₄⋊Dic₉	—	2	1	no	—
43	C₄⋊Dic₉	—	2	1	no	—
44	C₂×C₉⋊C₈	D₄	1	2	yes	—
45	C₂×C₉⋊C₈	D₄	1	2	yes	—
46	#1707	—	9	1	no	Sylow
47	C₃×C₄⋊Q₈	C₂×D₃	1	1	yes	characteristic
48	C₃×C₄⋊Q₈	C₂×D₃	1	1	yes	characteristic
49	C₂×#216	—	6	2	no	—
50	#869	—	3	1	no	—
51	#869	—	3	1	no	—
52	C₃×#103	—	2	2	no	—
53	C₃×C₄×C₈	C₂×D₃	1	1	yes	characteristic
54	#816	—	6	2	no	—
55	#808	—	6	2	no	—
56	#808	—	6	2	no	—
57	#807	—	6	2	no	—
58	C₄×C₃⋊C₈	—	3	1	no	—
59	C₄×C₃⋊C₈	—	3	1	no	—
60	Q₈×C₉	—	4	2	no	—
61	Q₈×C₉	—	4	2	no	—
62	C₂×C₄×C₉	—	4	1	no	—
63	C₂×C₄×C₉	—	4	1	no	—
64	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
65	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
66	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
67	C₂×Dic₉	—	4	1	no	—
68	C₂×Dic₉	—	4	1	no	—
69	Dic₁₈	—	4	2	no	—
70	Dic₁₈	—	4	2	no	—
71	C₈×C₉	—	2	2	no	—
72	C₉⋊C₈	SD₁₆	1	4	yes	—
73	C₉⋊C₈	Q₁₆	1	4	yes	complemented
74	#500	—	9	1	no	—
75	#498	—	9	1	no	—
76	#498	—	9	1	no	—
77	#494	—	9	1	no	—
78	#487	—	9	1	no	—
79	#487	D₉	1	1	yes	characteristic
80	C₈²	—	9	1	no	—
81	C₂×C₃×Q₈	—	2	1	no	—
82	C₂×C₃×Q₈	—	2	1	no	—
83	C₂×Dic₆	—	6	1	no	—
84	C₂×Dic₆	—	6	1	no	—
85	C₂×C₃×C₈	#67	1	2	yes	—
86	C₃×C₄⋊C₄	—	2	1	no	—
87	C₃×C₄⋊C₄	—	2	1	no	—
88	C₃×C₄⋊C₄	—	2	1	no	—
89	C₃×C₄⋊C₄	—	2	1	no	—
90	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
91	#216	—	12	4	no	—
92	C₄⋊Dic₃	—	6	1	no	—
93	C₄⋊Dic₃	—	6	1	no	—
94	C₄⋊Dic₃	—	6	1	no	—
95	C₄⋊Dic₃	—	6	1	no	—
96	C₂×C₃⋊C₈	—	3	2	no	—
97	C₂×C₃⋊C₈	—	3	2	no	—
98	C₂²×C₉	#127	1	1	yes	characteristic
99	C₄×C₉	—	8	1	no	—
100	C₄×C₉	—	8	1	no	—
101	C₄×C₉	C₂×SD₁₆	1	2	yes	—
102	C₄×C₉	C₂×Q₁₆	1	2	yes	—
103	C₄×C₉	#135	1	2	yes	—
104	Dic₉	—	8	1	no	—
105	Dic₉	—	8	1	no	—
106	C₂×Q₁₆	—	18	2	no	—
107	C₄⋊Q₈	—	9	1	no	—
108	C₄⋊Q₈	—	9	1	no	—
109	C₄⋊Q₈	C₂×D₉	1	1	yes	characteristic
110	C₄⋊Q₈	C₂×D₉	1	1	yes	characteristic
111	#107	—	18	2	no	—
112	#106	—	18	2	no	—
113	#106	—	18	2	no	—
114	#103	—	2	2	no	—
115	#103	—	18	2	no	—
116	C₄×C₈	—	9	1	no	—
117	C₄×C₈	—	9	1	no	—
118	C₄×C₈	C₂×D₉	1	1	yes	characteristic
119	C₃×Q₈	—	4	2	no	—
120	C₃×Q₈	—	4	2	no	—
121	C₂×C₃×C₄	—	4	1	no	—
122	C₂×C₃×C₄	—	4	1	no	—
123	C₂×C₃×C₄	D₃×D₄	1	1	yes	characteristic
124	C₂×C₃×C₄	D₃×D₄	1	1	yes	characteristic
125	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
126	C₂×Dic₃	—	12	1	no	—
127	C₂×Dic₃	—	12	1	no	—
128	Dic₆	—	12	2	no	—
129	Dic₆	—	12	2	no	—
130	C₃×C₈	—	2	2	no	—
131	C₃⋊C₈	—	3	4	no	—
132	C₃⋊C₈	—	3	4	no	—
133	C₂×C₉	#492	1	1	yes	characteristic
134	C₂×C₉	#494	1	1	yes	characteristic
135	C₂×C₉	#495	1	1	yes	characteristic
136	C₂×Q₈	—	2	1	no	—
137	C₂×Q₈	—	2	1	no	—
138	C₂×Q₈	—	18	1	no	—
139	C₂×Q₈	—	18	1	no	—
140	Q₁₆	—	36	4	no	—
141	C₂×C₈	—	9	2	no	—
142	C₂×C₈	—	9	2	no	—
143	C₂×C₈	#611	1	2	yes	—
144	C₄⋊C₄	—	2	1	no	—
145	C₄⋊C₄	—	2	1	no	—
146	C₄⋊C₄	—	2	1	no	—
147	C₄⋊C₄	—	2	1	no	—
148	C₄⋊C₄	—	18	1	no	—
149	C₄⋊C₄	—	18	1	no	—
150	C₄⋊C₄	—	18	1	no	—
151	C₄⋊C₄	—	18	1	no	—
152	C₄²	C₂²×D₉	1	1	yes	characteristic
153	C₂²×C₃	#940	1	1	yes	characteristic
154	C₃×C₄	—	8	1	no	—
155	C₃×C₄	—	8	1	no	—
156	C₃×C₄	D₃×SD₁₆	1	2	yes	—
157	C₃×C₄	D₃×Q₁₆	1	2	yes	—
158	C₃×C₄	#950	1	2	yes	—
159	Dic₃	—	24	1	no	—
160	Dic₃	—	24	1	no	—
161	C₉	#1707	1	1	yes	characteristic, complemented, Sylow
162	Q₈	—	4	2	no	—
163	Q₈	—	4	2	no	—
164	Q₈	—	36	2	no	—
165	Q₈	—	36	2	no	—
166	C₂×C₄	—	4	1	no	—
167	C₂×C₄	—	4	1	no	—
168	C₂×C₄	—	36	1	no	—
169	C₂×C₄	—	36	1	no	—
170	C₂×C₄	D₄×D₉	1	1	yes	characteristic
171	C₂×C₄	D₄×D₉	1	1	yes	characteristic
172	C₂×C₄	C₂×#611	1	1	yes	characteristic
173	C₈	—	2	2	no	—
174	C₈	—	9	4	no	—
175	C₈	—	9	4	no	—
176	C₂×C₃	#5059	1	1	yes	characteristic
177	C₂×C₃	#5175	1	1	yes	characteristic
178	C₂×C₃	#5183	1	1	yes	characteristic
179	C₂²	#63502	1	1	yes	characteristic
180	C₄	—	8	1	no	—
181	C₄	—	8	1	no	—
182	C₄	—	72	1	no	—
183	C₄	—	72	1	no	—
184	C₄	SD₁₆×D₉	1	2	yes	—
185	C₄	Q₁₆×D₉	1	2	yes	—
186	C₄	#63513	1	2	yes	—
187	C₃	#71212	1	1	yes	characteristic
188	C₂	#10588740	1	1	yes	characteristic
189	C₂	#10588855	1	1	yes	characteristic
190	C₂	#10588863	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₉×#487

C₂

yes

maximal, characteristic

#10588765

C₂

yes

maximal, characteristic

#10588760

C₂

yes

maximal, characteristic

#10588760

C₂

yes

maximal, characteristic

#10588758

C₂

yes

maximal, characteristic

#10588361

C₂

yes

maximal, characteristic

C₈×C₉⋊C₈

C₂

yes

maximal, characteristic

#71212

—

maximal

C₉×C₄⋊Q₈

C₂²

yes

characteristic

C₉×C₄⋊Q₈

C₂²

yes

characteristic

C₂×#3674

—

#63431

C₂²

yes

characteristic

#63431

C₂²

yes

characteristic

C₉×#103

—

C₄×C₈×C₉

C₂²

yes

—

—

—

—

C₄×C₉⋊C₈

C₂²

yes

characteristic

C₄×C₉⋊C₈

C₂²

yes

characteristic

C₃×#487

D₃

yes

—

—

—

—

—

C₈×C₃⋊C₈

—

C₂×Q₈×C₉

—

C₂×Q₈×C₉

—

C₂×Dic₁₈

—

C₂×Dic₁₈

—

C₂×C₈×C₉

D₄

yes

—

C₉×C₄⋊C₄

—

C₉×C₄⋊C₄

—

C₉×C₄⋊C₄

—

C₉×C₄⋊C₄

—

C₄²×C₉

C₂³

yes

characteristic

#3674

—

C₄⋊Dic₉

—

C₄⋊Dic₉

—

C₄⋊Dic₉

—

C₄⋊Dic₉

—

C₂×C₉⋊C₈

D₄

yes

—

C₂×C₉⋊C₈

D₄

yes

—

#1707

—

Sylow

C₃×C₄⋊Q₈

C₂×D₃

yes

characteristic

C₃×C₄⋊Q₈

C₂×D₃

yes

—

—

—

—

C₃×C₄×C₈

C₂×D₃

yes

—

—

—

—

C₄×C₃⋊C₈

—

C₄×C₃⋊C₈

—

Q₈×C₉

—

Q₈×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

C₂×D₄

yes

characteristic

C₂×C₄×C₉

C₂×D₄

yes

characteristic

C₂×C₄×C₉

C₂×D₄

yes

characteristic

C₂×Dic₉

—

C₂×Dic₉

—

Dic₁₈

—

Dic₁₈

—

C₈×C₉

—

C₉⋊C₈

SD₁₆

yes

—

C₉⋊C₈

Q₁₆

yes

—

—

—

—

—

D₉

yes

characteristic

C₈²

—

C₂×C₃×Q₈

—

C₂×C₃×Q₈

—

C₂×Dic₆

—

C₂×Dic₆

—

C₂×C₃×C₈

#67

yes

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄²

C₂²×D₃

yes

characteristic

#216

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂²×C₉

#127

yes

characteristic

C₄×C₉

—

100

C₄×C₉

—

101

C₄×C₉

C₂×SD₁₆

yes

—

102

C₄×C₉

C₂×Q₁₆

yes

—

103

C₄×C₉

#135

yes

—

104

Dic₉

—

105

Dic₉

—

106

C₂×Q₁₆

—

107

C₄⋊Q₈

—

108

C₄⋊Q₈

—

109

C₄⋊Q₈

C₂×D₉

yes

characteristic

110

C₄⋊Q₈

C₂×D₉

yes

characteristic

111

#107

—

112

#106

—

113

#106

—

114

#103

—

115

#103

—

116

C₄×C₈

—

117

C₄×C₈

—

118

C₄×C₈

C₂×D₉

yes

characteristic

119

C₃×Q₈

—

120

C₃×Q₈

—

121

C₂×C₃×C₄

—

122

C₂×C₃×C₄

—

123

C₂×C₃×C₄

D₃×D₄

yes

characteristic

124

C₂×C₃×C₄

D₃×D₄

yes

characteristic

125

C₂×C₃×C₄

C₂×#67

yes

characteristic

126

C₂×Dic₃

—

127

C₂×Dic₃

—

128

Dic₆

—

129

Dic₆

—

130

C₃×C₈

—

131

C₃⋊C₈

—

132

C₃⋊C₈

—

133

C₂×C₉

#492

yes

characteristic

134

C₂×C₉

#494

yes

characteristic

135

C₂×C₉

#495

yes

characteristic

136

C₂×Q₈

—

137

C₂×Q₈

—

138

C₂×Q₈

—

139

C₂×Q₈

—

140

Q₁₆

—

141

C₂×C₈

—

142

C₂×C₈

—

143

C₂×C₈

#611

yes

—

144

C₄⋊C₄

—

145

C₄⋊C₄

—

146

C₄⋊C₄

—

147

C₄⋊C₄

—

148

C₄⋊C₄

—

149

C₄⋊C₄

—

150

C₄⋊C₄

—

151

C₄⋊C₄

—

152

C₄²

C₂²×D₉

yes

characteristic

153

C₂²×C₃

#940

yes

characteristic

154

C₃×C₄

—

155

C₃×C₄

—

156

C₃×C₄

D₃×SD₁₆

yes

—

157

C₃×C₄

D₃×Q₁₆

yes

—

158

C₃×C₄

#950

yes

—

159

Dic₃

—

160

Dic₃

—

161

C₉

#1707

yes

characteristic, complemented, Sylow

162

Q₈

—

163

Q₈

—

164

Q₈

—

165

Q₈

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

D₄×D₉

yes

characteristic

171

C₂×C₄

D₄×D₉

yes

characteristic

172

C₂×C₄

C₂×#611

yes

characteristic

173

C₈

—

174

C₈

—

175

C₈

—

176

C₂×C₃

#5059

yes

characteristic

177

C₂×C₃

#5175

yes

characteristic

178

C₂×C₃

#5183

yes

characteristic

179

C₂²

#63502

yes

characteristic

180

C₄

—

181

C₄

—

182

C₄

—

183

C₄

—

184

C₄

SD₁₆×D₉

yes

—

185

C₄

Q₁₆×D₉

yes

—

186

C₄

#63513

yes

—

187

C₃

#71212

yes

characteristic

188

C₂

#10588740

yes

characteristic

189

C₂

#10588855

yes

characteristic

190

C₂

#10588863

yes

characteristic

	subgroup	G/H
center	C₂²	#63502
commutator, cosocle	C₄²×C₉	C₂³
Frattini	C₃×C₄²	C₂²×D₃
Fitting	C₉×#487	C₂
socle, prime radical	C₂²×C₃	#940
hypercenter	#487	D₉
hypocenter	C₉	#1707
LCS	G ▷ C₄²×C₉ ▷ C₂²×C₉ ▷ C₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #487

subgroup

G/H

center

C₂²

#63502

commutator, cosocle

C₄²×C₉

C₂³

Frattini

C₃×C₄²

C₂²×D₃

C₂

C₂²×C₃

D₉

C₉

G ▷ C₄²×C₉ ▷ C₂²×C₉ ▷ C₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #487

N	H ≅ G/N	type
73. C₉⋊C₈	Q₁₆	semidirect
161. C₉	#1707	semidirect

H ≅ G/N

type

73. C₉⋊C₈

Q₁₆

semidirect

161. C₉

#1707

semidirect

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

subgroup	index
C₉×#487	2
#10588765	2
#10588760	2
#10588758	2
#10588361	2
C₈×C₉⋊C₈	2
#71212	3

subgroup

index

C₈×C₉⋊C₈

Subgroups — #11782469

Conjugacy classes of subgroups

Complements

Maximal subgroups