Group #11785510 — subgroups

Subgroups — #11785510

Order 1152.

378 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 104 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×#63469	C₂	1	1	yes	maximal, characteristic
2	#10588814	C₂	1	2	yes	maximal
3	C₂×#63381	C₂	1	2	yes	maximal, complemented
4	C₉×#338	C₂	1	1	yes	maximal, characteristic, complemented
5	C₉⋊#338	C₂	1	1	yes	maximal, characteristic, complemented
6	#68724	—	3	1	no	maximal
7	C₂×#3697	C₂²	1	1	yes	characteristic
8	C₂×C₉×M₁₆	C₂²	1	2	yes	—
9	C₂×C₉×M₁₆	C₂²	1	1	yes	characteristic
10	C₂×#3667	C₂²	1	2	yes	—
11	C₂²×C₉⋊C₈	C₂²	1	1	yes	characteristic
12	#63469	—	2	4	no	—
13	C₂×#3663	C₄	1	2	yes	—
14	C₂×C₈×D₉	C₄	1	2	yes	—
15	#63383	—	2	2	no	—
16	#63382	—	2	2	no	—
17	#63381	—	2	4	no	—
18	C₂×#907	—	3	1	no	—
19	#5132	—	3	2	no	—
20	C₂×#822	—	3	2	no	—
21	C₃×#338	D₃	1	1	yes	characteristic
22	C₃⋊#338	—	3	1	no	—
23	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
24	C₂×#611	Q₈	1	2	yes	—
25	#3697	—	2	2	no	—
26	#3697	—	2	2	no	—
27	C₂×D₃₆	D₄	1	1	yes	characteristic
28	C₂×C₄×D₉	C₂×C₄	1	2	yes	—
29	C₂×Dic₁₈	D₄	1	1	yes	characteristic
30	C₉×M₁₆	—	2	2	no	—
31	C₉×M₁₆	—	2	2	no	—
32	C₉×M₁₆	—	2	2	no	—
33	C₂×C₈×C₉	—	2	2	no	—
34	C₂×C₈×C₉	C₂×C₄	1	2	yes	—
35	#3667	—	2	2	no	—
36	#3667	—	2	2	no	—
37	C₂×C₉⋊C₈	—	2	2	no	—
38	C₂×C₉⋊C₈	C₂×C₄	1	1	yes	characteristic
39	C₂×C₉⋊C₈	C₂×C₄	1	1	yes	characteristic
40	C₂×C₉⋊C₈	D₄	1	4	yes	—
41	#3663	—	2	4	no	—
42	C₈×D₉	—	2	4	no	—
43	#1813	—	9	1	no	Sylow
44	C₂×#235	—	3	1	no	—
45	C₂×C₃×M₁₆	C₂×D₃	1	2	yes	—
46	C₂×C₃×M₁₆	C₂×D₃	1	1	yes	characteristic
47	C₂×#208	—	3	2	no	—
48	C₂²×C₃⋊C₈	—	3	1	no	—
49	#907	—	6	4	no	—
50	C₂×ΓL(1,𝔽₂₅)	—	3	2	no	—
51	C₂×D₃×C₈	—	3	2	no	—
52	#824	—	6	2	no	—
53	#823	—	6	2	no	—
54	#822	—	6	4	no	—
55	C₂³×C₉	C₂×Q₈	1	1	yes	characteristic
56	C₂²×D₉	C₄⋊C₄	1	2	yes	—
57	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
59	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
60	C₂×C₄×C₉	#41	1	2	yes	—
61	#611	—	2	4	no	—
62	C₂×Dic₉	C₄⋊C₄	1	2	yes	—
63	D₃₆	—	2	2	no	—
64	C₄×D₉	—	2	2	no	—
65	C₄×D₉	—	2	2	no	—
66	Dic₁₈	—	2	2	no	—
67	C₈×C₉	—	2	2	no	—
68	C₈×C₉	—	4	2	no	—
69	C₉⋊C₈	—	4	2	no	—
70	C₉⋊C₈	C₂²⋊C₄	1	2	yes	—
71	C₉⋊C₈	C₂²⋊C₄	1	2	yes	—
72	C₂×#131	—	9	1	no	—
73	C₂×#108	—	9	2	no	—
74	#413	—	9	2	no	—
75	#338	—	9	1	no	—
76	#338	D₉	1	1	yes	characteristic, complemented
77	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
78	C₂×#67	—	3	2	no	—
79	#235	—	6	2	no	—
80	#235	—	6	2	no	—
81	C₂×D₁₂	—	3	1	no	—
82	C₂×C₄×D₃	—	3	2	no	—
83	C₂×Dic₆	—	3	1	no	—
84	C₃×M₁₆	—	2	2	no	—
85	C₃×M₁₆	—	2	2	no	—
86	C₃×M₁₆	—	2	2	no	—
87	C₂×C₃×C₈	—	2	2	no	—
88	C₂×C₃×C₈	C₄×D₃	1	2	yes	—
89	#208	—	6	2	no	—
90	#208	—	6	2	no	—
91	C₂×C₃⋊C₈	—	3	1	no	—
92	C₂×C₃⋊C₈	—	3	1	no	—
93	C₂×C₃⋊C₈	—	3	4	no	—
94	C₂×C₃⋊C₈	—	6	2	no	—
95	ΓL(1,𝔽₂₅)	—	6	4	no	—
96	D₃×C₈	—	6	4	no	—
97	C₂²×C₉	—	4	1	no	—
98	C₂²×C₉	C₂×C₄⋊C₄	1	1	yes	characteristic
99	C₂²×C₉	C₂²⋊Q₈	1	2	yes	—
100	C₂×D₉	—	2	2	no	—
101	C₂×D₉	—	4	2	no	—
102	C₄×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
103	C₄×C₉	#117	1	1	yes	characteristic
104	C₄×C₉	#121	1	2	yes	—
105	Dic₉	—	2	2	no	—
106	C₂×#41	—	9	1	no	—
107	#131	—	18	4	no	—
108	C₂×M₁₆	—	9	2	no	—
109	C₂×M₁₆	—	9	2	no	—
110	C₂×M₁₆	C₂×D₉	1	2	yes	—
111	C₂×M₁₆	C₂×D₉	1	1	yes	characteristic
112	C₂²×C₈	—	9	1	no	—
113	C₂²×C₈	—	9	2	no	—
114	#108	—	18	4	no	—
115	#101	—	18	2	no	—
116	#100	—	18	2	no	—
117	C₂³×C₃	D₃×Q₈	1	1	yes	characteristic
118	C₂²×D₃	—	3	2	no	—
119	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
120	C₂×C₃×C₄	#235	1	2	yes	—
121	C₂×C₃×C₄	D₃×D₄	1	2	yes	—
122	C₂×C₃×C₄	D₃×D₄	1	1	yes	characteristic
123	#67	—	6	4	no	—
124	C₂×Dic₃	—	3	2	no	—
125	D₁₂	—	6	2	no	—
126	C₄×D₃	—	6	2	no	—
127	C₄×D₃	—	6	2	no	—
128	Dic₆	—	6	2	no	—
129	C₃×C₈	—	2	2	no	—
130	C₃×C₈	—	4	2	no	—
131	C₃⋊C₈	—	3	2	no	—
132	C₃⋊C₈	—	3	2	no	—
133	C₃⋊C₈	—	12	2	no	—
134	C₂×C₉	—	2	1	no	—
135	C₂×C₉	—	2	2	no	—
136	C₂×C₉	#380	1	1	yes	characteristic
137	D₉	—	4	2	no	—
138	#41	—	18	2	no	—
139	#41	—	18	2	no	—
140	C₂×Q₈	—	9	1	no	—
141	C₂×D₄	—	9	2	no	—
142	C₂×D₄	—	9	1	no	—
143	C₂²×C₄	—	9	2	no	—
144	C₂²×C₄	C₂²×D₉	1	1	yes	characteristic
145	M₁₆	—	2	2	no	—
146	M₁₆	—	2	2	no	—
147	M₁₆	—	2	2	no	—
148	M₁₆	—	18	4	no	—
149	M₁₆	—	18	2	no	—
150	M₁₆	—	18	2	no	—
151	C₂×C₈	—	2	2	no	—
152	C₂×C₈	—	9	1	no	—
153	C₂×C₈	—	9	1	no	—
154	C₂×C₈	—	9	4	no	—
155	C₂×C₈	—	18	4	no	—
156	C₂×C₈	—	18	2	no	—
157	C₂×C₈	C₄×D₉	1	2	yes	—
158	C₂²×C₃	—	4	1	no	—
159	C₂²×C₃	D₃×C₄⋊C₄	1	1	yes	characteristic
160	C₂²×C₃	#896	1	2	yes	—
161	C₂×D₃	—	6	2	no	—
162	C₂×D₃	—	12	2	no	—
163	C₃×C₄	#872	1	1	yes	characteristic
164	C₃×C₄	D₃×C₂²⋊C₄	1	1	yes	characteristic
165	C₃×C₄	#884	1	2	yes	—
166	Dic₃	—	6	2	no	—
167	C₉	#1813	1	1	yes	characteristic, complemented, Sylow
168	C₂³	—	9	2	no	—
169	C₂³	Q₈×D₉	1	1	yes	characteristic
170	Q₈	—	18	2	no	—
171	D₄	—	18	4	no	—
172	D₄	—	18	2	no	—
173	C₂×C₄	—	9	2	no	—
174	C₂×C₄	—	18	2	no	—
175	C₂×C₄	—	18	2	no	—
176	C₂×C₄	C₂×C₄×D₉	1	1	yes	characteristic
177	C₂×C₄	#3697	1	2	yes	—
178	C₂×C₄	D₄×D₉	1	2	yes	—
179	C₂×C₄	D₄×D₉	1	1	yes	characteristic
180	C₈	—	2	2	no	—
181	C₈	—	4	2	no	—
182	C₈	—	9	2	no	—
183	C₈	—	9	2	no	—
184	C₈	—	36	2	no	—
185	C₂×C₃	—	2	1	no	—
186	C₂×C₃	—	2	2	no	—
187	C₂×C₃	#4667	1	1	yes	characteristic
188	D₃	—	12	2	no	—
189	C₂²	—	4	1	no	—
190	C₂²	—	18	2	no	—
191	C₂²	—	36	2	no	—
192	C₂²	C₄⋊C₄×D₉	1	1	yes	characteristic
193	C₂²	#63458	1	2	yes	—
194	C₄	—	18	2	no	—
195	C₄	#63434	1	1	yes	characteristic
196	C₄	C₂²⋊C₄×D₉	1	1	yes	characteristic
197	C₄	#63446	1	2	yes	—
198	C₃	#68724	1	1	yes	characteristic
199	C₂	—	2	1	no	—
200	C₂	—	2	2	no	—
201	C₂	—	36	2	no	—
202	C₂	#10588347	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#63469

C₂

yes

maximal, characteristic

#10588814

C₂

yes

maximal

C₂×#63381

C₂

yes

maximal, complemented

C₉×#338

C₂

yes

maximal, characteristic, complemented

C₉⋊#338

C₂

yes

maximal, characteristic, complemented

#68724

—

maximal

C₂×#3697

C₂²

yes

characteristic

C₂×C₉×M₁₆

C₂²

yes

—

C₂×C₉×M₁₆

C₂²

yes

characteristic

C₂×#3667

C₂²

yes

—

C₂²×C₉⋊C₈

C₂²

yes

characteristic

#63469

—

C₂×#3663

C₄

yes

—

C₂×C₈×D₉

C₄

yes

—

—

—

—

—

—

—

D₃

yes

characteristic

C₃⋊#338

—

C₂²×C₄×C₉

C₂³

yes

characteristic

C₂×#611

Q₈

yes

—

#3697

—

#3697

—

C₂×D₃₆

D₄

yes

characteristic

C₂×C₄×D₉

C₂×C₄

yes

—

C₂×Dic₁₈

D₄

yes

characteristic

C₉×M₁₆

—

C₉×M₁₆

—

C₉×M₁₆

—

C₂×C₈×C₉

—

C₂×C₈×C₉

C₂×C₄

yes

—

#3667

—

#3667

—

C₂×C₉⋊C₈

—

C₂×C₉⋊C₈

C₂×C₄

yes

characteristic

C₂×C₉⋊C₈

C₂×C₄

yes

characteristic

C₂×C₉⋊C₈

D₄

yes

—

#3663

—

C₈×D₉

—

#1813

—

Sylow

C₂×#235

—

C₂×C₃×M₁₆

C₂×D₃

yes

—

C₂×C₃×M₁₆

C₂×D₃

yes

characteristic

C₂×#208

—

C₂²×C₃⋊C₈

—

#907

—

C₂×ΓL(1,𝔽₂₅)

—

C₂×D₃×C₈

—

#824

—

#823

—

#822

—

C₂³×C₉

C₂×Q₈

yes

characteristic

C₂²×D₉

C₄⋊C₄

yes

—

C₂×C₄×C₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₉

C₂×D₄

yes

—

C₂×C₄×C₉

C₂×D₄

yes

characteristic

C₂×C₄×C₉

#41

yes

—

#611

—

C₂×Dic₉

C₄⋊C₄

yes

—

D₃₆

—

C₄×D₉

—

C₄×D₉

—

Dic₁₈

—

C₈×C₉

—

C₈×C₉

—

C₉⋊C₈

—

C₉⋊C₈

C₂²⋊C₄

yes

—

C₉⋊C₈

C₂²⋊C₄

yes

—

—

—

—

—

D₉

yes

characteristic, complemented

C₂²×C₃×C₄

C₂²×D₃

yes

characteristic

C₂×#67

—

#235

—

#235

—

C₂×D₁₂

—

C₂×C₄×D₃

—

C₂×Dic₆

—

C₃×M₁₆

—

C₃×M₁₆

—

C₃×M₁₆

—

C₂×C₃×C₈

—

C₂×C₃×C₈

C₄×D₃

yes

—

#208

—

#208

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

ΓL(1,𝔽₂₅)

—

D₃×C₈

—

C₂²×C₉

—

C₂²×C₉

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₉

C₂²⋊Q₈

yes

—

100

C₂×D₉

—

101

C₂×D₉

—

102

C₄×C₉

C₂×C₂²⋊C₄

yes

characteristic

103

C₄×C₉

#117

yes

characteristic

104

C₄×C₉

#121

yes

—

105

Dic₉

—

106

C₂×#41

—

107

#131

—

108

C₂×M₁₆

—

109

C₂×M₁₆

—

110

C₂×M₁₆

C₂×D₉

yes

—

111

C₂×M₁₆

C₂×D₉

yes

characteristic

112

C₂²×C₈

—

113

C₂²×C₈

—

114

#108

—

115

#101

—

116

#100

—

117

C₂³×C₃

D₃×Q₈

yes

characteristic

118

C₂²×D₃

—

119

C₂×C₃×C₄

C₂×C₄×D₃

yes

characteristic

120

C₂×C₃×C₄

#235

yes

—

121

C₂×C₃×C₄

D₃×D₄

yes

—

122

C₂×C₃×C₄

D₃×D₄

yes

characteristic

123

#67

—

124

C₂×Dic₃

—

125

D₁₂

—

126

C₄×D₃

—

127

C₄×D₃

—

128

Dic₆

—

129

C₃×C₈

—

130

C₃×C₈

—

131

C₃⋊C₈

—

132

C₃⋊C₈

—

133

C₃⋊C₈

—

134

C₂×C₉

—

135

C₂×C₉

—

136

C₂×C₉

#380

yes

characteristic

137

D₉

—

138

#41

—

139

#41

—

140

C₂×Q₈

—

141

C₂×D₄

—

142

C₂×D₄

—

143

C₂²×C₄

—

144

C₂²×C₄

C₂²×D₉

yes

characteristic

145

M₁₆

—

146

M₁₆

—

147

M₁₆

—

148

M₁₆

—

149

M₁₆

—

150

M₁₆

—

151

C₂×C₈

—

152

C₂×C₈

—

153

C₂×C₈

—

154

C₂×C₈

—

155

C₂×C₈

—

156

C₂×C₈

—

157

C₂×C₈

C₄×D₉

yes

—

158

C₂²×C₃

—

159

C₂²×C₃

D₃×C₄⋊C₄

yes

characteristic

160

C₂²×C₃

#896

yes

—

161

C₂×D₃

—

162

C₂×D₃

—

163

C₃×C₄

#872

yes

characteristic

164

C₃×C₄

D₃×C₂²⋊C₄

yes

characteristic

165

C₃×C₄

#884

yes

—

166

Dic₃

—

167

C₉

#1813

yes

characteristic, complemented, Sylow

168

C₂³

—

169

C₂³

Q₈×D₉

yes

characteristic

170

Q₈

—

171

D₄

—

172

D₄

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

C₂×C₄×D₉

yes

characteristic

177

C₂×C₄

#3697

yes

—

178

C₂×C₄

D₄×D₉

yes

—

179

C₂×C₄

D₄×D₉

yes

characteristic

180

C₈

—

181

C₈

—

182

C₈

—

183

C₈

—

184

C₈

—

185

C₂×C₃

—

186

C₂×C₃

—

187

C₂×C₃

#4667

yes

characteristic

188

D₃

—

189

C₂²

—

190

C₂²

—

191

C₂²

—

192

C₂²

C₄⋊C₄×D₉

yes

characteristic

193

C₂²

#63458

yes

—

194

C₄

—

195

C₄

#63434

yes

characteristic

196

C₄

C₂²⋊C₄×D₉

yes

characteristic

197

C₄

#63446

yes

—

198

C₃

#68724

yes

characteristic

199

C₂

—

200

C₂

—

201

C₂

—

202

C₂

#10588347

yes

characteristic

	subgroup	G/H
center	C₄	#63434
commutator	C₂×C₄×C₉	C₂²×C₄
Frattini	C₂²×C₃×C₄	C₂²×D₃
Fitting	C₉×#338	C₂
socle	C₂×C₃	#4667
prime radical	C₂×#3697	C₂²
cosocle	C₂²×C₄×C₉	C₂³
hypercenter	#338	D₉
hypocenter	C₉	#1813
LCS	G ▷ C₂×C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ #338

subgroup

G/H

center

C₄

#63434

commutator

C₂×C₄×C₉

C₂²×C₄

Frattini

C₂²×C₃×C₄

C₂²×D₃

C₂

C₂×C₃

C₂²

C₂²×C₄×C₉

C₂³

D₉

C₉

G ▷ C₂×C₄×C₉ ▷ C₂×C₉ ▷ C₉

UCS

C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ #338

N	H ≅ G/N	type
3. C₂×#63381	C₂	semidirect
4. C₉×#338	C₂	semidirect
5. C₉⋊#338	C₂	semidirect
76. #338	D₉	semidirect
167. C₉	#1813	semidirect

H ≅ G/N

type

3. C₂×#63381

C₂

semidirect

4. C₉×#338

C₂

semidirect

5. C₉⋊#338

C₂

semidirect

76. #338

D₉

semidirect

167. C₉

#1813

semidirect

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

subgroup	index
C₂×#63469	2
#10588814	2
C₂×#63381	2
C₉×#338	2
C₉⋊#338	2
#68724	3

subgroup

index

Subgroups — #11785510

Conjugacy classes of subgroups

Complements

Maximal subgroups