Group #36709741 — subgroups

Subgroups — C₂×#26022494 #36709741

Order 22784.

686 conjugacy classes of subgroups; only the 203 normal subgroups are shown.

normal subgroups only

This group has 686 conjugacy classes of subgroups — too many to draw. Only its 203 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×#19152000	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#19151975	C₂	1	1	yes	maximal, characteristic
3	#26022494	C₂	1	8	yes	maximal, direct product factor
4	C₂×#19151400	C₂	1	4	yes	maximal, complemented
5	C₂×#346×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₇₁₂	—	2	1	no	—
7	C₂²×D₇₁₂	C₂²	1	2	yes	—
8	C₂×#15668009	—	2	1	no	—
9	C₂×#15668004	—	2	1	no	—
10	#19152000	—	2	2	no	—
11	#19152000	C₂²	1	4	yes	complemented
12	#19151975	—	2	2	no	—
13	#19151975	C₂²	1	4	yes	—
14	C₂×#15667946	C₂²	1	2	yes	—
15	C₂×#15667944	—	2	1	no	—
16	#19151400	C₂²	1	16	yes	complemented
17	C₂²×C₁₆×C₈₉	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
19	#346×C₈₉	C₂²	1	4	yes	complemented
20	C₂²×D₃₅₆	—	2	2	no	—
21	C₂²×D₃₅₆	—	4	2	no	—
22	C₂×D₇₁₂	—	2	2	no	—
23	C₂×D₇₁₂	—	2	4	no	—
24	C₂×D₇₁₂	—	2	8	no	—
25	C₂×D₇₁₂	—	4	4	no	—
26	C₂×D₇₁₂	C₂³	1	4	yes	—
27	#15668009	—	2	4	no	—
28	#15668009	—	4	2	no	—
29	#15668004	—	2	4	no	—
30	#15668004	—	4	2	no	—
31	C₂×C₄⋊Dic₈₉	—	2	2	no	—
32	C₂×#13083553	—	4	2	no	—
33	#15667946	—	2	2	no	—
34	#15667946	C₂³	1	4	yes	—
35	#15667944	—	2	4	no	—
36	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
37	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
38	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
39	C₂×C₁₆×C₈₉	—	2	4	no	—
40	C₂×C₁₆×C₈₉	C₂³	1	4	yes	—
41	C₄×C₈×C₈₉	D₄	1	2	yes	—
42	C₄×C₈×C₈₉	C₂³	1	2	yes	—
43	C₂³×D₈₉	—	4	2	no	—
44	C₂×D₃₅₆	—	2	4	no	—
45	C₂×D₃₅₆	—	4	4	no	—
46	C₂×D₃₅₆	—	4	4	no	—
47	C₂×D₃₅₆	—	4	8	no	—
48	C₂×D₃₅₆	—	4	2	no	—
49	C₂×D₃₅₆	—	8	8	no	—
50	C₂²×Dic₈₉	—	4	2	no	—
51	D₇₁₂	—	2	8	no	—
52	D₇₁₂	—	4	4	no	—
53	D₇₁₂	—	4	8	no	—
54	C₄⋊Dic₈₉	—	2	4	no	—
55	C₄⋊Dic₈₉	—	4	2	no	—
56	#13083553	—	4	8	no	—
57	C₂²×C₄×C₈₉	D₈	1	2	yes	—
58	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
59	C₂×C₈×C₈₉	SD₁₆	1	4	yes	—
60	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
61	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
62	C₂×C₈×C₈₉	#41	1	4	yes	—
63	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
64	C₄²×C₈₉	D₈	1	2	yes	—
65	C₄²×C₈₉	C₂×D₄	1	2	yes	—
66	C₁₆×C₈₉	—	2	4	no	—
67	C₂³×C₈₉	C₂×D₈	1	1	yes	characteristic
68	C₂²×D₈₉	—	4	4	no	—
69	C₂²×D₈₉	—	8	4	no	—
70	C₂²×D₈₉	—	8	8	no	—
71	C₂×C₄×C₈₉	—	2	4	no	—
72	C₂×C₄×C₈₉	#124	1	2	yes	—
73	C₂×C₄×C₈₉	C₂×D₈	1	4	yes	—
74	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
75	C₂×C₄×C₈₉	C₂×SD₁₆	1	2	yes	—
76	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
77	C₂×Dic₈₉	—	4	4	no	—
78	C₂×Dic₈₉	—	8	4	no	—
79	D₃₅₆	—	4	4	no	—
80	D₃₅₆	—	4	4	no	—
81	D₃₅₆	—	8	16	no	—
82	D₃₅₆	—	8	4	no	—
83	C₈×C₈₉	#124	1	2	yes	—
84	C₈×C₈₉	C₂×SD₁₆	1	4	yes	—
85	C₈×C₈₉	C₂×#41	1	2	yes	—
86	C₂²×C₈₉	#486	1	2	yes	—
87	C₂²×C₈₉	#509	1	4	yes	—
88	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
89	C₂×D₈₉	—	8	4	no	—
90	C₂×D₈₉	—	8	8	no	—
91	C₂×D₈₉	—	16	8	no	—
92	C₄×C₈₉	—	2	4	no	—
93	C₄×C₈₉	#486	1	2	yes	—
94	C₄×C₈₉	C₂×#124	1	1	yes	characteristic
95	C₄×C₈₉	C₂²×SD₁₆	1	1	yes	characteristic
96	Dic₈₉	—	8	4	no	—
97	C₂×#2244	—	89	1	no	maximal, Sylow
98	C₂×C₈₉	#2244	1	4	yes	complemented
99	C₂×C₈₉	C₂×#486	1	1	yes	characteristic
100	C₂×C₈₉	C₂×#509	1	2	yes	—
101	D₈₉	—	16	4	no	—
102	C₂×#499	—	89	1	no	—
103	C₂×#493	—	89	1	no	—
104	#2244	—	89	8	no	—
105	C₂×#357	—	89	4	no	—
106	C₂×#346	D₈₉	1	1	yes	characteristic, complemented
107	C₈₉	C₂×#2244	1	1	yes	characteristic, complemented, Sylow
108	C₂²×D₈	—	89	2	no	—
109	C₂²×D₈	—	178	1	no	—
110	C₂×#127	—	178	1	no	—
111	C₂×#125	—	178	1	no	—
112	C₂²×C₁₆	C₂×D₈₉	1	2	yes	—
113	#499	—	89	4	no	—
114	#499	—	178	2	no	—
115	#493	—	89	4	no	—
116	#493	—	178	2	no	—
117	C₂×#107	—	89	2	no	—
118	C₂×#106	—	178	1	no	—
119	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
120	#357	—	89	16	no	—
121	#346	C₂×D₈₉	1	4	yes	complemented
122	C₂²×D₄	—	356	2	no	—
123	C₂²×D₄	—	178	2	no	—
124	C₂×D₈	—	89	4	no	—
125	C₂×D₈	—	356	4	no	—
126	C₂×D₈	—	178	2	no	—
127	C₂×D₈	—	178	4	no	—
128	C₂×D₈	—	178	8	no	—
129	C₂²×C₈	D₃₅₆	1	1	yes	characteristic
130	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
131	#127	—	356	2	no	—
132	#127	—	178	4	no	—
133	#125	—	356	2	no	—
134	#125	—	178	4	no	—
135	C₂×C₄⋊C₄	—	356	2	no	—
136	C₂×C₄⋊C₄	—	178	2	no	—
137	C₂×C₄²	D₃₅₆	1	1	yes	characteristic
138	C₂×C₁₆	—	2	4	no	—
139	C₂×C₁₆	C₂²×D₈₉	1	4	yes	—
140	#107	—	89	4	no	—
141	#107	—	178	2	no	—
142	#106	—	178	4	no	—
143	C₄×C₈	D₃₅₆	1	2	yes	—
144	C₄×C₈	C₂²×D₈₉	1	2	yes	—
145	C₂⁴	—	356	2	no	—
146	C₂×D₄	—	356	4	no	—
147	C₂×D₄	—	356	2	no	—
148	C₂×D₄	—	356	8	no	—
149	C₂×D₄	—	356	4	no	—
150	C₂×D₄	—	178	4	no	—
151	C₂×D₄	—	712	8	no	—
152	C₂²×C₄	—	356	2	no	—
153	C₂²×C₄	D₇₁₂	1	2	yes	—
154	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
155	D₈	—	356	4	no	—
156	D₈	—	356	8	no	—
157	D₈	—	178	8	no	—
158	C₂×C₈	#13083563	1	4	yes	—
159	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
160	C₂×C₈	C₂×D₃₅₆	1	2	yes	—
161	C₂×C₈	#13083574	1	4	yes	—
162	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
163	C₄⋊C₄	—	356	8	no	—
164	C₄⋊C₄	—	356	2	no	—
165	C₄⋊C₄	—	178	4	no	—
166	C₄²	D₇₁₂	1	2	yes	—
167	C₄²	C₂×D₃₅₆	1	2	yes	—
168	C₁₆	—	2	4	no	—
169	C₂³	—	356	4	no	—
170	C₂³	—	712	4	no	—
171	C₂³	—	712	8	no	—
172	C₂³	C₂×D₇₁₂	1	1	yes	characteristic
173	D₄	—	356	4	no	—
174	D₄	—	356	4	no	—
175	D₄	—	712	16	no	—
176	D₄	—	712	4	no	—
177	C₂×C₄	—	2	4	no	—
178	C₂×C₄	—	356	4	no	—
179	C₂×C₄	—	712	4	no	—
180	C₂×C₄	#15668000	1	2	yes	—
181	C₂×C₄	C₂×D₇₁₂	1	4	yes	—
182	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
183	C₂×C₄	C₂×#13083563	1	2	yes	—
184	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
185	C₈	#15668000	1	2	yes	—
186	C₈	C₂×#13083563	1	4	yes	—
187	C₈	C₂×#13083574	1	2	yes	—
188	C₂²	—	1424	8	no	—
189	C₂²	—	712	4	no	—
190	C₂²	—	712	8	no	—
191	C₂²	#19151945	1	2	yes	—
192	C₂²	#19152036	1	4	yes	—
193	C₂²	C₂²×D₇₁₂	1	1	yes	characteristic
194	C₄	—	2	4	no	—
195	C₄	—	712	4	no	—
196	C₄	#19151945	1	2	yes	—
197	C₄	C₂×#15668000	1	1	yes	characteristic
198	C₄	C₂²×#13083563	1	1	yes	characteristic
199	C₂	—	1424	4	no	—
200	C₂	#26022494	1	4	yes	direct product factor
201	C₂	C₂×#19151945	1	1	yes	characteristic
202	C₂	C₂×#19152036	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19152000

C₂

yes

maximal, characteristic, complemented

C₂×#19151975

C₂

yes

maximal, characteristic

#26022494

C₂

yes

maximal, direct product factor

C₂×#19151400

C₂

yes

maximal, complemented

C₂×#346×C₈₉

C₂

yes

maximal, characteristic, complemented

C₂²×D₇₁₂

—

C₂²×D₇₁₂

C₂²

yes

—

—

—

—

C₂²

yes

complemented

#19151975

—

#19151975

C₂²

yes

—

C₂×#15667946

C₂²

yes

—

C₂×#15667944

—

#19151400

C₂²

yes

complemented

C₂²×C₁₆×C₈₉

C₂²

yes

—

C₂×C₄×C₈×C₈₉

C₂²

yes

characteristic

#346×C₈₉

C₂²

yes

complemented

C₂²×D₃₅₆

—

C₂²×D₃₅₆

—

C₂×D₇₁₂

—

C₂×D₇₁₂

—

C₂×D₇₁₂

—

C₂×D₇₁₂

—

C₂×D₇₁₂

C₂³

yes

—

—

—

—

—

C₂×C₄⋊Dic₈₉

—

C₂×#13083553

—

#15667946

—

#15667946

C₂³

yes

—

#15667944

—

C₂²×C₈×C₈₉

D₄

yes

characteristic

C₂²×C₈×C₈₉

C₂³

yes

characteristic

C₂×C₄²×C₈₉

D₄

yes

characteristic

C₂×C₁₆×C₈₉

—

C₂×C₁₆×C₈₉

C₂³

yes

—

C₄×C₈×C₈₉

D₄

yes

—

C₄×C₈×C₈₉

C₂³

yes

—

C₂³×D₈₉

—

C₂×D₃₅₆

—

C₂×D₃₅₆

—

C₂×D₃₅₆

—

C₂×D₃₅₆

—

C₂×D₃₅₆

—

C₂×D₃₅₆

—

C₂²×Dic₈₉

—

D₇₁₂

—

D₇₁₂

—

D₇₁₂

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

#13083553

—

C₂²×C₄×C₈₉

D₈

yes

—

C₂²×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₈×C₈₉

SD₁₆

yes

—

C₂×C₈×C₈₉

C₂×D₄

yes

characteristic

C₂×C₈×C₈₉

C₂×D₄

yes

—

C₂×C₈×C₈₉

#41

yes

—

C₂×C₈×C₈₉

C₂⁴

yes

characteristic

C₄²×C₈₉

D₈

yes

—

C₄²×C₈₉

C₂×D₄

yes

—

C₁₆×C₈₉

—

C₂³×C₈₉

C₂×D₈

yes

characteristic

C₂²×D₈₉

—

C₂²×D₈₉

—

C₂²×D₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

#124

yes

—

C₂×C₄×C₈₉

C₂×D₈

yes

—

C₂×C₄×C₈₉

C₂×D₈

yes

characteristic

C₂×C₄×C₈₉

C₂×SD₁₆

yes

—

C₂×C₄×C₈₉

C₂²×D₄

yes

characteristic

C₂×Dic₈₉

—

C₂×Dic₈₉

—

D₃₅₆

—

D₃₅₆

—

D₃₅₆

—

D₃₅₆

—

C₈×C₈₉

#124

yes

—

C₈×C₈₉

C₂×SD₁₆

yes

—

C₈×C₈₉

C₂×#41

yes

—

C₂²×C₈₉

#486

yes

—

C₂²×C₈₉

#509

yes

—

C₂²×C₈₉

C₂²×D₈

yes

characteristic

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₄×C₈₉

—

C₄×C₈₉

#486

yes

—

C₄×C₈₉

C₂×#124

yes

characteristic

C₄×C₈₉

C₂²×SD₁₆

yes

characteristic

Dic₈₉

—

—

C₂×C₈₉

yes

C₂×C₈₉

yes

100

C₂×C₈₉

C₂×#509

yes

—

101

D₈₉

—

102

C₂×#499

—

103

C₂×#493

—

104

#2244

—

105

C₂×#357

—

106

C₂×#346

D₈₉

yes

characteristic, complemented

107

C₈₉

C₂×#2244

yes

characteristic, complemented, Sylow

108

C₂²×D₈

—

109

C₂²×D₈

—

178

—

110

C₂×#127

—

178

—

111

C₂×#125

—

178

—

112

C₂²×C₁₆

C₂×D₈₉

yes

—

113

#499

—

114

#499

—

178

—

115

#493

—

116

#493

—

178

—

117

C₂×#107

—

118

C₂×#106

—

178

—

119

C₂×C₄×C₈

C₂×D₈₉

yes

characteristic

120

#357

—

121

#346

C₂×D₈₉

yes

complemented

122

C₂²×D₄

—

356

—

123

C₂²×D₄

—

178

—

124

C₂×D₈

—

125

C₂×D₈

—

356

—

126

C₂×D₈

—

178

—

127

C₂×D₈

—

178

—

128

C₂×D₈

—

178

—

129

C₂²×C₈

D₃₅₆

yes

characteristic

130

C₂²×C₈

C₂²×D₈₉

yes

characteristic

131

#127

—

356

—

132

#127

—

178

—

133

#125

—

356

—

134

#125

—

178

—

135

C₂×C₄⋊C₄

—

356

—

136

C₂×C₄⋊C₄

—

178

—

137

C₂×C₄²

D₃₅₆

yes

characteristic

138

C₂×C₁₆

—

139

C₂×C₁₆

C₂²×D₈₉

yes

—

140

#107

—

141

#107

—

178

—

142

#106

—

178

—

143

C₄×C₈

D₃₅₆

yes

—

144

C₄×C₈

C₂²×D₈₉

yes

—

145

C₂⁴

—

356

—

146

C₂×D₄

—

356

—

147

C₂×D₄

—

356

—

148

C₂×D₄

—

356

—

149

C₂×D₄

—

356

—

150

C₂×D₄

—

178

—

151

C₂×D₄

—

712

—

152

C₂²×C₄

—

356

—

153

C₂²×C₄

D₇₁₂

yes

—

154

C₂²×C₄

C₂×D₃₅₆

yes

characteristic

155

D₈

—

356

—

156

D₈

—

356

—

157

D₈

—

178

—

158

C₂×C₈

#13083563

yes

—

159

C₂×C₈

C₂×D₃₅₆

yes

characteristic

160

C₂×C₈

C₂×D₃₅₆

yes

—

161

C₂×C₈

#13083574

yes

—

162

C₂×C₈

C₂³×D₈₉

yes

characteristic

163

C₄⋊C₄

—

356

—

164

C₄⋊C₄

—

356

—

165

C₄⋊C₄

—

178

—

166

C₄²

D₇₁₂

yes

—

167

C₄²

C₂×D₃₅₆

yes

—

168

C₁₆

—

169

C₂³

—

356

—

170

C₂³

—

712

—

171

C₂³

—

712

—

172

C₂³

C₂×D₇₁₂

yes

characteristic

173

D₄

—

356

—

174

D₄

—

356

—

175

D₄

—

712

—

176

D₄

—

712

—

177

C₂×C₄

—

178

C₂×C₄

—

356

—

179

C₂×C₄

—

712

—

180

C₂×C₄

#15668000

yes

—

181

C₂×C₄

C₂×D₇₁₂

yes

—

182

C₂×C₄

C₂×D₇₁₂

yes

characteristic

183

C₂×C₄

C₂×#13083563

yes

—

184

C₂×C₄

C₂²×D₃₅₆

yes

characteristic

185

C₈

#15668000

yes

—

186

C₈

C₂×#13083563

yes

—

187

C₈

C₂×#13083574

yes

—

188

C₂²

—

1424

—

189

C₂²

—

712

—

190

C₂²

—

712

—

191

C₂²

#19151945

yes

—

192

C₂²

#19152036

yes

—

193

C₂²

C₂²×D₇₁₂

yes

characteristic

194

C₄

—

195

C₄

—

712

—

196

C₄

#19151945

yes

—

197

C₄

C₂×#15668000

yes

characteristic

198

C₄

C₂²×#13083563

yes

characteristic

199

C₂

—

1424

—

200

C₂

#26022494

yes

direct product factor

201

C₂

C₂×#19151945

yes

characteristic

202

C₂

C₂×#19152036

yes

—

	subgroup	G/H
center	C₂³	C₂×D₇₁₂
commutator, cosocle	C₂×C₈×C₈₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₂×#346×C₈₉	C₂
socle	C₂³×C₈₉	C₂×D₈
prime radical	C₂×#19151975	C₂
hypercenter	C₂×#346	D₈₉
hypocenter	C₈₉	C₂×#2244
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

subgroup

G/H

center

C₂³

C₂×D₇₁₂

commutator, cosocle

C₂×C₈×C₈₉

C₂⁴

Frattini

C₂×C₈

C₂³×D₈₉

Fitting

C₂×#346×C₈₉

C₂

socle

C₂³×C₈₉

C₂×D₈

C₂

D₈₉

C₈₉

G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

N	H ≅ G/N	type
1. C₂×#19152000	C₂	semidirect
3. #26022494	C₂	direct (N × H)
4. C₂×#19151400	C₂	semidirect
5. C₂×#346×C₈₉	C₂	semidirect
11. #19152000	C₂²	semidirect
16. #19151400	C₂²	semidirect
19. #346×C₈₉	C₂²	semidirect
98. C₂×C₈₉	#2244	semidirect
106. C₂×#346	D₈₉	semidirect
107. C₈₉	C₂×#2244	semidirect
121. #346	C₂×D₈₉	semidirect
200. C₂	#26022494	direct (N × H)

H ≅ G/N

type

1. C₂×#19152000

C₂

semidirect

3. #26022494

C₂

direct (N × H)

4. C₂×#19151400

C₂

semidirect

5. C₂×#346×C₈₉

C₂

semidirect

11. #19152000

C₂²

semidirect

16. #19151400

C₂²

semidirect

19. #346×C₈₉

C₂²

semidirect

98. C₂×C₈₉

#2244

semidirect

106. C₂×#346

D₈₉

semidirect

107. C₈₉

C₂×#2244

semidirect

121. #346

C₂×D₈₉

semidirect

200. C₂

#26022494

direct (N × H)

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

subgroup	index
C₂×#19152000	2
C₂×#19151975	2
#26022494	2
C₂×#19151400	2
C₂×#346×C₈₉	2
C₂×#2244	89

subgroup

index

C₂×#346×C₈₉

Subgroups — C₂×#26022494 #36709741

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#26022494 #36709741

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#26022494 #36709741