Group #36712069 — subgroups

Subgroups — #36712069

Order 22784.

562 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 562 conjugacy classes of subgroups — too many to draw. Only its 165 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	Q₈×D₇₁₂	C₂	1	1	yes	maximal, characteristic, complemented
2	#26022475	C₂	1	3	yes	maximal, complemented
3	#26022389	C₂	1	3	yes	maximal, complemented
4	#26022284	C₂	1	3	yes	maximal, complemented
5	Q₈×#13083549	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×#15667953	C₂	1	3	yes	maximal, complemented
7	C₈₉×#3391	C₂	1	1	yes	maximal, characteristic, complemented
8	Q₈×D₃₅₆	—	2	1	no	—
9	C₂×#15667953	C₂²	1	3	yes	—
10	#19152003	C₂²	1	3	yes	—
11	#19151879	—	2	3	no	—
12	C₄×D₇₁₂	C₂²	1	3	yes	—
13	#19151423	C₂²	1	3	yes	—
14	C₄⋊#13083549	C₂²	1	3	yes	—
15	#19151399	C₂²	1	3	yes	—
16	#19151399	C₂²	1	6	yes	—
17	#19151398	C₂²	1	1	yes	characteristic
18	C₄×#13083549	C₂²	1	3	yes	—
19	#550×C₈₉	—	2	1	no	—
20	#493×C₈₉	C₂²	1	3	yes	—
21	#459×C₈₉	—	2	3	no	—
22	C₈×Q₈×C₈₉	C₂²	1	1	yes	characteristic
23	C₄×D₈×C₈₉	C₂²	1	3	yes	—
24	C₂×Q₈×D₈₉	—	4	1	no	—
25	C₂×D₇₁₂	C₂³	1	1	yes	characteristic
26	#15668012	—	2	3	no	—
27	#15667992	—	4	3	no	—
28	C₄×D₃₅₆	—	2	3	no	—
29	#15667953	—	2	6	no	—
30	C₂×#13083549	C₂³	1	1	yes	characteristic
31	C₂×#13083549	C₂³	1	3	yes	—
32	#15667946	—	2	3	no	—
33	#15667946	C₂³	1	3	yes	—
34	#15667934	—	2	3	no	—
35	C₂×#41×C₈₉	—	4	1	no	—
36	C₂×D₈×C₈₉	—	2	3	no	—
37	C₂×D₈×C₈₉	C₂³	1	3	yes	—
38	#127×C₈₉	—	2	3	no	—
39	#121×C₈₉	—	4	3	no	—
40	C₄×Q₈×C₈₉	D₄	1	1	yes	characteristic
41	C₄×D₄×C₈₉	—	2	3	no	—
42	#107×C₈₉	C₂³	1	1	yes	characteristic
43	C₄⋊C₈×C₈₉	D₄	1	3	yes	—
44	#102×C₈₉	—	2	3	no	—
45	C₄×C₈×C₈₉	C₂³	1	3	yes	—
46	C₂×Dic₁₇₈	—	4	3	no	—
47	Q₈×D₈₉	—	8	4	no	—
48	C₂×D₃₅₆	—	2	1	no	—
49	C₂×C₄×D₈₉	—	4	3	no	—
50	D₇₁₂	—	2	1	no	—
51	D₇₁₂	#41	1	2	yes	complemented
52	C₄⋊Dic₈₉	—	2	3	no	—
53	C₄⋊Dic₈₉	—	4	3	no	—
54	#13083551	—	4	3	no	—
55	#13083549	—	2	3	no	—
56	#13083549	—	2	1	no	—
57	#41×C₈₉	—	8	4	no	—
58	C₂×Q₈×C₈₉	D₈	1	1	yes	characteristic
59	C₂×D₄×C₈₉	—	2	3	no	—
60	C₂×D₄×C₈₉	—	4	3	no	—
61	C₂×D₄×C₈₉	—	4	3	no	—
62	C₂²×C₄×C₈₉	—	4	3	no	—
63	D₈×C₈₉	—	2	3	no	—
64	D₈×C₈₉	—	2	3	no	—
65	D₈×C₈₉	—	8	3	no	—
66	C₂×C₈×C₈₉	C₂×D₄	1	3	yes	—
67	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
68	C₄⋊C₄×C₈₉	—	2	1	no	—
69	C₄⋊C₄×C₈₉	D₈	1	3	yes	—
70	C₂²⋊C₄×C₈₉	—	4	3	no	—
71	C₄²×C₈₉	C₂×D₄	1	3	yes	—
72	C₂³×C₈₉	—	4	3	no	—
73	C₂²×D₈₉	—	4	1	no	—
74	Q₈×C₈₉	D₁₆	1	4	yes	complemented
75	D₄×C₈₉	—	4	3	no	—
76	D₄×C₈₉	—	4	3	no	—
77	D₄×C₈₉	—	8	6	no	—
78	D₄×C₈₉	—	16	3	no	—
79	C₂×C₄×C₈₉	—	4	1	no	—
80	C₂×C₄×C₈₉	—	8	6	no	—
81	C₂×C₄×C₈₉	C₂×D₈	1	3	yes	—
82	C₂×C₄×C₈₉	C₂×D₈	1	3	yes	—
83	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
84	C₂×Dic₈₉	—	4	3	no	—
85	D₃₅₆	—	2	2	no	—
86	D₃₅₆	—	4	1	no	—
87	C₄×D₈₉	—	8	6	no	—
88	Dic₁₇₈	—	8	6	no	—
89	C₈×C₈₉	—	2	3	no	—
90	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
91	C₈×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
92	C₂²×C₈₉	—	8	3	no	—
93	C₂²×C₈₉	—	8	3	no	—
94	C₂²×C₈₉	—	8	3	no	—
95	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
96	C₂×D₈₉	—	4	2	no	—
97	C₂×D₈₉	—	8	1	no	—
98	C₂×D₈₉	—	8	1	no	—
99	C₄×C₈₉	—	2	3	no	—
100	C₄×C₈₉	—	8	1	no	—
101	C₄×C₈₉	C₂×D₁₆	1	6	yes	—
102	C₄×C₈₉	#548	1	1	yes	characteristic
103	C₄×C₈₉	#577	1	1	yes	characteristic
104	Dic₈₉	—	8	3	no	—
105	#33925	—	89	1	no	maximal, Sylow
106	C₂×C₈₉	—	16	3	no	—
107	C₂×C₈₉	#3358	1	1	yes	characteristic
108	C₂×C₈₉	C₂²×D₁₆	1	1	yes	characteristic
109	C₂×C₈₉	#3416	1	1	yes	characteristic
110	D₈₉	—	8	2	no	—
111	#3391	D₈₉	1	1	yes	characteristic, complemented
112	Q₈×D₈	—	89	1	no	—
113	#2240	—	89	3	no	—
114	#2224	—	89	3	no	—
115	#2206	—	89	3	no	—
116	Q₈×C₁₆	—	89	1	no	—
117	C₄×D₁₆	—	89	3	no	—
118	C₈₉	#33925	1	1	yes	characteristic, complemented, Sylow
119	#550	—	2	1	no	—
120	D₄×Q₈	—	178	1	no	—
121	C₂×D₁₆	—	89	3	no	—
122	#500	—	89	3	no	—
123	#493	C₂×D₈₉	1	3	yes	—
124	#474	—	178	3	no	—
125	#459	—	2	3	no	—
126	C₈×Q₈	C₂×D₈₉	1	1	yes	characteristic
127	C₄×D₈	—	89	3	no	—
128	C₄×D₈	C₂×D₈₉	1	3	yes	—
129	#366	—	89	3	no	—
130	C₄⋊C₁₆	—	89	3	no	—
131	#357	—	89	3	no	—
132	#357	—	89	6	no	—
133	#357	—	89	1	no	—
134	C₄×C₁₆	—	89	3	no	—
135	C₂×#41	—	4	1	no	—
136	C₂²×Q₈	—	356	1	no	—
137	C₂×D₈	—	2	3	no	—
138	C₂×D₈	—	89	1	no	—
139	C₂×D₈	C₂²×D₈₉	1	3	yes	—
140	C₄⋊Q₈	—	178	3	no	—
141	#127	—	2	3	no	—
142	C₂²⋊Q₈	—	356	3	no	—
143	#121	—	4	3	no	—
144	C₄×Q₈	C₂²⋊D₈₉	1	1	yes	characteristic
145	C₄×D₄	—	2	3	no	—
146	C₄×D₄	—	178	3	no	—
147	D₁₆	—	178	6	no	—
148	C₂×C₁₆	—	89	1	no	—
149	C₂×C₁₆	—	89	3	no	—
150	#107	—	89	3	no	—
151	#107	—	178	3	no	—
152	#107	C₂²×D₈₉	1	1	yes	characteristic
153	C₄⋊C₈	C₂²⋊D₈₉	1	3	yes	—
154	#102	—	2	3	no	—
155	#102	—	178	3	no	—
156	C₄×C₈	C₂²×D₈₉	1	3	yes	—
157	#41	—	8	4	no	—
158	C₂×Q₈	—	356	3	no	—
159	C₂×Q₈	—	712	4	no	—
160	C₂×Q₈	#13083559	1	1	yes	characteristic
161	C₂×D₄	—	2	3	no	—
162	C₂×D₄	—	4	3	no	—
163	C₂×D₄	—	4	3	no	—
164	C₂×D₄	—	178	1	no	—
165	C₂²×C₄	—	4	3	no	—
166	C₂²×C₄	—	356	3	no	—
167	D₈	—	2	3	no	—
168	D₈	—	2	3	no	—
169	D₈	—	8	3	no	—
170	D₈	—	89	2	no	—
171	D₈	—	178	1	no	—
172	C₂×C₈	C₂×C₂²⋊D₈₉	1	3	yes	—
173	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
174	C₄⋊C₄	—	2	1	no	—
175	C₄⋊C₄	—	356	3	no	—
176	C₄⋊C₄	—	178	3	no	—
177	C₄⋊C₄	#13083559	1	3	yes	—
178	C₂²⋊C₄	—	4	3	no	—
179	C₂²⋊C₄	—	356	3	no	—
180	C₄²	C₂×C₂²⋊D₈₉	1	3	yes	—
181	C₁₆	—	178	3	no	—
182	C₁₆	—	178	1	no	—
183	C₂³	—	4	3	no	—
184	C₂³	—	356	1	no	—
185	Q₈	—	712	6	no	—
186	Q₈	#15667953	1	4	yes	complemented
187	D₄	—	4	3	no	—
188	D₄	—	4	3	no	—
189	D₄	—	8	6	no	—
190	D₄	—	16	3	no	—
191	D₄	—	356	1	no	—
192	D₄	—	178	2	no	—
193	C₂×C₄	—	4	1	no	—
194	C₂×C₄	—	8	6	no	—
195	C₂×C₄	—	356	3	no	—
196	C₂×C₄	—	712	6	no	—
197	C₂×C₄	C₂×#13083559	1	3	yes	—
198	C₂×C₄	C₂×#13083559	1	3	yes	—
199	C₂×C₄	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
200	C₈	—	2	3	no	—
201	C₈	#41×D₈₉	1	1	yes	characteristic
202	C₈	#15668059	1	1	yes	characteristic
203	C₂²	—	8	3	no	—
204	C₂²	—	8	3	no	—
205	C₂²	—	8	3	no	—
206	C₂²	—	356	2	no	—
207	C₂²	—	712	1	no	—
208	C₂²	—	712	1	no	—
209	C₂²	C₂²×#13083559	1	1	yes	characteristic
210	C₄	—	2	3	no	—
211	C₄	—	8	1	no	—
212	C₄	—	712	3	no	—
213	C₄	C₂×#15667953	1	6	yes	—
214	C₄	#19152255	1	1	yes	characteristic
215	C₄	#19152409	1	1	yes	characteristic
216	C₂	—	16	3	no	—
217	C₂	—	712	2	no	—
218	C₂	#26032883	1	1	yes	characteristic
219	C₂	C₂²×#15667953	1	1	yes	characteristic
220	C₂	#26033399	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×D₇₁₂

C₂

yes

maximal, characteristic, complemented

#26022475

C₂

yes

maximal, complemented

#26022389

C₂

yes

maximal, complemented

#26022284

C₂

yes

maximal, complemented

Q₈×#13083549

C₂

yes

maximal, characteristic, complemented

C₄×#15667953

C₂

yes

maximal, complemented

C₈₉×#3391

C₂

yes

maximal, characteristic, complemented

Q₈×D₃₅₆

—

C₂×#15667953

C₂²

yes

—

#19152003

C₂²

yes

—

#19151879

—

C₄×D₇₁₂

C₂²

yes

—

#19151423

C₂²

yes

—

C₄⋊#13083549

C₂²

yes

—

#19151399

C₂²

yes

—

#19151399

C₂²

yes

—

#19151398

C₂²

yes

characteristic

C₄×#13083549

C₂²

yes

—

#550×C₈₉

—

#493×C₈₉

C₂²

yes

—

#459×C₈₉

—

C₈×Q₈×C₈₉

C₂²

yes

characteristic

C₄×D₈×C₈₉

C₂²

yes

—

C₂×Q₈×D₈₉

—

C₂×D₇₁₂

C₂³

yes

characteristic

#15668012

—

#15667992

—

C₄×D₃₅₆

—

#15667953

—

C₂×#13083549

C₂³

yes

characteristic

C₂×#13083549

C₂³

yes

—

#15667946

—

#15667946

C₂³

yes

—

#15667934

—

C₂×#41×C₈₉

—

C₂×D₈×C₈₉

—

C₂×D₈×C₈₉

C₂³

yes

—

#127×C₈₉

—

#121×C₈₉

—

C₄×Q₈×C₈₉

D₄

yes

characteristic

C₄×D₄×C₈₉

—

#107×C₈₉

C₂³

yes

characteristic

C₄⋊C₈×C₈₉

D₄

yes

—

#102×C₈₉

—

C₄×C₈×C₈₉

C₂³

yes

—

C₂×Dic₁₇₈

—

Q₈×D₈₉

—

C₂×D₃₅₆

—

C₂×C₄×D₈₉

—

D₇₁₂

—

D₇₁₂

#41

yes

complemented

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

#13083551

—

#13083549

—

#13083549

—

#41×C₈₉

—

C₂×Q₈×C₈₉

D₈

yes

characteristic

C₂×D₄×C₈₉

—

C₂×D₄×C₈₉

—

C₂×D₄×C₈₉

—

C₂²×C₄×C₈₉

—

D₈×C₈₉

—

D₈×C₈₉

—

D₈×C₈₉

—

C₂×C₈×C₈₉

C₂×D₄

yes

—

C₂×C₈×C₈₉

C₂⁴

yes

characteristic

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

D₈

yes

—

C₂²⋊C₄×C₈₉

—

C₄²×C₈₉

C₂×D₄

yes

—

C₂³×C₈₉

—

C₂²×D₈₉

—

Q₈×C₈₉

D₁₆

yes

complemented

D₄×C₈₉

—

D₄×C₈₉

—

D₄×C₈₉

—

D₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×D₈

yes

—

C₂×C₄×C₈₉

C₂×D₈

yes

—

C₂×C₄×C₈₉

C₂²×D₄

yes

characteristic

C₂×Dic₈₉

—

D₃₅₆

—

D₃₅₆

—

C₄×D₈₉

—

Dic₁₇₈

—

C₈×C₈₉

—

C₈×C₈₉

C₂×#41

yes

characteristic

C₈×C₈₉

2₋¹⁺⁴

yes

characteristic

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

C₂²×D₈

yes

characteristic

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₄×C₈₉

—

100

C₄×C₈₉

—

101

C₄×C₈₉

C₂×D₁₆

yes

—

102

C₄×C₈₉

#548

yes

characteristic

103

C₄×C₈₉

#577

yes

characteristic

104

Dic₈₉

—

105

#33925

—

maximal, Sylow

106

C₂×C₈₉

—

107

C₂×C₈₉

#3358

yes

characteristic

108

C₂×C₈₉

C₂²×D₁₆

yes

characteristic

109

C₂×C₈₉

#3416

yes

characteristic

110

D₈₉

—

111

#3391

D₈₉

yes

characteristic, complemented

112

Q₈×D₈

—

113

#2240

—

114

#2224

—

115

#2206

—

116

Q₈×C₁₆

—

117

C₄×D₁₆

—

118

C₈₉

#33925

yes

characteristic, complemented, Sylow

119

#550

—

120

D₄×Q₈

—

178

—

121

C₂×D₁₆

—

122

#500

—

123

#493

C₂×D₈₉

yes

—

124

#474

—

178

—

125

#459

—

126

C₈×Q₈

C₂×D₈₉

yes

characteristic

127

C₄×D₈

—

128

C₄×D₈

C₂×D₈₉

yes

—

129

#366

—

130

C₄⋊C₁₆

—

131

#357

—

132

#357

—

133

#357

—

134

C₄×C₁₆

—

135

C₂×#41

—

136

C₂²×Q₈

—

356

—

137

C₂×D₈

—

138

C₂×D₈

—

139

C₂×D₈

C₂²×D₈₉

yes

—

140

C₄⋊Q₈

—

178

—

141

#127

—

142

C₂²⋊Q₈

—

356

—

143

#121

—

144

C₄×Q₈

C₂²⋊D₈₉

yes

characteristic

145

C₄×D₄

—

146

C₄×D₄

—

178

—

147

D₁₆

—

178

—

148

C₂×C₁₆

—

149

C₂×C₁₆

—

150

#107

—

151

#107

—

178

—

152

#107

C₂²×D₈₉

yes

characteristic

153

C₄⋊C₈

C₂²⋊D₈₉

yes

—

154

#102

—

155

#102

—

178

—

156

C₄×C₈

C₂²×D₈₉

yes

—

157

#41

—

158

C₂×Q₈

—

356

—

159

C₂×Q₈

—

712

—

160

C₂×Q₈

#13083559

yes

characteristic

161

C₂×D₄

—

162

C₂×D₄

—

163

C₂×D₄

—

164

C₂×D₄

—

178

—

165

C₂²×C₄

—

166

C₂²×C₄

—

356

—

167

D₈

—

168

D₈

—

169

D₈

—

170

D₈

—

171

D₈

—

178

—

172

C₂×C₈

C₂×C₂²⋊D₈₉

yes

—

173

C₂×C₈

C₂³×D₈₉

yes

characteristic

174

C₄⋊C₄

—

175

C₄⋊C₄

—

356

—

176

C₄⋊C₄

—

178

—

177

C₄⋊C₄

#13083559

yes

—

178

C₂²⋊C₄

—

179

C₂²⋊C₄

—

356

—

180

C₄²

C₂×C₂²⋊D₈₉

yes

—

181

C₁₆

—

178

—

182

C₁₆

—

178

—

183

C₂³

—

184

C₂³

—

356

—

185

Q₈

—

712

—

186

Q₈

#15667953

yes

complemented

187

D₄

—

188

D₄

—

189

D₄

—

190

D₄

—

191

D₄

—

356

—

192

D₄

—

178

—

193

C₂×C₄

—

194

C₂×C₄

—

195

C₂×C₄

—

356

—

196

C₂×C₄

—

712

—

197

C₂×C₄

C₂×#13083559

yes

—

198

C₂×C₄

C₂×#13083559

yes

—

199

C₂×C₄

C₂²×C₂²⋊D₈₉

yes

characteristic

200

C₈

—

201

C₈

#41×D₈₉

yes

characteristic

202

C₈

#15668059

yes

characteristic

203

C₂²

—

204

C₂²

—

205

C₂²

—

206

C₂²

—

356

—

207

C₂²

—

712

—

208

C₂²

—

712

—

209

C₂²

C₂²×#13083559

yes

characteristic

210

C₄

—

211

C₄

—

212

C₄

—

712

—

213

C₄

C₂×#15667953

yes

—

214

C₄

#19152255

yes

characteristic

215

C₄

#19152409

yes

characteristic

216

C₂

—

217

C₂

—

712

—

218

C₂

#26032883

yes

characteristic

219

C₂

C₂²×#15667953

yes

characteristic

220

C₂

#26033399

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×#13083559
commutator, cosocle	C₂×C₈×C₈₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₈₉×#3391	C₂
socle	C₂²×C₈₉	C₂²×D₈
hypercenter	#3391	D₈₉
hypocenter	C₈₉	#33925
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈ ◁ #3391

subgroup

G/H

center

C₂²

C₂²×#13083559

commutator, cosocle

C₂×C₈×C₈₉

C₂⁴

Frattini

C₂×C₈

C₂³×D₈₉

Fitting

C₈₉×#3391

C₂

socle

C₂²×C₈₉

C₂²×D₈

D₈₉

C₈₉

G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈ ◁ #3391

N	H ≅ G/N	type
1. Q₈×D₇₁₂	C₂	semidirect
2. #26022475	C₂	semidirect
3. #26022389	C₂	semidirect
4. #26022284	C₂	semidirect
5. Q₈×#13083549	C₂	semidirect
6. C₄×#15667953	C₂	semidirect
7. C₈₉×#3391	C₂	semidirect
51. D₇₁₂	#41	semidirect
74. Q₈×C₈₉	D₁₆	semidirect
111. #3391	D₈₉	semidirect
118. C₈₉	#33925	semidirect
186. Q₈	#15667953	semidirect

H ≅ G/N

type

1. Q₈×D₇₁₂

C₂

semidirect

2. #26022475

C₂

semidirect

3. #26022389

C₂

semidirect

4. #26022284

C₂

semidirect

5. Q₈×#13083549

C₂

semidirect

6. C₄×#15667953

C₂

semidirect

7. C₈₉×#3391

C₂

semidirect

51. D₇₁₂

#41

semidirect

74. Q₈×C₈₉

D₁₆

semidirect

111. #3391

D₈₉

semidirect

118. C₈₉

#33925

semidirect

186. Q₈

#15667953

semidirect

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

subgroup	index
Q₈×D₇₁₂	2
#26022475	2
#26022389	2
#26022284	2
Q₈×#13083549	2
C₄×#15667953	2
C₈₉×#3391	2
#33925	89

subgroup

index

Q₈×D₇₁₂

C₈₉×#3391

Subgroups — #36712069

Conjugacy classes of subgroups

Complements

Maximal subgroups