Group #36712076 — subgroups

Subgroups — #36712076

Order 22784.

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

normal subgroups only

This group has 506 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
2	#26022479	C₂	1	3	yes	maximal, complemented
3	#26022394	C₂	1	3	yes	maximal, complemented
4	#26022294	C₂	1	3	yes	maximal
5	Q₈×#13083549	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×#15667955	C₂	1	3	yes	maximal
7	C₈₉×#3395	C₂	1	1	yes	maximal, characteristic, complemented
8	Q₈×D₃₅₆	—	2	1	no	—
9	C₂×#15667955	C₂²	1	3	yes	—
10	#19152003	C₂²	1	3	yes	—
11	#19151879	—	2	3	no	—
12	C₄×D₇₁₂	C₂²	1	3	yes	—
13	#19151425	C₂²	1	3	yes	—
14	C₄⋊#13083549	C₂²	1	3	yes	—
15	#19151402	C₂²	1	3	yes	—
16	#19151402	C₂²	1	6	yes	—
17	#19151398	C₂²	1	1	yes	characteristic
18	C₄×#13083549	C₂²	1	3	yes	—
19	#557×C₈₉	—	2	1	no	—
20	#494×C₈₉	C₂²	1	3	yes	—
21	#462×C₈₉	—	2	3	no	—
22	C₈×Q₈×C₈₉	C₂²	1	1	yes	characteristic
23	C₄×Q₁₆×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	#15667955	—	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×Q₁₆×C₈₉	—	2	3	no	—
36	C₂×Q₁₆×C₈₉	C₂³	1	3	yes	—
37	C₄⋊Q₈×C₈₉	—	2	3	no	—
38	#125×C₈₉	—	4	3	no	—
39	C₄×Q₈×C₈₉	—	2	3	no	—
40	C₄×Q₈×C₈₉	—	4	1	no	—
41	C₄×Q₈×C₈₉	D₄	1	1	yes	characteristic
42	#107×C₈₉	C₂³	1	1	yes	characteristic
43	C₄⋊C₈×C₈₉	D₄	1	3	yes	—
44	#103×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	—
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	C₂×Q₈×C₈₉	—	2	3	no	—
58	C₂×Q₈×C₈₉	D₈	1	1	yes	characteristic
59	Q₁₆×C₈₉	—	2	3	no	—
60	Q₁₆×C₈₉	—	2	3	no	—
61	Q₁₆×C₈₉	—	8	3	no	—
62	C₂×C₈×C₈₉	C₂×D₄	1	3	yes	—
63	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
64	C₄⋊C₄×C₈₉	—	2	1	no	—
65	C₄⋊C₄×C₈₉	—	4	3	no	—
66	C₄⋊C₄×C₈₉	—	4	3	no	—
67	C₄⋊C₄×C₈₉	—	4	3	no	—
68	C₄⋊C₄×C₈₉	D₈	1	3	yes	—
69	C₄²×C₈₉	—	4	3	no	—
70	C₄²×C₈₉	C₂×D₄	1	3	yes	—
71	C₂²×D₈₉	—	4	1	no	—
72	Q₈×C₈₉	—	4	3	no	—
73	Q₈×C₈₉	—	4	3	no	—
74	Q₈×C₈₉	SD₃₂	1	4	yes	complemented
75	C₂×C₄×C₈₉	—	4	1	no	—
76	C₂×C₄×C₈₉	—	4	3	no	—
77	C₂×C₄×C₈₉	C₂×D₈	1	3	yes	—
78	C₂×C₄×C₈₉	C₂×D₈	1	3	yes	—
79	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
80	C₂×Dic₈₉	—	4	3	no	—
81	D₃₅₆	—	2	2	no	—
82	D₃₅₆	—	4	1	no	—
83	C₄×D₈₉	—	8	6	no	—
84	Dic₁₇₈	—	8	6	no	—
85	C₈×C₈₉	—	2	3	no	—
86	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
87	C₈×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
88	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
89	C₂×D₈₉	—	4	2	no	—
90	C₂×D₈₉	—	8	1	no	—
91	C₂×D₈₉	—	8	1	no	—
92	C₄×C₈₉	—	2	3	no	—
93	C₄×C₈₉	—	8	3	no	—
94	C₄×C₈₉	—	8	1	no	—
95	C₄×C₈₉	C₂×SD₃₂	1	6	yes	—
96	C₄×C₈₉	#548	1	1	yes	characteristic
97	C₄×C₈₉	#577	1	1	yes	characteristic
98	Dic₈₉	—	8	3	no	—
99	#33926	—	89	1	no	maximal, Sylow
100	C₂×C₈₉	#3358	1	1	yes	characteristic
101	C₂×C₈₉	C₂²×SD₃₂	1	1	yes	characteristic
102	C₂×C₈₉	#3417	1	1	yes	characteristic
103	D₈₉	—	8	2	no	—
104	#3395	D₈₉	1	1	yes	characteristic, complemented
105	Q₈×D₈	—	89	1	no	—
106	#2241	—	89	3	no	—
107	#2225	—	89	3	no	—
108	#2208	—	89	3	no	—
109	Q₈×C₁₆	—	89	1	no	—
110	C₄×SD₃₂	—	89	3	no	—
111	C₈₉	#33926	1	1	yes	characteristic, complemented, Sylow
112	#557	—	2	1	no	—
113	D₄×Q₈	—	178	1	no	—
114	C₂×SD₃₂	—	89	3	no	—
115	#500	—	89	3	no	—
116	#494	C₂×D₈₉	1	3	yes	—
117	#474	—	178	3	no	—
118	#462	—	2	3	no	—
119	C₈×Q₈	C₂×D₈₉	1	1	yes	characteristic
120	C₄×Q₁₆	C₂×D₈₉	1	3	yes	—
121	C₄×D₈	—	89	3	no	—
122	#367	—	89	3	no	—
123	C₄⋊C₁₆	—	89	3	no	—
124	#358	—	89	3	no	—
125	#358	—	89	6	no	—
126	#357	—	89	1	no	—
127	C₄×C₁₆	—	89	3	no	—
128	C₂²×Q₈	—	356	1	no	—
129	C₂×Q₁₆	—	2	3	no	—
130	C₂×Q₁₆	C₂²×D₈₉	1	3	yes	—
131	C₂×D₈	—	89	1	no	—
132	C₄⋊Q₈	—	2	3	no	—
133	C₄⋊Q₈	—	178	3	no	—
134	#125	—	4	3	no	—
135	C₂²⋊Q₈	—	356	3	no	—
136	C₄×Q₈	—	2	3	no	—
137	C₄×Q₈	—	4	1	no	—
138	C₄×Q₈	C₂²⋊D₈₉	1	1	yes	characteristic
139	C₄×D₄	—	178	3	no	—
140	SD₃₂	—	178	6	no	—
141	C₂×C₁₆	—	89	1	no	—
142	C₂×C₁₆	—	89	3	no	—
143	#107	—	89	3	no	—
144	#107	—	178	3	no	—
145	#107	C₂²×D₈₉	1	1	yes	characteristic
146	C₄⋊C₈	C₂²⋊D₈₉	1	3	yes	—
147	#103	—	2	3	no	—
148	#102	—	178	3	no	—
149	C₄×C₈	C₂²×D₈₉	1	3	yes	—
150	C₂×Q₈	—	2	3	no	—
151	C₂×Q₈	—	356	3	no	—
152	C₂×Q₈	—	712	4	no	—
153	C₂×Q₈	#13083559	1	1	yes	characteristic
154	C₂×D₄	—	178	1	no	—
155	C₂²×C₄	—	356	3	no	—
156	Q₁₆	—	2	3	no	—
157	Q₁₆	—	2	3	no	—
158	Q₁₆	—	8	3	no	—
159	D₈	—	89	2	no	—
160	D₈	—	178	1	no	—
161	C₂×C₈	C₂×C₂²⋊D₈₉	1	3	yes	—
162	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
163	C₄⋊C₄	—	2	1	no	—
164	C₄⋊C₄	—	4	3	no	—
165	C₄⋊C₄	—	4	3	no	—
166	C₄⋊C₄	—	4	3	no	—
167	C₄⋊C₄	—	356	3	no	—
168	C₄⋊C₄	—	178	3	no	—
169	C₄⋊C₄	#13083559	1	3	yes	—
170	C₂²⋊C₄	—	356	3	no	—
171	C₄²	—	4	3	no	—
172	C₄²	C₂×C₂²⋊D₈₉	1	3	yes	—
173	C₁₆	—	178	3	no	—
174	C₁₆	—	178	1	no	—
175	C₂³	—	356	1	no	—
176	Q₈	—	4	3	no	—
177	Q₈	—	4	3	no	—
178	Q₈	—	712	6	no	—
179	Q₈	#15667955	1	4	yes	complemented
180	D₄	—	356	1	no	—
181	D₄	—	178	2	no	—
182	C₂×C₄	—	4	1	no	—
183	C₂×C₄	—	4	3	no	—
184	C₂×C₄	—	356	3	no	—
185	C₂×C₄	—	712	6	no	—
186	C₂×C₄	C₂×#13083559	1	3	yes	—
187	C₂×C₄	C₂×#13083559	1	3	yes	—
188	C₂×C₄	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
189	C₈	—	2	3	no	—
190	C₈	#41×D₈₉	1	1	yes	characteristic
191	C₈	#15668059	1	1	yes	characteristic
192	C₂²	—	356	2	no	—
193	C₂²	—	712	1	no	—
194	C₂²	—	712	1	no	—
195	C₂²	C₂²×#13083559	1	1	yes	characteristic
196	C₄	—	2	3	no	—
197	C₄	—	8	3	no	—
198	C₄	—	8	1	no	—
199	C₄	—	712	3	no	—
200	C₄	C₂×#15667955	1	6	yes	—
201	C₄	#19152255	1	1	yes	characteristic
202	C₄	#19152409	1	1	yes	characteristic
203	C₂	—	712	2	no	—
204	C₂	#26032883	1	1	yes	characteristic
205	C₂	C₂²×#15667955	1	1	yes	characteristic
206	C₂	#26033403	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×D₇₁₂

C₂

yes

maximal, characteristic

#26022479

C₂

yes

maximal, complemented

#26022394

C₂

yes

maximal, complemented

#26022294

C₂

yes

maximal

Q₈×#13083549

C₂

yes

maximal, characteristic, complemented

C₄×#15667955

C₂

yes

maximal

C₈₉×#3395

C₂

yes

maximal, characteristic, complemented

Q₈×D₃₅₆

—

C₂×#15667955

C₂²

yes

—

#19152003

C₂²

yes

—

#19151879

—

C₄×D₇₁₂

C₂²

yes

—

#19151425

C₂²

yes

—

C₄⋊#13083549

C₂²

yes

—

#19151402

C₂²

yes

—

#19151402

C₂²

yes

—

#19151398

C₂²

yes

characteristic

C₄×#13083549

C₂²

yes

—

#557×C₈₉

—

#494×C₈₉

C₂²

yes

—

#462×C₈₉

—

C₈×Q₈×C₈₉

C₂²

yes

characteristic

C₄×Q₁₆×C₈₉

C₂²

yes

—

C₂×Q₈×D₈₉

—

C₂×D₇₁₂

C₂³

yes

characteristic

#15668012

—

#15667992

—

C₄×D₃₅₆

—

#15667955

—

C₂×#13083549

C₂³

yes

characteristic

C₂×#13083549

C₂³

yes

—

#15667946

—

#15667946

C₂³

yes

—

#15667934

—

C₂×Q₁₆×C₈₉

—

C₂×Q₁₆×C₈₉

C₂³

yes

—

C₄⋊Q₈×C₈₉

—

#125×C₈₉

—

C₄×Q₈×C₈₉

—

C₄×Q₈×C₈₉

—

C₄×Q₈×C₈₉

D₄

yes

characteristic

#107×C₈₉

C₂³

yes

characteristic

C₄⋊C₈×C₈₉

D₄

yes

—

#103×C₈₉

—

C₄×C₈×C₈₉

C₂³

yes

—

C₂×Dic₁₇₈

—

Q₈×D₈₉

—

C₂×D₃₅₆

—

C₂×C₄×D₈₉

—

D₇₁₂

—

D₇₁₂

#41

yes

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

#13083551

—

#13083549

—

#13083549

—

C₂×Q₈×C₈₉

—

C₂×Q₈×C₈₉

D₈

yes

characteristic

Q₁₆×C₈₉

—

Q₁₆×C₈₉

—

Q₁₆×C₈₉

—

C₂×C₈×C₈₉

C₂×D₄

yes

—

C₂×C₈×C₈₉

C₂⁴

yes

characteristic

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

D₈

yes

—

C₄²×C₈₉

—

C₄²×C₈₉

C₂×D₄

yes

—

C₂²×D₈₉

—

Q₈×C₈₉

—

Q₈×C₈₉

—

Q₈×C₈₉

SD₃₂

yes

complemented

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×D₈

yes

characteristic

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

C₂×SD₃₂

yes

—

C₄×C₈₉

yes

C₄×C₈₉

yes

Dic₈₉

—

#33926

—

maximal, Sylow

100

C₂×C₈₉

#3358

yes

characteristic

101

C₂×C₈₉

C₂²×SD₃₂

yes

characteristic

102

C₂×C₈₉

#3417

yes

characteristic

103

D₈₉

—

104

#3395

D₈₉

yes

characteristic, complemented

105

Q₈×D₈

—

106

#2241

—

107

#2225

—

108

#2208

—

109

Q₈×C₁₆

—

110

C₄×SD₃₂

—

111

C₈₉

#33926

yes

characteristic, complemented, Sylow

112

#557

—

113

D₄×Q₈

—

178

—

114

C₂×SD₃₂

—

115

#500

—

116

#494

C₂×D₈₉

yes

—

117

#474

—

178

—

118

#462

—

119

C₈×Q₈

C₂×D₈₉

yes

characteristic

120

C₄×Q₁₆

C₂×D₈₉

yes

—

121

C₄×D₈

—

122

#367

—

123

C₄⋊C₁₆

—

124

#358

—

125

#358

—

126

#357

—

127

C₄×C₁₆

—

128

C₂²×Q₈

—

356

—

129

C₂×Q₁₆

—

130

C₂×Q₁₆

C₂²×D₈₉

yes

—

131

C₂×D₈

—

132

C₄⋊Q₈

—

133

C₄⋊Q₈

—

178

—

134

#125

—

135

C₂²⋊Q₈

—

356

—

136

C₄×Q₈

—

137

C₄×Q₈

—

138

C₄×Q₈

C₂²⋊D₈₉

yes

characteristic

139

C₄×D₄

—

178

—

140

SD₃₂

—

178

—

141

C₂×C₁₆

—

142

C₂×C₁₆

—

143

#107

—

144

#107

—

178

—

145

#107

C₂²×D₈₉

yes

characteristic

146

C₄⋊C₈

C₂²⋊D₈₉

yes

—

147

#103

—

148

#102

—

178

—

149

C₄×C₈

C₂²×D₈₉

yes

—

150

C₂×Q₈

—

151

C₂×Q₈

—

356

—

152

C₂×Q₈

—

712

—

153

C₂×Q₈

#13083559

yes

characteristic

154

C₂×D₄

—

178

—

155

C₂²×C₄

—

356

—

156

Q₁₆

—

157

Q₁₆

—

158

Q₁₆

—

159

D₈

—

160

D₈

—

178

—

161

C₂×C₈

C₂×C₂²⋊D₈₉

yes

—

162

C₂×C₈

C₂³×D₈₉

yes

characteristic

163

C₄⋊C₄

—

164

C₄⋊C₄

—

165

C₄⋊C₄

—

166

C₄⋊C₄

—

167

C₄⋊C₄

—

356

—

168

C₄⋊C₄

—

178

—

169

C₄⋊C₄

#13083559

yes

—

170

C₂²⋊C₄

—

356

—

171

C₄²

—

172

C₄²

C₂×C₂²⋊D₈₉

yes

—

173

C₁₆

—

178

—

174

C₁₆

—

178

—

175

C₂³

—

356

—

176

Q₈

—

177

Q₈

—

178

Q₈

—

712

—

179

Q₈

#15667955

yes

complemented

180

D₄

—

356

—

181

D₄

—

178

—

182

C₂×C₄

—

183

C₂×C₄

—

184

C₂×C₄

—

356

—

185

C₂×C₄

—

712

—

186

C₂×C₄

C₂×#13083559

yes

—

187

C₂×C₄

C₂×#13083559

yes

—

188

C₂×C₄

C₂²×C₂²⋊D₈₉

yes

characteristic

189

C₈

—

190

C₈

#41×D₈₉

yes

characteristic

191

C₈

#15668059

yes

characteristic

192

C₂²

—

356

—

193

C₂²

—

712

—

194

C₂²

—

712

—

195

C₂²

C₂²×#13083559

yes

characteristic

196

C₄

—

197

C₄

—

198

C₄

—

199

C₄

—

712

—

200

C₄

C₂×#15667955

yes

—

201

C₄

#19152255

yes

characteristic

202

C₄

#19152409

yes

characteristic

203

C₂

—

712

—

204

C₂

#26032883

yes

characteristic

205

C₂

C₂²×#15667955

yes

characteristic

206

C₂

#26033403

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×#13083559
commutator, cosocle	C₂×C₈×C₈₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₈₉×#3395	C₂
socle	C₂²×C₈₉	C₂²×D₈
prime radical	C₂×D₇₁₂	C₂³
hypercenter	#3395	D₈₉
hypocenter	C₈₉	#33926
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈ ◁ #3395

subgroup

G/H

center

C₂²

C₂²×#13083559

commutator, cosocle

C₂×C₈×C₈₉

C₂⁴

Frattini

C₂×C₈

C₂³×D₈₉

Fitting

C₈₉×#3395

C₂

socle

C₂²×C₈₉

C₂²×D₈

prime radical

C₂×D₇₁₂

C₂³

D₈₉

C₈₉

G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈ ◁ #3395

N	H ≅ G/N	type
2. #26022479	C₂	semidirect
3. #26022394	C₂	semidirect
5. Q₈×#13083549	C₂	semidirect
7. C₈₉×#3395	C₂	semidirect
74. Q₈×C₈₉	SD₃₂	semidirect
104. #3395	D₈₉	semidirect
111. C₈₉	#33926	semidirect
179. Q₈	#15667955	semidirect

H ≅ G/N

type

2. #26022479

C₂

semidirect

3. #26022394

C₂

semidirect

5. Q₈×#13083549

C₂

semidirect

7. C₈₉×#3395

C₂

semidirect

74. Q₈×C₈₉

SD₃₂

semidirect

104. #3395

D₈₉

semidirect

111. C₈₉

#33926

semidirect

179. Q₈

#15667955

semidirect

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

subgroup	index
Q₈×D₇₁₂	2
#26022479	2
#26022394	2
#26022294	2
Q₈×#13083549	2
C₄×#15667955	2
C₈₉×#3395	2
#33926	89

subgroup

index

Q₈×D₇₁₂

C₈₉×#3395

Subgroups — #36712076

Conjugacy classes of subgroups

Complements

Maximal subgroups