Group #36457509 — subgroups

Subgroups — #36457509

Order 22784.

392 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 91 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	#26028527	C₂	1	1	yes	maximal, characteristic
2	#26021226	C₂	1	2	yes	maximal
3	#26018134	C₂	1	2	yes	maximal
4	#26018133	C₂	1	1	yes	maximal, characteristic
5	C₈₉×#1960	C₂	1	1	yes	maximal, characteristic
6	C₂×#15667923	C₂²	1	2	yes	—
7	#19151532	—	2	2	no	—
8	#19151531	—	2	2	no	—
9	#19151530	—	2	2	no	—
10	#19151464	C₂²	1	2	yes	—
11	#19151360	—	2	2	no	—
12	#19151360	—	2	2	no	—
13	C₂×C₄⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
14	C₂×#103×C₈₉	—	2	2	no	—
15	C₂×#97×C₈₉	C₂²	1	1	yes	characteristic
16	#382×C₈₉	C₂²	1	1	yes	characteristic
17	C₂²×#10623494	—	2	2	no	—
18	C₂×C₄⋊Dic₈₉	—	2	2	no	—
19	C₂×C₄⋊Dic₈₉	D₄	1	2	yes	—
20	C₂×C₄⋊Dic₈₉	D₄	1	2	yes	—
21	C₂×#13083553	—	2	2	no	—
22	C₂×#13083553	—	2	2	no	—
23	#15667923	—	2	2	no	—
24	#15667923	—	2	2	no	—
25	#15667919	—	2	2	no	—
26	#15667919	—	2	2	no	—
27	#15667919	—	2	2	no	—
28	C₂²×Q₈×C₈₉	—	2	1	no	—
29	C₂²×C₈×C₈₉	—	2	1	no	—
30	C₄⋊Q₈×C₈₉	—	2	2	no	—
31	C₄⋊Q₈×C₈₉	—	2	1	no	—
32	C₄⋊Q₈×C₈₉	Q₈	1	2	yes	—
33	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
34	C₂×C₄⋊C₄×C₈₉	D₄	1	2	yes	—
35	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
36	#103×C₈₉	—	4	4	no	—
37	#97×C₈₉	—	2	2	no	—
38	#95×C₈₉	—	2	2	no	—
39	C₂²×Dic₈₉	—	2	2	no	—
40	C₂²×Dic₈₉	—	2	2	no	—
41	C₂×#10623494	—	2	4	no	—
42	C₂×#10623494	—	4	2	no	—
43	C₂×#10623494	—	4	2	no	—
44	C₄⋊Dic₈₉	—	4	2	no	—
45	C₄⋊Dic₈₉	—	4	2	no	—
46	C₄⋊Dic₈₉	—	8	2	no	—
47	#13083553	—	8	2	no	—
48	#13083553	—	8	2	no	—
49	C₂×Q₈×C₈₉	—	2	2	no	—
50	C₂×Q₈×C₈₉	—	4	2	no	—
51	C₂×Q₈×C₈₉	—	4	2	no	—
52	C₂×Q₈×C₈₉	—	4	1	no	—
53	C₂²×C₄×C₈₉	—	2	2	no	—
54	C₂²×C₄×C₈₉	—	4	1	no	—
55	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
56	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
57	C₂×C₈×C₈₉	—	2	2	no	—
58	C₂×C₈×C₈₉	—	4	1	no	—
59	C₂×C₈×C₈₉	—	4	1	no	—
60	C₄⋊C₄×C₈₉	—	2	4	no	—
61	C₄⋊C₄×C₈₉	—	4	2	no	—
62	C₄⋊C₄×C₈₉	—	4	2	no	—
63	C₄²×C₈₉	C₂×Q₈	1	1	yes	characteristic
64	C₄²×C₈₉	#41	1	2	yes	—
65	C₄²×C₈₉	#41	1	1	yes	characteristic
66	C₂³×C₈₉	#120	1	1	yes	characteristic
67	Q₈×C₈₉	—	4	4	no	—
68	Q₈×C₈₉	—	8	2	no	—
69	C₂×C₄×C₈₉	—	2	1	no	—
70	C₂×C₄×C₈₉	—	2	1	no	—
71	C₂×C₄×C₈₉	—	2	2	no	—
72	C₂×C₄×C₈₉	—	2	2	no	—
73	C₂×C₄×C₈₉	—	4	2	no	—
74	C₂×C₄×C₈₉	—	4	2	no	—
75	C₂×C₄×C₈₉	—	4	2	no	—
76	C₂×C₄×C₈₉	—	4	2	no	—
77	C₂×C₄×C₈₉	—	8	1	no	—
78	C₂×C₄×C₈₉	—	8	1	no	—
79	C₂×C₄×C₈₉	C₂²⋊Q₈	1	2	yes	—
80	C₂×C₄×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
81	C₂×C₄×C₈₉	#124	1	2	yes	—
82	C₂×C₄×C₈₉	#124	1	1	yes	characteristic
83	C₂×Dic₈₉	—	4	2	no	—
84	C₂×Dic₈₉	—	4	2	no	—
85	C₂×Dic₈₉	—	4	2	no	—
86	C₂×Dic₈₉	—	4	2	no	—
87	C₂×Dic₈₉	—	4	2	no	—
88	C₂×Dic₈₉	—	4	2	no	—
89	C₈×C₈₉	—	4	2	no	—
90	#10623494	—	4	4	no	—
91	C₂²×C₈₉	#393	1	1	yes	characteristic
92	C₂²×C₈₉	#453	1	1	yes	characteristic
93	C₂²×C₈₉	#454	1	1	yes	characteristic
94	C₂²×C₈₉	#455	1	2	yes	—
95	C₂²×C₈₉	#456	1	2	yes	—
96	C₄×C₈₉	—	2	1	no	—
97	C₄×C₈₉	—	2	1	no	—
98	C₄×C₈₉	—	2	2	no	—
99	C₄×C₈₉	—	2	2	no	—
100	C₄×C₈₉	—	8	2	no	—
101	C₄×C₈₉	—	8	2	no	—
102	Dic₈₉	—	8	2	no	—
103	Dic₈₉	—	8	2	no	—
104	#10361	—	89	1	no	maximal, Sylow
105	C₂×C₈₉	#1661	1	2	yes	—
106	C₂×C₈₉	#1662	1	1	yes	characteristic
107	C₂×C₈₉	#1663	1	1	yes	characteristic
108	C₂×C₈₉	#2027	1	1	yes	characteristic
109	C₂×C₈₉	#2028	1	2	yes	—
110	#2845	—	89	1	no	—
111	#1993	—	89	2	no	—
112	#1960	D₈₉	1	1	yes	characteristic
113	#1295	—	89	2	no	—
114	#1295	—	89	1	no	—
115	C₈₉	#10361	1	1	yes	characteristic, complemented, Sylow
116	C₂×C₄⋊Q₈	C₂×D₈₉	1	1	yes	characteristic
117	C₂×#103	—	2	2	no	—
118	C₂×#97	—	89	2	no	—
119	C₂×#97	C₂×D₈₉	1	1	yes	characteristic
120	#398	—	178	2	no	—
121	#398	—	178	2	no	—
122	#398	—	178	2	no	—
123	#382	—	89	2	no	—
124	#382	C₂×D₈₉	1	1	yes	characteristic
125	#340	—	178	2	no	—
126	#340	—	178	2	no	—
127	C₂²×Q₈	—	2	1	no	—
128	C₂²×C₈	—	2	1	no	—
129	C₂²×C₈	—	178	2	no	—
130	C₄⋊Q₈	—	2	2	no	—
131	C₄⋊Q₈	—	2	1	no	—
132	C₄⋊Q₈	Dic₁₇₈	1	2	yes	—
133	C₂×C₄⋊C₄	—	2	2	no	—
134	C₂×C₄⋊C₄	—	89	2	no	—
135	C₂×C₄⋊C₄	—	89	2	no	—
136	C₂×C₄⋊C₄	—	178	2	no	—
137	C₂×C₄⋊C₄	—	178	2	no	—
138	C₂×C₄⋊C₄	—	178	2	no	—
139	C₂×C₄⋊C₄	C₂²⋊D₈₉	1	2	yes	—
140	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
141	#103	—	4	4	no	—
142	#97	—	2	2	no	—
143	#97	—	178	2	no	—
144	#97	—	178	2	no	—
145	#95	—	2	2	no	—
146	#95	—	178	2	no	—
147	#95	—	178	2	no	—
148	#95	—	178	2	no	—
149	C₂×Q₈	—	2	2	no	—
150	C₂×Q₈	—	4	2	no	—
151	C₂×Q₈	—	4	2	no	—
152	C₂×Q₈	—	4	1	no	—
153	C₂²×C₄	—	2	2	no	—
154	C₂²×C₄	—	4	1	no	—
155	C₂²×C₄	—	178	2	no	—
156	C₂²×C₄	—	178	2	no	—
157	C₂²×C₄	D₄×D₈₉	1	2	yes	—
158	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
159	C₂×C₈	—	2	2	no	—
160	C₂×C₈	—	4	1	no	—
161	C₂×C₈	—	4	1	no	—
162	C₂×C₈	—	356	2	no	—
163	C₂×C₈	—	356	2	no	—
164	C₂×C₈	—	178	4	no	—
165	C₄⋊C₄	—	2	4	no	—
166	C₄⋊C₄	—	4	2	no	—
167	C₄⋊C₄	—	4	2	no	—
168	C₄⋊C₄	—	356	2	no	—
169	C₄⋊C₄	—	356	2	no	—
170	C₄⋊C₄	—	712	2	no	—
171	C₄⋊C₄	—	712	2	no	—
172	C₄⋊C₄	—	712	2	no	—
173	C₄²	C₂×Dic₁₇₈	1	1	yes	characteristic
174	C₄²	#13083573	1	2	yes	—
175	C₄²	#13083574	1	1	yes	characteristic
176	C₂³	#15667980	1	1	yes	characteristic
177	Q₈	—	4	4	no	—
178	Q₈	—	8	2	no	—
179	C₂×C₄	—	2	1	no	—
180	C₂×C₄	—	2	1	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	4	2	no	—
184	C₂×C₄	—	4	2	no	—
185	C₂×C₄	—	4	2	no	—
186	C₂×C₄	—	4	2	no	—
187	C₂×C₄	—	8	1	no	—
188	C₂×C₄	—	8	1	no	—
189	C₂×C₄	—	356	2	no	—
190	C₂×C₄	—	356	2	no	—
191	C₂×C₄	—	356	2	no	—
192	C₂×C₄	—	356	2	no	—
193	C₂×C₄	—	356	2	no	—
194	C₂×C₄	—	356	2	no	—
195	C₂×C₄	#15667989	1	2	yes	—
196	C₂×C₄	#15667991	1	1	yes	characteristic
197	C₂×C₄	#15667998	1	2	yes	—
198	C₂×C₄	#15667999	1	1	yes	characteristic
199	C₈	—	4	2	no	—
200	C₈	—	356	4	no	—
201	C₂²	#19151510	1	1	yes	characteristic
202	C₂²	#19151765	1	1	yes	characteristic
203	C₂²	#19151770	1	1	yes	characteristic
204	C₂²	#19151774	1	2	yes	—
205	C₂²	#19151779	1	2	yes	—
206	C₄	—	2	1	no	—
207	C₄	—	2	1	no	—
208	C₄	—	2	2	no	—
209	C₄	—	2	2	no	—
210	C₄	—	8	2	no	—
211	C₄	—	8	2	no	—
212	C₄	—	712	2	no	—
213	C₄	—	712	2	no	—
214	C₂	#26019566	1	2	yes	—
215	C₂	#26019573	1	1	yes	characteristic
216	C₂	#26019578	1	1	yes	characteristic
217	C₂	#26021408	1	1	yes	characteristic
218	C₂	#26021411	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#26028527

C₂

yes

maximal, characteristic

#26021226

C₂

yes

maximal

#26018134

C₂

yes

maximal

#26018133

C₂

yes

maximal, characteristic

C₈₉×#1960

C₂

yes

maximal, characteristic

C₂×#15667923

C₂²

yes

—

—

—

—

C₂²

yes

—

#19151360

—

#19151360

—

C₂×C₄⋊Q₈×C₈₉

C₂²

yes

characteristic

C₂×#103×C₈₉

—

C₂×#97×C₈₉

C₂²

yes

characteristic

#382×C₈₉

C₂²

yes

characteristic

C₂²×#10623494

—

C₂×C₄⋊Dic₈₉

—

C₂×C₄⋊Dic₈₉

D₄

yes

—

C₂×C₄⋊Dic₈₉

D₄

yes

—

—

—

—

—

—

—

—

C₂²×Q₈×C₈₉

—

C₂²×C₈×C₈₉

—

C₄⋊Q₈×C₈₉

—

C₄⋊Q₈×C₈₉

—

C₄⋊Q₈×C₈₉

Q₈

yes

—

C₂×C₄⋊C₄×C₈₉

—

C₂×C₄⋊C₄×C₈₉

D₄

yes

—

C₂×C₄²×C₈₉

C₂³

yes

characteristic

#103×C₈₉

—

#97×C₈₉

—

#95×C₈₉

—

C₂²×Dic₈₉

—

C₂²×Dic₈₉

—

C₂×#10623494

—

C₂×#10623494

—

C₂×#10623494

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

#13083553

—

#13083553

—

C₂×Q₈×C₈₉

—

C₂×Q₈×C₈₉

—

C₂×Q₈×C₈₉

—

C₂×Q₈×C₈₉

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

C₂×D₄

yes

—

C₂²×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄²×C₈₉

C₂×Q₈

yes

characteristic

C₄²×C₈₉

#41

yes

—

C₄²×C₈₉

yes

C₂³×C₈₉

yes

Q₈×C₈₉

—

Q₈×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂²⋊Q₈

yes

—

C₂×C₄×C₈₉

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₈₉

#124

yes

—

C₂×C₄×C₈₉

#124

yes

characteristic

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₈×C₈₉

—

#10623494

—

C₂²×C₈₉

yes

C₂²×C₈₉

yes

C₂²×C₈₉

yes

C₂²×C₈₉

yes

—

C₂²×C₈₉

#456

yes

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

100

C₄×C₈₉

—

101

C₄×C₈₉

—

102

Dic₈₉

—

103

Dic₈₉

—

104

#10361

—

maximal, Sylow

105

C₂×C₈₉

#1661

yes

—

106

C₂×C₈₉

#1662

yes

characteristic

107

C₂×C₈₉

#1663

yes

characteristic

108

C₂×C₈₉

#2027

yes

characteristic

109

C₂×C₈₉

#2028

yes

—

110

#2845

—

111

#1993

—

112

#1960

D₈₉

yes

characteristic

113

#1295

—

114

#1295

—

115

C₈₉

#10361

yes

characteristic, complemented, Sylow

116

C₂×C₄⋊Q₈

C₂×D₈₉

yes

characteristic

117

C₂×#103

—

118

C₂×#97

—

119

C₂×#97

C₂×D₈₉

yes

characteristic

120

#398

—

178

—

121

#398

—

178

—

122

#398

—

178

—

123

#382

—

124

#382

C₂×D₈₉

yes

characteristic

125

#340

—

178

—

126

#340

—

178

—

127

C₂²×Q₈

—

128

C₂²×C₈

—

129

C₂²×C₈

—

178

—

130

C₄⋊Q₈

—

131

C₄⋊Q₈

—

132

C₄⋊Q₈

Dic₁₇₈

yes

—

133

C₂×C₄⋊C₄

—

134

C₂×C₄⋊C₄

—

135

C₂×C₄⋊C₄

—

136

C₂×C₄⋊C₄

—

178

—

137

C₂×C₄⋊C₄

—

178

—

138

C₂×C₄⋊C₄

—

178

—

139

C₂×C₄⋊C₄

C₂²⋊D₈₉

yes

—

140

C₂×C₄²

C₂²×D₈₉

yes

characteristic

141

#103

—

142

#97

—

143

#97

—

178

—

144

#97

—

178

—

145

#95

—

146

#95

—

178

—

147

#95

—

178

—

148

#95

—

178

—

149

C₂×Q₈

—

150

C₂×Q₈

—

151

C₂×Q₈

—

152

C₂×Q₈

—

153

C₂²×C₄

—

154

C₂²×C₄

—

155

C₂²×C₄

—

178

—

156

C₂²×C₄

—

178

—

157

C₂²×C₄

D₄×D₈₉

yes

—

158

C₂²×C₄

C₂×C₂²⋊D₈₉

yes

characteristic

159

C₂×C₈

—

160

C₂×C₈

—

161

C₂×C₈

—

162

C₂×C₈

—

356

—

163

C₂×C₈

—

356

—

164

C₂×C₈

—

178

—

165

C₄⋊C₄

—

166

C₄⋊C₄

—

167

C₄⋊C₄

—

168

C₄⋊C₄

—

356

—

169

C₄⋊C₄

—

356

—

170

C₄⋊C₄

—

712

—

171

C₄⋊C₄

—

712

—

172

C₄⋊C₄

—

712

—

173

C₄²

C₂×Dic₁₇₈

yes

characteristic

174

C₄²

#13083573

yes

—

175

C₄²

#13083574

yes

characteristic

176

C₂³

#15667980

yes

characteristic

177

Q₈

—

178

Q₈

—

179

C₂×C₄

—

180

C₂×C₄

—

181

C₂×C₄

—

182

C₂×C₄

—

183

C₂×C₄

—

184

C₂×C₄

—

185

C₂×C₄

—

186

C₂×C₄

—

187

C₂×C₄

—

188

C₂×C₄

—

189

C₂×C₄

—

356

—

190

C₂×C₄

—

356

—

191

C₂×C₄

—

356

—

192

C₂×C₄

—

356

—

193

C₂×C₄

—

356

—

194

C₂×C₄

—

356

—

195

C₂×C₄

#15667989

yes

—

196

C₂×C₄

#15667991

yes

characteristic

197

C₂×C₄

#15667998

yes

—

198

C₂×C₄

#15667999

yes

characteristic

199

C₈

—

200

C₈

—

356

—

201

C₂²

#19151510

yes

characteristic

202

C₂²

#19151765

yes

characteristic

203

C₂²

#19151770

yes

characteristic

204

C₂²

#19151774

yes

—

205

C₂²

#19151779

yes

—

206

C₄

—

207

C₄

—

208

C₄

—

209

C₄

—

210

C₄

—

211

C₄

—

212

C₄

—

712

—

213

C₄

—

712

—

214

C₂

#26019566

yes

—

215

C₂

#26019573

yes

characteristic

216

C₂

#26019578

yes

characteristic

217

C₂

#26021408

yes

characteristic

218

C₂

#26021411

yes

—

	subgroup	G/H
center	C₂³	#15667980
commutator, cosocle	C₂×C₄²×C₈₉	C₂³
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₈₉×#1960	C₂
socle, prime radical	C₂³×C₈₉	#120
hypercenter	#1960	D₈₉
hypocenter	C₈₉	#10361
LCS	G ▷ C₂×C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1960

subgroup

G/H

center

C₂³

#15667980

commutator, cosocle

C₂×C₄²×C₈₉

C₂³

Frattini

C₂×C₄²

C₂²×D₈₉

Fitting

C₈₉×#1960

C₂

C₂³×C₈₉

D₈₉

C₈₉

G ▷ C₂×C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1960

N	H ≅ G/N	type
115. C₈₉	#10361	semidirect

H ≅ G/N

type

115. C₈₉

#10361

semidirect

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

subgroup	index
#26028527	2
#26021226	2
#26018134	2
#26018133	2
C₈₉×#1960	2
#10361	89

subgroup

index

C₈₉×#1960

Subgroups — #36457509

Conjugacy classes of subgroups

Complements

Maximal subgroups