Group #36473915 — subgroups

Subgroups — #36473915

Order 22784.

402 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 73 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	#26021947	C₂	1	2	yes	maximal
2	#26020805	C₂	1	2	yes	maximal
3	#26020471	C₂	1	1	yes	maximal, characteristic
4	#26018375	C₂	1	1	yes	maximal, characteristic
5	C₈₉×#3202	C₂	1	1	yes	maximal, characteristic
6	#19152365	C₂²	1	1	yes	characteristic
7	#19151646	—	2	1	no	—
8	#19151582	C₂²	1	2	yes	—
9	#19151582	C₂²	1	1	yes	characteristic
10	C₂×#15667931	—	2	2	no	—
11	#19151397	—	2	4	no	—
12	#19151368	—	2	2	no	—
13	#19151363	—	2	1	no	—
14	#536×C₈₉	C₂²	1	2	yes	—
15	#485×C₈₉	—	2	2	no	—
16	#470×C₈₉	—	2	1	no	—
17	#467×C₈₉	—	2	1	no	—
18	#451×C₈₉	—	2	2	no	—
19	#408×C₈₉	C₂²	1	1	yes	characteristic
20	#15668019	—	4	2	no	—
21	C₂×#13083558	—	2	1	no	—
22	C₂×#13083558	—	2	1	no	—
23	C₂×#13083558	—	2	2	no	—
24	C₂×#13083558	—	2	2	no	—
25	#15667947	—	4	2	no	—
26	#15667931	—	2	2	no	—
27	#15667931	—	4	4	no	—
28	#15667931	D₄	1	2	yes	—
29	#15667929	—	2	1	no	—
30	#15667929	—	2	2	no	—
31	C₂×#41×C₈₉	C₂³	1	1	yes	characteristic
32	C₂²×Q₈×C₈₉	D₄	1	2	yes	—
33	C₂×Q₁₆×C₈₉	—	2	1	no	—
34	C₂×Q₁₆×C₈₉	—	2	1	no	—
35	C₂×M₁₆×C₈₉	D₄	1	1	yes	characteristic
36	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
37	C₄⋊Q₈×C₈₉	—	2	2	no	—
38	#124×C₈₉	—	2	2	no	—
39	#123×C₈₉	—	2	2	no	—
40	#123×C₈₉	—	2	2	no	—
41	C₂²⋊Q₈×C₈₉	—	2	2	no	—
42	C₂²⋊Q₈×C₈₉	—	2	2	no	—
43	#117×C₈₉	D₄	1	2	yes	—
44	#107×C₈₉	—	2	1	no	—
45	#106×C₈₉	—	2	1	no	—
46	#103×C₈₉	—	2	2	no	—
47	#103×C₈₉	—	2	2	no	—
48	C₂²⋊C₈×C₈₉	—	2	2	no	—
49	#13083558	—	2	4	no	—
50	#13083558	—	2	2	no	—
51	#13083558	—	2	2	no	—
52	#13083558	—	4	1	no	—
53	#13083558	—	4	1	no	—
54	#13083558	—	4	2	no	—
55	#13083558	—	4	2	no	—
56	C₂×#10623494	—	4	1	no	—
57	C₂×#10623494	—	4	2	no	—
58	C₂×#10623494	—	4	1	no	—
59	C₂×#10623494	—	4	2	no	—
60	#41×C₈₉	—	4	1	no	—
61	#41×C₈₉	—	4	1	no	—
62	C₂×Q₈×C₈₉	—	2	2	no	—
63	C₂×Q₈×C₈₉	—	2	2	no	—
64	C₂×Q₈×C₈₉	—	4	2	no	—
65	C₂×Q₈×C₈₉	—	4	2	no	—
66	C₂×Q₈×C₈₉	C₂×D₄	1	1	yes	characteristic
67	C₂×D₄×C₈₉	#41	1	1	yes	characteristic
68	C₂×D₄×C₈₉	#41	1	2	yes	—
69	C₂²×C₄×C₈₉	—	2	2	no	—
70	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
71	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
72	Q₁₆×C₈₉	—	8	1	no	—
73	Q₁₆×C₈₉	—	8	1	no	—
74	M₁₆×C₈₉	—	4	1	no	—
75	C₂×C₈×C₈₉	—	2	1	no	—
76	C₂×C₈×C₈₉	—	2	1	no	—
77	C₂×C₈×C₈₉	—	4	1	no	—
78	C₄⋊C₄×C₈₉	—	2	2	no	—
79	C₄⋊C₄×C₈₉	—	4	2	no	—
80	C₄⋊C₄×C₈₉	—	4	2	no	—
81	C₂²⋊C₄×C₈₉	—	2	2	no	—
82	C₂²⋊C₄×C₈₉	—	4	2	no	—
83	C₂²⋊C₄×C₈₉	—	4	2	no	—
84	C₄²×C₈₉	—	2	2	no	—
85	C₂³×C₈₉	#123	1	2	yes	—
86	C₂³×C₈₉	#124	1	1	yes	characteristic
87	Q₈×C₈₉	—	4	1	no	—
88	Q₈×C₈₉	—	4	2	no	—
89	Q₈×C₈₉	—	4	2	no	—
90	Q₈×C₈₉	—	8	2	no	—
91	D₄×C₈₉	—	4	1	no	—
92	D₄×C₈₉	—	4	2	no	—
93	C₂×C₄×C₈₉	—	2	1	no	—
94	C₂×C₄×C₈₉	—	2	2	no	—
95	C₂×C₄×C₈₉	—	2	1	no	—
96	C₂×C₄×C₈₉	—	2	2	no	—
97	C₂×C₄×C₈₉	—	4	4	no	—
98	C₂×C₄×C₈₉	—	4	2	no	—
99	C₂×C₄×C₈₉	—	4	2	no	—
100	C₂×C₄×C₈₉	#120	1	1	yes	characteristic
101	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
102	C₂×C₄×C₈₉	#121	1	2	yes	—
103	C₈×C₈₉	—	4	1	no	—
104	C₈×C₈₉	—	4	1	no	—
105	#10623494	—	4	2	no	—
106	#10623494	—	4	2	no	—
107	#10623494	—	8	2	no	—
108	C₂²×C₈₉	—	2	2	no	—
109	C₂²×C₈₉	—	2	1	no	—
110	C₂²×C₈₉	—	4	2	no	—
111	C₂²×C₈₉	—	4	1	no	—
112	C₂²×C₈₉	#394	1	1	yes	characteristic
113	C₄×C₈₉	—	2	1	no	—
114	C₄×C₈₉	—	2	2	no	—
115	C₄×C₈₉	—	8	2	no	—
116	C₄×C₈₉	—	8	2	no	—
117	C₄×C₈₉	B(4,𝔽₂)	1	1	yes	characteristic
118	C₄×C₈₉	#458	1	1	yes	characteristic
119	#13431	—	89	1	no	maximal, Sylow
120	C₂×C₈₉	—	2	1	no	—
121	C₂×C₈₉	—	4	2	no	—
122	C₂×C₈₉	—	4	1	no	—
123	C₂×C₈₉	#2044	1	1	yes	characteristic
124	#3202	D₈₉	1	1	yes	characteristic
125	#2135	—	89	2	no	—
126	#1915	—	89	2	no	—
127	#1858	—	89	1	no	—
128	#1392	—	89	1	no	—
129	C₈₉	#13431	1	1	yes	characteristic, complemented, Sylow
130	#568	—	89	1	no	—
131	#536	C₂×D₈₉	1	2	yes	—
132	#485	—	2	2	no	—
133	#470	—	2	1	no	—
134	#467	—	2	1	no	—
135	#451	—	2	2	no	—
136	#430	—	178	1	no	—
137	#413	—	89	1	no	—
138	#413	—	89	2	no	—
139	C₂×#101	—	178	2	no	—
140	#408	C₂×D₈₉	1	1	yes	characteristic
141	#356	—	178	4	no	—
142	#344	—	178	2	no	—
143	#341	—	178	1	no	—
144	C₂×#41	C₂²×D₈₉	1	1	yes	characteristic
145	C₂²×Q₈	C₂²⋊D₈₉	1	2	yes	—
146	C₂×Q₁₆	—	2	1	no	—
147	C₂×Q₁₆	—	2	1	no	—
148	#131	—	356	2	no	—
149	C₂×M₁₆	—	178	2	no	—
150	C₂×M₁₆	—	178	1	no	—
151	C₂×M₁₆	—	178	2	no	—
152	C₂×M₁₆	—	178	1	no	—
153	C₂×M₁₆	C₂²⋊D₈₉	1	1	yes	characteristic
154	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
155	C₄⋊Q₈	—	2	2	no	—
156	#124	—	2	2	no	—
157	#123	—	2	2	no	—
158	#123	—	2	2	no	—
159	C₂²⋊Q₈	—	2	2	no	—
160	C₂²⋊Q₈	—	2	2	no	—
161	#117	C₂²⋊D₈₉	1	2	yes	—
162	#108	—	356	2	no	—
163	#107	—	2	1	no	—
164	#106	—	2	1	no	—
165	#103	—	2	2	no	—
166	#103	—	2	2	no	—
167	#101	—	89	2	no	—
168	#101	—	356	4	no	—
169	#101	—	178	2	no	—
170	#100	—	178	2	no	—
171	#100	—	178	1	no	—
172	C₂²⋊C₈	—	2	2	no	—
173	#41	—	4	1	no	—
174	#41	—	4	1	no	—
175	C₂×Q₈	—	2	2	no	—
176	C₂×Q₈	—	2	2	no	—
177	C₂×Q₈	—	4	2	no	—
178	C₂×Q₈	—	4	2	no	—
179	C₂×Q₈	D₄×D₈₉	1	1	yes	characteristic
180	C₂×D₄	#13083573	1	1	yes	characteristic
181	C₂×D₄	#13083573	1	2	yes	—
182	C₂²×C₄	—	2	2	no	—
183	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
184	C₂²×C₄	C₂×C₂²⋊D₈₉	1	2	yes	—
185	Q₁₆	—	8	1	no	—
186	Q₁₆	—	8	1	no	—
187	M₁₆	—	4	1	no	—
188	M₁₆	—	356	1	no	—
189	M₁₆	—	356	1	no	—
190	M₁₆	—	356	2	no	—
191	M₁₆	—	356	2	no	—
192	M₁₆	—	178	4	no	—
193	M₁₆	—	178	2	no	—
194	M₁₆	—	178	2	no	—
195	C₂×C₈	—	2	1	no	—
196	C₂×C₈	—	2	1	no	—
197	C₂×C₈	—	4	1	no	—
198	C₂×C₈	—	356	1	no	—
199	C₂×C₈	—	356	1	no	—
200	C₂×C₈	—	356	2	no	—
201	C₂×C₈	—	356	2	no	—
202	C₄⋊C₄	—	2	2	no	—
203	C₄⋊C₄	—	4	2	no	—
204	C₄⋊C₄	—	4	2	no	—
205	C₂²⋊C₄	—	2	2	no	—
206	C₂²⋊C₄	—	4	2	no	—
207	C₂²⋊C₄	—	4	2	no	—
208	C₄²	—	2	2	no	—
209	C₂³	#15667993	1	2	yes	—
210	C₂³	#15667999	1	1	yes	characteristic
211	Q₈	—	4	1	no	—
212	Q₈	—	4	2	no	—
213	Q₈	—	4	2	no	—
214	Q₈	—	8	2	no	—
215	D₄	—	4	1	no	—
216	D₄	—	4	2	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	4	4	no	—
222	C₂×C₄	—	4	2	no	—
223	C₂×C₄	—	4	2	no	—
224	C₂×C₄	#15667981	1	1	yes	characteristic
225	C₂×C₄	#15667983	1	1	yes	characteristic
226	C₂×C₄	#15667984	1	2	yes	—
227	C₈	—	4	1	no	—
228	C₈	—	4	1	no	—
229	C₈	—	356	2	no	—
230	C₈	—	356	2	no	—
231	C₈	—	712	2	no	—
232	C₂²	—	2	2	no	—
233	C₂²	—	2	1	no	—
234	C₂²	—	4	2	no	—
235	C₂²	—	4	1	no	—
236	C₂²	#19151512	1	1	yes	characteristic
237	C₄	—	2	1	no	—
238	C₄	—	2	2	no	—
239	C₄	—	8	2	no	—
240	C₄	—	8	2	no	—
241	C₄	#19151785	1	1	yes	characteristic
242	C₄	#19151788	1	1	yes	characteristic
243	C₂	—	2	1	no	—
244	C₂	—	4	2	no	—
245	C₂	—	4	1	no	—
246	C₂	#26021500	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26021947

C₂

yes

maximal

#26020805

C₂

yes

maximal

#26020471

C₂

yes

maximal, characteristic

#26018375

C₂

yes

maximal, characteristic

C₈₉×#3202

C₂

yes

maximal, characteristic

#19152365

C₂²

yes

characteristic

#19151646

—

#19151582

C₂²

yes

—

#19151582

C₂²

yes

—

—

—

—

#536×C₈₉

C₂²

yes

—

#485×C₈₉

—

#470×C₈₉

—

#467×C₈₉

—

#451×C₈₉

—

#408×C₈₉

C₂²

yes

—

—

—

—

—

—

—

—

D₄

yes

—

#15667929

—

#15667929

—

C₂×#41×C₈₉

C₂³

yes

characteristic

C₂²×Q₈×C₈₉

D₄

yes

—

C₂×Q₁₆×C₈₉

—

C₂×Q₁₆×C₈₉

—

C₂×M₁₆×C₈₉

D₄

yes

characteristic

C₂²×C₈×C₈₉

D₄

yes

characteristic

C₄⋊Q₈×C₈₉

—

#124×C₈₉

—

#123×C₈₉

—

#123×C₈₉

—

C₂²⋊Q₈×C₈₉

—

C₂²⋊Q₈×C₈₉

—

#117×C₈₉

D₄

yes

—

#107×C₈₉

—

#106×C₈₉

—

#103×C₈₉

—

#103×C₈₉

—

C₂²⋊C₈×C₈₉

—

—

—

—

—

—

—

—

—

—

—

—

#41×C₈₉

—

#41×C₈₉

—

C₂×Q₈×C₈₉

—

C₂×Q₈×C₈₉

—

C₂×Q₈×C₈₉

—

C₂×Q₈×C₈₉

—

C₂×Q₈×C₈₉

C₂×D₄

yes

C₂×D₄×C₈₉

yes

C₂×D₄×C₈₉

yes

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

C₂×D₄

yes

—

C₂²×C₄×C₈₉

C₂×D₄

yes

characteristic

Q₁₆×C₈₉

—

Q₁₆×C₈₉

—

M₁₆×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₂²⋊C₄×C₈₉

—

C₂²⋊C₄×C₈₉

—

C₂²⋊C₄×C₈₉

—

C₄²×C₈₉

—

C₂³×C₈₉

#123

yes

—

C₂³×C₈₉

#124

yes

characteristic

Q₈×C₈₉

—

Q₈×C₈₉

—

Q₈×C₈₉

—

Q₈×C₈₉

—

D₄×C₈₉

—

D₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

100

C₂×C₄×C₈₉

#120

yes

characteristic

101

C₂×C₄×C₈₉

#121

yes

characteristic

102

C₂×C₄×C₈₉

#121

yes

—

103

C₈×C₈₉

—

104

C₈×C₈₉

—

105

#10623494

—

106

#10623494

—

107

#10623494

—

108

C₂²×C₈₉

—

109

C₂²×C₈₉

—

110

C₂²×C₈₉

—

111

C₂²×C₈₉

—

112

C₂²×C₈₉

#394

yes

characteristic

113

C₄×C₈₉

—

114

C₄×C₈₉

—

115

C₄×C₈₉

—

116

C₄×C₈₉

—

117

C₄×C₈₉

B(4,𝔽₂)

yes

characteristic

118

C₄×C₈₉

yes

119

—

120

C₂×C₈₉

—

121

C₂×C₈₉

—

122

C₂×C₈₉

—

123

C₂×C₈₉

#2044

yes

characteristic

124

#3202

D₈₉

yes

characteristic

125

#2135

—

126

#1915

—

127

#1858

—

128

#1392

—

129

C₈₉

#13431

yes

characteristic, complemented, Sylow

130

#568

—

131

#536

C₂×D₈₉

yes

—

132

#485

—

133

#470

—

134

#467

—

135

#451

—

136

#430

—

178

—

137

#413

—

138

#413

—

139

C₂×#101

—

178

—

140

#408

C₂×D₈₉

yes

characteristic

141

#356

—

178

—

142

#344

—

178

—

143

#341

—

178

—

144

C₂×#41

C₂²×D₈₉

yes

characteristic

145

C₂²×Q₈

C₂²⋊D₈₉

yes

—

146

C₂×Q₁₆

—

147

C₂×Q₁₆

—

148

#131

—

356

—

149

C₂×M₁₆

—

178

—

150

C₂×M₁₆

—

178

—

151

C₂×M₁₆

—

178

—

152

C₂×M₁₆

—

178

—

153

C₂×M₁₆

C₂²⋊D₈₉

yes

characteristic

154

C₂²×C₈

C₂²⋊D₈₉

yes

characteristic

155

C₄⋊Q₈

—

156

#124

—

157

#123

—

158

#123

—

159

C₂²⋊Q₈

—

160

C₂²⋊Q₈

—

161

#117

C₂²⋊D₈₉

yes

—

162

#108

—

356

—

163

#107

—

164

#106

—

165

#103

—

166

#103

—

167

#101

—

168

#101

—

356

—

169

#101

—

178

—

170

#100

—

178

—

171

#100

—

178

—

172

C₂²⋊C₈

—

173

#41

—

174

#41

—

175

C₂×Q₈

—

176

C₂×Q₈

—

177

C₂×Q₈

—

178

C₂×Q₈

—

179

C₂×Q₈

D₄×D₈₉

yes

characteristic

180

C₂×D₄

#13083573

yes

characteristic

181

C₂×D₄

#13083573

yes

—

182

C₂²×C₄

—

183

C₂²×C₄

C₂×C₂²⋊D₈₉

yes

characteristic

184

C₂²×C₄

C₂×C₂²⋊D₈₉

yes

—

185

Q₁₆

—

186

Q₁₆

—

187

M₁₆

—

188

M₁₆

—

356

—

189

M₁₆

—

356

—

190

M₁₆

—

356

—

191

M₁₆

—

356

—

192

M₁₆

—

178

—

193

M₁₆

—

178

—

194

M₁₆

—

178

—

195

C₂×C₈

—

196

C₂×C₈

—

197

C₂×C₈

—

198

C₂×C₈

—

356

—

199

C₂×C₈

—

356

—

200

C₂×C₈

—

356

—

201

C₂×C₈

—

356

—

202

C₄⋊C₄

—

203

C₄⋊C₄

—

204

C₄⋊C₄

—

205

C₂²⋊C₄

—

206

C₂²⋊C₄

—

207

C₂²⋊C₄

—

208

C₄²

—

209

C₂³

#15667993

yes

—

210

C₂³

#15667999

yes

characteristic

211

Q₈

—

212

Q₈

—

213

Q₈

—

214

Q₈

—

215

D₄

—

216

D₄

—

217

C₂×C₄

—

218

C₂×C₄

—

219

C₂×C₄

—

220

C₂×C₄

—

221

C₂×C₄

—

222

C₂×C₄

—

223

C₂×C₄

—

224

C₂×C₄

#15667981

yes

characteristic

225

C₂×C₄

#15667983

yes

characteristic

226

C₂×C₄

#15667984

yes

—

227

C₈

—

228

C₈

—

229

C₈

—

356

—

230

C₈

—

356

—

231

C₈

—

712

—

232

C₂²

—

233

C₂²

—

234

C₂²

—

235

C₂²

—

236

C₂²

#19151512

yes

characteristic

237

C₄

—

238

C₄

—

239

C₄

—

240

C₄

—

241

C₄

#19151785

yes

characteristic

242

C₄

#19151788

yes

characteristic

243

C₂

—

244

C₂

—

245

C₂

—

246

C₂

#26021500

yes

characteristic

	subgroup	G/H
center	C₂	#26021500
commutator, prime radical, cosocle	C₂×#41×C₈₉	C₂³
Frattini	C₂×#41	C₂²×D₈₉
Fitting	C₈₉×#3202	C₂
socle	C₂×C₈₉	#2044
hypercenter	#3202	D₈₉
hypocenter	C₈₉	#13431
LCS	G ▷ C₂×#41×C₈₉ ▷ C₂²×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#41 ◁ #3202

subgroup

G/H

center

C₂

#26021500

commutator, prime radical, cosocle

C₂×#41×C₈₉

C₂³

Frattini

C₂×#41

C₂²×D₈₉

Fitting

C₈₉×#3202

C₂

C₂×C₈₉

D₈₉

C₈₉

G ▷ C₂×#41×C₈₉ ▷ C₂²×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#41 ◁ #3202

N	H ≅ G/N	type
129. C₈₉	#13431	semidirect

H ≅ G/N

type

129. C₈₉

#13431

semidirect

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

subgroup	index
#26021947	2
#26020805	2
#26020471	2
#26018375	2
C₈₉×#3202	2
#13431	89

subgroup

index

C₈₉×#3202

Subgroups — #36473915

Conjugacy classes of subgroups

Complements

Maximal subgroups