Group #36459477 — subgroups

Subgroups — #36459477

Order 22784.

526 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 155 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	#26021059	C₂	1	2	yes	maximal
2	#26020412	C₂	1	1	yes	maximal, characteristic, complemented
3	#26020411	C₂	1	2	yes	maximal, complemented
4	#26018079	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈₉×#1957	C₂	1	1	yes	maximal, characteristic
6	C₂×#15667946	—	2	2	no	—
7	C₂×#15667945	—	2	4	no	—
8	C₂×#15667933	—	2	4	no	—
9	C₂×C₄×#10623494	C₂²	1	2	yes	—
10	#19151468	C₂²	1	2	yes	—
11	C₂×#127×C₈₉	C₂²	1	1	yes	characteristic
12	C₂×#102×C₈₉	—	2	2	no	—
13	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
14	#384×C₈₉	C₂²	1	1	yes	characteristic
15	C₂²×#10623494	D₄	1	4	yes	—
16	C₂×C₄⋊Dic₈₉	—	2	2	no	—
17	C₂×C₄⋊Dic₈₉	—	2	2	no	—
18	C₂×C₄⋊Dic₈₉	—	2	2	no	—
19	#15667946	—	4	4	no	—
20	#15667945	—	4	8	no	—
21	#15667933	—	4	8	no	—
22	C₄×#10623494	—	2	2	no	—
23	C₄×#10623494	—	2	2	no	—
24	C₂²×D₄×C₈₉	—	2	1	no	—
25	C₂²×D₄×C₈₉	—	2	2	no	—
26	C₂²×C₈×C₈₉	D₄	1	2	yes	—
27	#127×C₈₉	—	2	2	no	—
28	#127×C₈₉	—	2	1	no	—
29	#127×C₈₉	Q₈	1	2	yes	—
30	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
31	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
32	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
33	#102×C₈₉	—	4	4	no	—
34	C₄×C₈×C₈₉	—	2	2	no	—
35	C₂²×Dic₈₉	—	4	2	no	—
36	C₂×#10623494	—	2	4	no	—
37	C₂×#10623494	—	2	4	no	—
38	C₂×#10623494	D₈	1	8	yes	—
39	C₄⋊Dic₈₉	—	4	4	no	—
40	C₄⋊Dic₈₉	—	4	4	no	—
41	C₄⋊Dic₈₉	—	4	4	no	—
42	C₂⁴×C₈₉	—	4	1	no	—
43	C₂×D₄×C₈₉	—	2	4	no	—
44	C₂×D₄×C₈₉	—	2	2	no	—
45	C₂×D₄×C₈₉	—	4	1	no	—
46	C₂×D₄×C₈₉	—	4	2	no	—
47	C₂×D₄×C₈₉	—	4	2	no	—
48	C₂×D₄×C₈₉	—	4	4	no	—
49	C₂×D₄×C₈₉	—	4	4	no	—
50	C₂×D₄×C₈₉	—	4	2	no	—
51	C₂²×C₄×C₈₉	—	4	1	no	—
52	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
53	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
54	C₂×C₈×C₈₉	—	2	2	no	—
55	C₂×C₈×C₈₉	—	2	2	no	—
56	C₂×C₈×C₈₉	D₈	1	4	yes	—
57	C₄⋊C₄×C₈₉	—	4	2	no	—
58	C₄⋊C₄×C₈₉	—	4	4	no	—
59	C₄²×C₈₉	C₂×Q₈	1	1	yes	characteristic
60	C₄²×C₈₉	#41	1	2	yes	—
61	C₄²×C₈₉	#41	1	1	yes	characteristic
62	C₂³×C₈₉	—	4	2	no	—
63	C₂³×C₈₉	—	8	2	no	—
64	C₂³×C₈₉	—	8	1	no	—
65	C₂³×C₈₉	—	8	2	no	—
66	C₂³×C₈₉	—	8	1	no	—
67	C₂³×C₈₉	#127	1	1	yes	characteristic
68	D₄×C₈₉	—	4	8	no	—
69	D₄×C₈₉	—	4	4	no	—
70	D₄×C₈₉	—	8	2	no	—
71	D₄×C₈₉	—	8	4	no	—
72	C₂×C₄×C₈₉	—	8	2	no	—
73	C₂×C₄×C₈₉	—	8	1	no	—
74	C₂×C₄×C₈₉	C₂²⋊Q₈	1	2	yes	—
75	C₂×C₄×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
76	C₂×C₄×C₈₉	#123	1	1	yes	characteristic
77	C₂×C₄×C₈₉	#123	1	2	yes	—
78	C₂×C₄×C₈₉	C₂×D₈	1	4	yes	—
79	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
80	C₂×C₄×C₈₉	#137	1	4	yes	—
81	C₂×C₄×C₈₉	#137	1	2	yes	—
82	C₂×Dic₈₉	—	8	2	no	—
83	C₂×Dic₈₉	—	8	2	no	—
84	C₂×Dic₈₉	—	8	2	no	—
85	C₈×C₈₉	—	2	4	no	—
86	#10623494	—	2	8	no	—
87	C₂²×C₈₉	—	8	4	no	—
88	C₂²×C₈₉	—	8	2	no	—
89	C₂²×C₈₉	—	16	1	no	—
90	C₂²×C₈₉	—	16	1	no	—
91	C₂²×C₈₉	—	16	2	no	—
92	C₂²×C₈₉	#399	1	1	yes	characteristic
93	C₂²×C₈₉	#493	1	2	yes	—
94	C₂²×C₈₉	#493	1	1	yes	characteristic
95	C₂²×C₈₉	#497	1	2	yes	—
96	C₂²×C₈₉	#497	1	1	yes	characteristic
97	C₄×C₈₉	—	16	1	no	—
98	C₄×C₈₉	#474	1	4	yes	—
99	C₄×C₈₉	#474	1	2	yes	—
100	C₄×C₈₉	#480	1	2	yes	—
101	C₄×C₈₉	#480	1	4	yes	—
102	Dic₈₉	—	16	2	no	—
103	#10673	—	89	1	no	maximal, Sylow
104	C₂×C₈₉	—	16	2	no	—
105	C₂×C₈₉	#1712	1	1	yes	characteristic
106	C₂×C₈₉	#1722	1	1	yes	characteristic
107	C₂×C₈₉	#1722	1	2	yes	—
108	C₂×C₈₉	#2093	1	2	yes	—
109	C₂×C₈₉	#2093	1	1	yes	characteristic
110	#1957	—	89	2	no	—
111	#1957	D₈₉	1	1	yes	characteristic
112	#1845	—	89	1	no	—
113	#1845	—	89	2	no	—
114	#1273	—	89	1	no	—
115	C₈₉	#10673	1	1	yes	characteristic, complemented, Sylow
116	C₂×#127	C₂×D₈₉	1	1	yes	characteristic
117	C₂×#107	—	178	4	no	—
118	C₂×#107	—	178	2	no	—
119	C₂×#102	—	2	2	no	—
120	C₂×#102	—	178	4	no	—
121	C₂×C₄×C₈	—	89	2	no	—
122	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
123	#384	—	89	2	no	—
124	#384	C₂×D₈₉	1	1	yes	characteristic
125	C₂²×D₄	—	2	2	no	—
126	C₂²×D₄	—	2	1	no	—
127	C₂²×C₈	—	89	4	no	—
128	C₂²×C₈	C₂²⋊D₈₉	1	2	yes	—
129	#127	—	2	2	no	—
130	#127	—	2	1	no	—
131	#127	Dic₁₇₈	1	2	yes	—
132	C₂×C₄⋊C₄	—	2	2	no	—
133	C₂×C₄⋊C₄	—	2	1	no	—
134	C₂×C₄⋊C₄	—	178	2	no	—
135	C₂×C₄⋊C₄	—	178	2	no	—
136	C₂×C₄⋊C₄	—	178	2	no	—
137	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
138	#107	—	356	8	no	—
139	#107	—	356	4	no	—
140	#102	—	4	4	no	—
141	#102	—	356	8	no	—
142	C₄×C₈	—	2	2	no	—
143	C₄×C₈	—	178	2	no	—
144	C₄×C₈	—	178	2	no	—
145	C₂⁴	—	4	1	no	—
146	C₂×D₄	—	2	4	no	—
147	C₂×D₄	—	2	2	no	—
148	C₂×D₄	—	4	1	no	—
149	C₂×D₄	—	4	2	no	—
150	C₂×D₄	—	4	4	no	—
151	C₂×D₄	—	4	2	no	—
152	C₂×D₄	—	4	4	no	—
153	C₂×D₄	—	4	2	no	—
154	C₂²×C₄	—	4	1	no	—
155	C₂²×C₄	—	356	2	no	—
156	C₂²×C₄	D₄×D₈₉	1	2	yes	—
157	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
158	C₂×C₈	—	2	2	no	—
159	C₂×C₈	—	2	2	no	—
160	C₂×C₈	—	89	8	no	—
161	C₂×C₈	—	178	4	no	—
162	C₂×C₈	—	178	4	no	—
163	C₂×C₈	#13083559	1	4	yes	—
164	C₄⋊C₄	—	4	4	no	—
165	C₄⋊C₄	—	4	2	no	—
166	C₄⋊C₄	—	356	4	no	—
167	C₄⋊C₄	—	356	4	no	—
168	C₄⋊C₄	—	356	4	no	—
169	C₄²	C₂×Dic₁₇₈	1	1	yes	characteristic
170	C₄²	#13083573	1	2	yes	—
171	C₄²	#13083574	1	1	yes	characteristic
172	C₂³	—	4	2	no	—
173	C₂³	—	8	2	no	—
174	C₂³	—	8	1	no	—
175	C₂³	—	8	2	no	—
176	C₂³	—	8	1	no	—
177	C₂³	#15668008	1	1	yes	characteristic
178	D₄	—	4	8	no	—
179	D₄	—	4	4	no	—
180	D₄	—	8	2	no	—
181	D₄	—	8	4	no	—
182	C₂×C₄	—	8	2	no	—
183	C₂×C₄	—	8	1	no	—
184	C₂×C₄	—	712	2	no	—
185	C₂×C₄	—	712	2	no	—
186	C₂×C₄	—	712	2	no	—
187	C₂×C₄	#15667989	1	2	yes	—
188	C₂×C₄	#15667991	1	1	yes	characteristic
189	C₂×C₄	#15667993	1	1	yes	characteristic
190	C₂×C₄	#15667994	1	2	yes	—
191	C₂×C₄	D₈×D₈₉	1	4	yes	—
192	C₂×C₄	C₂×#13083559	1	2	yes	—
193	C₂×C₄	#15668041	1	4	yes	—
194	C₂×C₄	#15668042	1	2	yes	—
195	C₈	—	2	4	no	—
196	C₈	—	178	8	no	—
197	C₂²	—	8	4	no	—
198	C₂²	—	8	2	no	—
199	C₂²	—	16	1	no	—
200	C₂²	—	16	1	no	—
201	C₂²	—	16	2	no	—
202	C₂²	#19151534	1	1	yes	characteristic
203	C₂²	#19151973	1	2	yes	—
204	C₂²	#19151974	1	1	yes	characteristic
205	C₂²	#19151989	1	2	yes	—
206	C₂²	#19151993	1	1	yes	characteristic
207	C₄	—	16	1	no	—
208	C₄	—	1424	2	no	—
209	C₄	#19151876	1	4	yes	—
210	C₄	#19151878	1	2	yes	—
211	C₄	#19151909	1	2	yes	—
212	C₄	#19151910	1	4	yes	—
213	C₂	—	16	2	no	—
214	C₂	#26019866	1	1	yes	characteristic
215	C₂	#26019921	1	1	yes	characteristic
216	C₂	#26019925	1	2	yes	—
217	C₂	#26021778	1	2	yes	—
218	C₂	#26021780	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26021059

C₂

yes

maximal

#26020412

C₂

yes

maximal, characteristic, complemented

#26020411

C₂

yes

maximal, complemented

#26018079

C₂

yes

maximal, characteristic, complemented

C₈₉×#1957

C₂

yes

maximal, characteristic

—

—

—

C₂²

yes

—

#19151468

C₂²

yes

—

C₂×#127×C₈₉

C₂²

yes

characteristic

C₂×#102×C₈₉

—

C₂×C₄×C₈×C₈₉

C₂²

yes

characteristic

#384×C₈₉

C₂²

yes

characteristic

C₂²×#10623494

D₄

yes

—

C₂×C₄⋊Dic₈₉

—

C₂×C₄⋊Dic₈₉

—

C₂×C₄⋊Dic₈₉

—

—

—

—

—

—

C₂²×D₄×C₈₉

—

C₂²×D₄×C₈₉

—

C₂²×C₈×C₈₉

D₄

yes

—

#127×C₈₉

—

#127×C₈₉

—

#127×C₈₉

Q₈

yes

—

C₂×C₄⋊C₄×C₈₉

—

C₂×C₄⋊C₄×C₈₉

—

C₂×C₄²×C₈₉

C₂³

yes

characteristic

#102×C₈₉

—

C₄×C₈×C₈₉

—

C₂²×Dic₈₉

—

C₂×#10623494

—

C₂×#10623494

—

C₂×#10623494

D₈

yes

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₂⁴×C₈₉

—

C₂×D₄×C₈₉

—

C₂×D₄×C₈₉

—

C₂×D₄×C₈₉

—

C₂×D₄×C₈₉

—

C₂×D₄×C₈₉

—

C₂×D₄×C₈₉

—

C₂×D₄×C₈₉

—

C₂×D₄×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

D₈

yes

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄²×C₈₉

C₂×Q₈

yes

characteristic

C₄²×C₈₉

#41

yes

—

C₄²×C₈₉

#41

yes

characteristic

C₂³×C₈₉

—

C₂³×C₈₉

—

C₂³×C₈₉

—

C₂³×C₈₉

—

C₂³×C₈₉

—

C₂³×C₈₉

#127

yes

characteristic

D₄×C₈₉

—

D₄×C₈₉

—

D₄×C₈₉

—

D₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂²⋊Q₈

yes

—

C₂×C₄×C₈₉

C₂²⋊Q₈

yes

C₂×C₄×C₈₉

yes

C₂×C₄×C₈₉

yes

—

C₂×C₄×C₈₉

C₂×D₈

yes

—

C₂×C₄×C₈₉

C₂×D₈

yes

—

C₂×C₄×C₈₉

#137

yes

—

C₂×C₄×C₈₉

#137

yes

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₈×C₈₉

—

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

#399

yes

characteristic

C₂²×C₈₉

#493

yes

—

C₂²×C₈₉

#493

yes

characteristic

C₂²×C₈₉

#497

yes

—

C₂²×C₈₉

#497

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

#474

yes

—

C₄×C₈₉

#474

yes

—

100

C₄×C₈₉

#480

yes

—

101

C₄×C₈₉

#480

yes

—

102

Dic₈₉

—

103

#10673

—

maximal, Sylow

104

C₂×C₈₉

—

105

C₂×C₈₉

#1712

yes

characteristic

106

C₂×C₈₉

#1722

yes

characteristic

107

C₂×C₈₉

#1722

yes

—

108

C₂×C₈₉

#2093

yes

—

109

C₂×C₈₉

#2093

yes

characteristic

110

#1957

—

111

#1957

D₈₉

yes

characteristic

112

#1845

—

113

#1845

—

114

#1273

—

115

C₈₉

#10673

yes

characteristic, complemented, Sylow

116

C₂×#127

C₂×D₈₉

yes

characteristic

117

C₂×#107

—

178

—

118

C₂×#107

—

178

—

119

C₂×#102

—

120

C₂×#102

—

178

—

121

C₂×C₄×C₈

—

122

C₂×C₄×C₈

C₂×D₈₉

yes

characteristic

123

#384

—

124

#384

C₂×D₈₉

yes

characteristic

125

C₂²×D₄

—

126

C₂²×D₄

—

127

C₂²×C₈

—

128

C₂²×C₈

C₂²⋊D₈₉

yes

—

129

#127

—

130

#127

—

131

#127

Dic₁₇₈

yes

—

132

C₂×C₄⋊C₄

—

133

C₂×C₄⋊C₄

—

134

C₂×C₄⋊C₄

—

178

—

135

C₂×C₄⋊C₄

—

178

—

136

C₂×C₄⋊C₄

—

178

—

137

C₂×C₄²

C₂²×D₈₉

yes

characteristic

138

#107

—

356

—

139

#107

—

356

—

140

#102

—

141

#102

—

356

—

142

C₄×C₈

—

143

C₄×C₈

—

178

—

144

C₄×C₈

—

178

—

145

C₂⁴

—

146

C₂×D₄

—

147

C₂×D₄

—

148

C₂×D₄

—

149

C₂×D₄

—

150

C₂×D₄

—

151

C₂×D₄

—

152

C₂×D₄

—

153

C₂×D₄

—

154

C₂²×C₄

—

155

C₂²×C₄

—

356

—

156

C₂²×C₄

D₄×D₈₉

yes

—

157

C₂²×C₄

C₂×C₂²⋊D₈₉

yes

characteristic

158

C₂×C₈

—

159

C₂×C₈

—

160

C₂×C₈

—

161

C₂×C₈

—

178

—

162

C₂×C₈

—

178

—

163

C₂×C₈

#13083559

yes

—

164

C₄⋊C₄

—

165

C₄⋊C₄

—

166

C₄⋊C₄

—

356

—

167

C₄⋊C₄

—

356

—

168

C₄⋊C₄

—

356

—

169

C₄²

C₂×Dic₁₇₈

yes

characteristic

170

C₄²

#13083573

yes

—

171

C₄²

#13083574

yes

characteristic

172

C₂³

—

173

C₂³

—

174

C₂³

—

175

C₂³

—

176

C₂³

—

177

C₂³

#15668008

yes

characteristic

178

D₄

—

179

D₄

—

180

D₄

—

181

D₄

—

182

C₂×C₄

—

183

C₂×C₄

—

184

C₂×C₄

—

712

—

185

C₂×C₄

—

712

—

186

C₂×C₄

—

712

—

187

C₂×C₄

#15667989

yes

—

188

C₂×C₄

#15667991

yes

characteristic

189

C₂×C₄

#15667993

yes

characteristic

190

C₂×C₄

#15667994

yes

—

191

C₂×C₄

D₈×D₈₉

yes

—

192

C₂×C₄

C₂×#13083559

yes

—

193

C₂×C₄

#15668041

yes

—

194

C₂×C₄

#15668042

yes

—

195

C₈

—

196

C₈

—

178

—

197

C₂²

—

198

C₂²

—

199

C₂²

—

200

C₂²

—

201

C₂²

—

202

C₂²

#19151534

yes

characteristic

203

C₂²

#19151973

yes

—

204

C₂²

#19151974

yes

characteristic

205

C₂²

#19151989

yes

—

206

C₂²

#19151993

yes

characteristic

207

C₄

—

208

C₄

—

1424

—

209

C₄

#19151876

yes

—

210

C₄

#19151878

yes

—

211

C₄

#19151909

yes

—

212

C₄

#19151910

yes

—

213

C₂

—

214

C₂

#26019866

yes

characteristic

215

C₂

#26019921

yes

characteristic

216

C₂

#26019925

yes

—

217

C₂

#26021778

yes

—

218

C₂

#26021780

yes

characteristic

	subgroup	G/H
center	C₂³	#15668008
commutator, cosocle	C₂×C₄²×C₈₉	C₂³
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₈₉×#1957	C₂
socle	C₂³×C₈₉	#127
prime radical	C₂×#127×C₈₉	C₂²
hypercenter	#1957	D₈₉
hypocenter	C₈₉	#10673
LCS	G ▷ C₂×C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1957

subgroup

G/H

center

C₂³

#15668008

commutator, cosocle

C₂×C₄²×C₈₉

C₂³

Frattini

C₂×C₄²

C₂²×D₈₉

Fitting

C₈₉×#1957

C₂

socle

C₂³×C₈₉

#127

prime radical

C₂×#127×C₈₉

C₂²

D₈₉

C₈₉

G ▷ C₂×C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1957

N	H ≅ G/N	type
2. #26020412	C₂	semidirect
3. #26020411	C₂	semidirect
4. #26018079	C₂	semidirect
115. C₈₉	#10673	semidirect

H ≅ G/N

type

2. #26020412

C₂

semidirect

3. #26020411

C₂

semidirect

4. #26018079

C₂

semidirect

115. C₈₉

#10673

semidirect

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

subgroup	index
#26021059	2
#26020412	2
#26020411	2
#26018079	2
C₈₉×#1957	2
#10673	89

subgroup

index

C₈₉×#1957

Subgroups — #36459477

Conjugacy classes of subgroups

Complements

Maximal subgroups