Group #36459500 — subgroups

Subgroups — #36459500

Order 22784.

494 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 147 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	#26020409	C₂	1	1	yes	maximal, characteristic, complemented
3	#26020379	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×#15667944	—	2	2	no	—
7	C₂×#15667933	—	2	4	no	—
8	C₂×C₄×#10623494	C₂²	1	2	yes	—
9	#19151468	C₂²	1	2	yes	—
10	#19151360	—	2	4	no	—
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	#15667944	—	4	4	no	—
20	#15667933	—	4	8	no	—
21	C₄×#10623494	—	2	2	no	—
22	C₄×#10623494	—	2	2	no	—
23	C₂²×D₄×C₈₉	—	2	1	no	—
24	C₂²×D₄×C₈₉	—	2	2	no	—
25	C₂²×C₈×C₈₉	D₄	1	2	yes	—
26	#127×C₈₉	—	2	2	no	—
27	#127×C₈₉	—	2	1	no	—
28	#127×C₈₉	Q₈	1	2	yes	—
29	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
30	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
31	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
32	#102×C₈₉	—	4	4	no	—
33	C₄×C₈×C₈₉	—	2	2	no	—
34	C₂²×Dic₈₉	—	4	2	no	—
35	C₂×#10623494	—	2	4	no	—
36	C₂×#10623494	—	2	4	no	—
37	C₂×#10623494	—	2	4	no	—
38	C₄⋊Dic₈₉	—	4	4	no	—
39	C₄⋊Dic₈₉	—	4	4	no	—
40	C₄⋊Dic₈₉	—	4	4	no	—
41	C₂⁴×C₈₉	—	4	1	no	—
42	C₂×D₄×C₈₉	—	2	4	no	—
43	C₂×D₄×C₈₉	—	2	2	no	—
44	C₂×D₄×C₈₉	—	4	1	no	—
45	C₂×D₄×C₈₉	—	4	2	no	—
46	C₂×D₄×C₈₉	—	4	2	no	—
47	C₂×D₄×C₈₉	—	4	4	no	—
48	C₂×D₄×C₈₉	—	4	2	no	—
49	C₂×D₄×C₈₉	—	4	4	no	—
50	C₂²×C₄×C₈₉	—	4	1	no	—
51	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
52	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
53	C₂×C₈×C₈₉	—	2	2	no	—
54	C₂×C₈×C₈₉	—	2	2	no	—
55	C₂×C₈×C₈₉	SD₁₆	1	4	yes	—
56	C₄⋊C₄×C₈₉	—	4	2	no	—
57	C₄⋊C₄×C₈₉	—	4	4	no	—
58	C₄²×C₈₉	C₂×Q₈	1	1	yes	characteristic
59	C₄²×C₈₉	#41	1	2	yes	—
60	C₄²×C₈₉	#41	1	1	yes	characteristic
61	C₂³×C₈₉	—	4	2	no	—
62	C₂³×C₈₉	—	8	1	no	—
63	C₂³×C₈₉	—	8	2	no	—
64	C₂³×C₈₉	—	8	2	no	—
65	C₂³×C₈₉	—	8	1	no	—
66	C₂³×C₈₉	#127	1	1	yes	characteristic
67	D₄×C₈₉	—	4	4	no	—
68	D₄×C₈₉	—	4	8	no	—
69	D₄×C₈₉	—	8	2	no	—
70	D₄×C₈₉	—	8	4	no	—
71	C₂×C₄×C₈₉	—	8	1	no	—
72	C₂×C₄×C₈₉	—	8	2	no	—
73	C₂×C₄×C₈₉	C₂²⋊Q₈	1	2	yes	—
74	C₂×C₄×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
75	C₂×C₄×C₈₉	#123	1	1	yes	characteristic
76	C₂×C₄×C₈₉	#123	1	2	yes	—
77	C₂×C₄×C₈₉	C₂×SD₁₆	1	2	yes	—
78	C₂×C₄×C₈₉	#135	1	4	yes	—
79	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	4	yes	—
80	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	2	yes	—
81	C₂×Dic₈₉	—	8	2	no	—
82	C₂×Dic₈₉	—	8	2	no	—
83	C₂×Dic₈₉	—	8	2	no	—
84	C₈×C₈₉	—	2	4	no	—
85	#10623494	—	4	4	no	—
86	C₂²×C₈₉	—	8	4	no	—
87	C₂²×C₈₉	—	8	2	no	—
88	C₂²×C₈₉	—	16	1	no	—
89	C₂²×C₈₉	—	16	1	no	—
90	C₂²×C₈₉	—	16	2	no	—
91	C₂²×C₈₉	#399	1	1	yes	characteristic
92	C₂²×C₈₉	#492	1	1	yes	characteristic
93	C₂²×C₈₉	#495	1	2	yes	—
94	C₂²×C₈₉	#496	1	2	yes	—
95	C₂²×C₈₉	#496	1	1	yes	characteristic
96	C₄×C₈₉	—	16	1	no	—
97	C₄×C₈₉	#476	1	2	yes	—
98	C₄×C₈₉	#478	1	4	yes	—
99	C₄×C₈₉	#481	1	2	yes	—
100	C₄×C₈₉	#482	1	4	yes	—
101	Dic₈₉	—	16	2	no	—
102	#10677	—	89	1	no	maximal, Sylow
103	C₂×C₈₉	—	16	2	no	—
104	C₂×C₈₉	#1708	1	1	yes	characteristic
105	C₂×C₈₉	#1715	1	1	yes	characteristic
106	C₂×C₈₉	#1717	1	2	yes	—
107	C₂×C₈₉	#2092	1	2	yes	—
108	C₂×C₈₉	#2094	1	1	yes	characteristic
109	#1957	—	89	2	no	—
110	#1957	D₈₉	1	1	yes	characteristic
111	#1844	—	89	1	no	—
112	#1834	—	89	2	no	—
113	#1273	—	89	1	no	—
114	C₈₉	#10677	1	1	yes	characteristic, complemented, Sylow
115	C₂×#127	C₂×D₈₉	1	1	yes	characteristic
116	C₂×#106	—	178	2	no	—
117	C₂×#102	—	2	2	no	—
118	C₂×#102	—	178	4	no	—
119	C₂×C₄×C₈	—	89	2	no	—
120	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
121	#384	—	89	2	no	—
122	#384	C₂×D₈₉	1	1	yes	characteristic
123	#340	—	178	4	no	—
124	C₂²×D₄	—	2	2	no	—
125	C₂²×D₄	—	2	1	no	—
126	C₂²×C₈	—	89	4	no	—
127	C₂²×C₈	C₂²⋊D₈₉	1	2	yes	—
128	#127	—	2	2	no	—
129	#127	—	2	1	no	—
130	#127	Dic₁₇₈	1	2	yes	—
131	C₂×C₄⋊C₄	—	2	2	no	—
132	C₂×C₄⋊C₄	—	2	1	no	—
133	C₂×C₄⋊C₄	—	178	2	no	—
134	C₂×C₄⋊C₄	—	178	2	no	—
135	C₂×C₄⋊C₄	—	178	2	no	—
136	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
137	#106	—	356	4	no	—
138	#102	—	4	4	no	—
139	#102	—	356	8	no	—
140	C₄×C₈	—	2	2	no	—
141	C₄×C₈	—	178	2	no	—
142	C₄×C₈	—	178	2	no	—
143	C₂⁴	—	4	1	no	—
144	C₂×D₄	—	2	4	no	—
145	C₂×D₄	—	2	2	no	—
146	C₂×D₄	—	4	1	no	—
147	C₂×D₄	—	4	2	no	—
148	C₂×D₄	—	4	2	no	—
149	C₂×D₄	—	4	4	no	—
150	C₂×D₄	—	4	4	no	—
151	C₂×D₄	—	4	2	no	—
152	C₂²×C₄	—	4	1	no	—
153	C₂²×C₄	—	356	2	no	—
154	C₂²×C₄	D₄×D₈₉	1	2	yes	—
155	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
156	C₂×C₈	—	2	2	no	—
157	C₂×C₈	—	2	2	no	—
158	C₂×C₈	—	178	4	no	—
159	C₂×C₈	—	178	4	no	—
160	C₂×C₈	—	178	4	no	—
161	C₂×C₈	#13083562	1	4	yes	—
162	C₄⋊C₄	—	4	2	no	—
163	C₄⋊C₄	—	4	4	no	—
164	C₄⋊C₄	—	356	4	no	—
165	C₄⋊C₄	—	356	4	no	—
166	C₄⋊C₄	—	356	4	no	—
167	C₄²	C₂×Dic₁₇₈	1	1	yes	characteristic
168	C₄²	#13083573	1	2	yes	—
169	C₄²	#13083574	1	1	yes	characteristic
170	C₂³	—	4	2	no	—
171	C₂³	—	8	1	no	—
172	C₂³	—	8	2	no	—
173	C₂³	—	8	2	no	—
174	C₂³	—	8	1	no	—
175	C₂³	#15668008	1	1	yes	characteristic
176	D₄	—	4	4	no	—
177	D₄	—	4	8	no	—
178	D₄	—	8	2	no	—
179	D₄	—	8	4	no	—
180	C₂×C₄	—	8	1	no	—
181	C₂×C₄	—	8	2	no	—
182	C₂×C₄	—	712	2	no	—
183	C₂×C₄	—	712	2	no	—
184	C₂×C₄	—	712	2	no	—
185	C₂×C₄	#15667989	1	2	yes	—
186	C₂×C₄	#15667991	1	1	yes	characteristic
187	C₂×C₄	#15667993	1	1	yes	characteristic
188	C₂×C₄	#15667994	1	2	yes	—
189	C₂×C₄	C₂×#13083562	1	2	yes	—
190	C₂×C₄	#15668033	1	4	yes	—
191	C₂×C₄	#15668036	1	4	yes	—
192	C₂×C₄	#15668037	1	2	yes	—
193	C₈	—	2	4	no	—
194	C₈	—	356	4	no	—
195	C₂²	—	8	4	no	—
196	C₂²	—	8	2	no	—
197	C₂²	—	16	1	no	—
198	C₂²	—	16	1	no	—
199	C₂²	—	16	2	no	—
200	C₂²	#19151534	1	1	yes	characteristic
201	C₂²	#19151970	1	1	yes	characteristic
202	C₂²	#19151979	1	2	yes	—
203	C₂²	#19151986	1	2	yes	—
204	C₂²	#19151988	1	1	yes	characteristic
205	C₄	—	16	1	no	—
206	C₄	—	1424	2	no	—
207	C₄	#19151888	1	2	yes	—
208	C₄	#19151900	1	4	yes	—
209	C₄	#19151914	1	2	yes	—
210	C₄	#19151921	1	4	yes	—
211	C₂	—	16	2	no	—
212	C₂	#26019851	1	1	yes	characteristic
213	C₂	#26019878	1	1	yes	characteristic
214	C₂	#26019893	1	2	yes	—
215	C₂	#26021775	1	2	yes	—
216	C₂	#26021785	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26021059

C₂

yes

maximal

#26020409

C₂

yes

maximal, characteristic, complemented

#26020379

C₂

yes

maximal, complemented

#26018079

C₂

yes

maximal, characteristic, complemented

C₈₉×#1957

C₂

yes

maximal, characteristic

C₂×#15667944

—

C₂×#15667933

—

C₂×C₄×#10623494

C₂²

yes

—

#19151468

C₂²

yes

—

#19151360

—

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

—

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

characteristic

C₂²×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

SD₁₆

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×SD₁₆

yes

—

C₂×C₄×C₈₉

#135

yes

—

C₂×C₄×C₈₉

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₈₉

AGL(1,ℤ/8)

yes

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₈×C₈₉

—

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

yes

C₂²×C₈₉

yes

C₂²×C₈₉

yes

—

C₂²×C₈₉

#496

yes

—

C₂²×C₈₉

#496

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

#476

yes

—

C₄×C₈₉

#478

yes

—

C₄×C₈₉

#481

yes

—

100

C₄×C₈₉

#482

yes

—

101

Dic₈₉

—

102

#10677

—

maximal, Sylow

103

C₂×C₈₉

—

104

C₂×C₈₉

#1708

yes

characteristic

105

C₂×C₈₉

#1715

yes

characteristic

106

C₂×C₈₉

#1717

yes

—

107

C₂×C₈₉

#2092

yes

—

108

C₂×C₈₉

#2094

yes

characteristic

109

#1957

—

110

#1957

D₈₉

yes

characteristic

111

#1844

—

112

#1834

—

113

#1273

—

114

C₈₉

#10677

yes

characteristic, complemented, Sylow

115

C₂×#127

C₂×D₈₉

yes

characteristic

116

C₂×#106

—

178

—

117

C₂×#102

—

118

C₂×#102

—

178

—

119

C₂×C₄×C₈

—

120

C₂×C₄×C₈

C₂×D₈₉

yes

characteristic

121

#384

—

122

#384

C₂×D₈₉

yes

characteristic

123

#340

—

178

—

124

C₂²×D₄

—

125

C₂²×D₄

—

126

C₂²×C₈

—

127

C₂²×C₈

C₂²⋊D₈₉

yes

—

128

#127

—

129

#127

—

130

#127

Dic₁₇₈

yes

—

131

C₂×C₄⋊C₄

—

132

C₂×C₄⋊C₄

—

133

C₂×C₄⋊C₄

—

178

—

134

C₂×C₄⋊C₄

—

178

—

135

C₂×C₄⋊C₄

—

178

—

136

C₂×C₄²

C₂²×D₈₉

yes

characteristic

137

#106

—

356

—

138

#102

—

139

#102

—

356

—

140

C₄×C₈

—

141

C₄×C₈

—

178

—

142

C₄×C₈

—

178

—

143

C₂⁴

—

144

C₂×D₄

—

145

C₂×D₄

—

146

C₂×D₄

—

147

C₂×D₄

—

148

C₂×D₄

—

149

C₂×D₄

—

150

C₂×D₄

—

151

C₂×D₄

—

152

C₂²×C₄

—

153

C₂²×C₄

—

356

—

154

C₂²×C₄

D₄×D₈₉

yes

—

155

C₂²×C₄

C₂×C₂²⋊D₈₉

yes

characteristic

156

C₂×C₈

—

157

C₂×C₈

—

158

C₂×C₈

—

178

—

159

C₂×C₈

—

178

—

160

C₂×C₈

—

178

—

161

C₂×C₈

#13083562

yes

—

162

C₄⋊C₄

—

163

C₄⋊C₄

—

164

C₄⋊C₄

—

356

—

165

C₄⋊C₄

—

356

—

166

C₄⋊C₄

—

356

—

167

C₄²

C₂×Dic₁₇₈

yes

characteristic

168

C₄²

#13083573

yes

—

169

C₄²

#13083574

yes

characteristic

170

C₂³

—

171

C₂³

—

172

C₂³

—

173

C₂³

—

174

C₂³

—

175

C₂³

#15668008

yes

characteristic

176

D₄

—

177

D₄

—

178

D₄

—

179

D₄

—

180

C₂×C₄

—

181

C₂×C₄

—

182

C₂×C₄

—

712

—

183

C₂×C₄

—

712

—

184

C₂×C₄

—

712

—

185

C₂×C₄

#15667989

yes

—

186

C₂×C₄

#15667991

yes

characteristic

187

C₂×C₄

#15667993

yes

characteristic

188

C₂×C₄

#15667994

yes

—

189

C₂×C₄

C₂×#13083562

yes

—

190

C₂×C₄

#15668033

yes

—

191

C₂×C₄

#15668036

yes

—

192

C₂×C₄

#15668037

yes

—

193

C₈

—

194

C₈

—

356

—

195

C₂²

—

196

C₂²

—

197

C₂²

—

198

C₂²

—

199

C₂²

—

200

C₂²

#19151534

yes

characteristic

201

C₂²

#19151970

yes

characteristic

202

C₂²

#19151979

yes

—

203

C₂²

#19151986

yes

—

204

C₂²

#19151988

yes

characteristic

205

C₄

—

206

C₄

—

1424

—

207

C₄

#19151888

yes

—

208

C₄

#19151900

yes

—

209

C₄

#19151914

yes

—

210

C₄

#19151921

yes

—

211

C₂

—

212

C₂

#26019851

yes

characteristic

213

C₂

#26019878

yes

characteristic

214

C₂

#26019893

yes

—

215

C₂

#26021775

yes

—

216

C₂

#26021785

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	#10677
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. #26020409	C₂	semidirect
3. #26020379	C₂	semidirect
4. #26018079	C₂	semidirect
114. C₈₉	#10677	semidirect

H ≅ G/N

type

2. #26020409

C₂

semidirect

3. #26020379

C₂

semidirect

4. #26018079

C₂

semidirect

114. C₈₉

#10677

semidirect

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

subgroup	index
#26021059	2
#26020409	2
#26020379	2
#26018079	2
C₈₉×#1957	2
#10677	89

subgroup

index

C₈₉×#1957

Subgroups — #36459500

Conjugacy classes of subgroups

Complements

Maximal subgroups