Group #36459177 — subgroups

Subgroups — #36459177

Order 22784.

350 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 123 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	#26020380	C₂	1	1	yes	maximal, characteristic
2	#26020257	C₂	1	2	yes	maximal
3	#26018136	C₂	1	2	yes	maximal
4	#26018087	C₂	1	1	yes	maximal, characteristic
5	C₈₉×#1984	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#10623494	C₂²	1	2	yes	—
7	C₂×#15667936	—	2	2	no	—
8	#19151468	C₂²	1	2	yes	—
9	#19151361	—	2	2	no	—
10	#19151360	—	2	2	no	—
11	C₂×C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
14	#336×C₈₉	—	2	2	no	—
15	C₂²×#10623494	—	2	2	no	—
16	C₂×C₄⋊Dic₈₉	—	2	2	no	—
17	C₂×C₄⋊Dic₈₉	—	2	2	no	—
18	C₂×C₄⋊Dic₈₉	—	2	2	no	—
19	C₄⋊#10623494	—	2	2	no	—
20	C₄⋊#10623494	—	2	2	no	—
21	#15667936	—	4	4	no	—
22	C₂²×Q₈×C₈₉	D₄	1	1	yes	characteristic
23	C₂²×C₈×C₈₉	—	2	1	no	—
24	C₂²×C₈×C₈₉	D₄	1	2	yes	—
25	C₄×Q₈×C₈₉	—	2	2	no	—
26	C₄×Q₈×C₈₉	—	2	2	no	—
27	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
28	C₂×C₄⋊C₄×C₈₉	D₄	1	1	yes	characteristic
29	C₂×C₄²×C₈₉	—	2	1	no	—
30	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
31	C₄⋊C₈×C₈₉	—	2	2	no	—
32	C₄×C₈×C₈₉	—	2	2	no	—
33	C₂²×Dic₈₉	—	4	2	no	—
34	C₂×#10623494	—	4	2	no	—
35	C₂×#10623494	—	4	2	no	—
36	C₂×#10623494	—	4	2	no	—
37	C₄⋊Dic₈₉	—	4	2	no	—
38	C₄⋊Dic₈₉	—	4	2	no	—
39	C₄⋊Dic₈₉	—	4	2	no	—
40	C₄⋊Dic₈₉	—	4	2	no	—
41	C₄⋊Dic₈₉	—	8	2	no	—
42	C₂×Q₈×C₈₉	—	2	1	no	—
43	C₂×Q₈×C₈₉	—	2	2	no	—
44	C₂×Q₈×C₈₉	—	2	2	no	—
45	C₂×Q₈×C₈₉	Q₁₆	1	2	yes	—
46	C₂²×C₄×C₈₉	—	2	1	no	—
47	C₂²×C₄×C₈₉	—	2	1	no	—
48	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
51	C₂×C₈×C₈₉	—	2	2	no	—
52	C₂×C₈×C₈₉	—	2	2	no	—
53	C₂×C₈×C₈₉	—	2	2	no	—
54	C₂×C₈×C₈₉	—	4	1	no	—
55	C₂×C₈×C₈₉	—	4	2	no	—
56	C₄⋊C₄×C₈₉	—	2	2	no	—
57	C₄⋊C₄×C₈₉	—	4	2	no	—
58	C₄²×C₈₉	—	4	1	no	—
59	C₄²×C₈₉	—	4	1	no	—
60	C₄²×C₈₉	#41	1	2	yes	—
61	C₄²×C₈₉	#41	1	2	yes	—
62	C₂³×C₈₉	#121	1	1	yes	characteristic
63	Q₈×C₈₉	—	2	4	no	—
64	Q₈×C₈₉	—	4	2	no	—
65	C₂×C₄×C₈₉	—	2	2	no	—
66	C₂×C₄×C₈₉	—	2	2	no	—
67	C₂×C₄×C₈₉	—	4	1	no	—
68	C₂×C₄×C₈₉	—	4	1	no	—
69	C₂×C₄×C₈₉	—	4	1	no	—
70	C₂×C₄×C₈₉	—	4	2	no	—
71	C₂×C₄×C₈₉	#123	1	1	yes	characteristic
72	C₂×C₄×C₈₉	#123	1	2	yes	—
73	C₂×C₄×C₈₉	#124	1	1	yes	characteristic
74	C₂×C₄×C₈₉	#126	1	2	yes	—
75	C₂×C₄×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
76	C₂×C₄×C₈₉	#135	1	2	yes	—
77	C₂×C₄×C₈₉	#135	1	1	yes	characteristic
78	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	2	yes	—
79	C₂×C₄×C₈₉	#137	1	2	yes	—
80	C₂×Dic₈₉	—	8	2	no	—
81	C₂×Dic₈₉	—	8	2	no	—
82	C₂×Dic₈₉	—	8	2	no	—
83	C₈×C₈₉	—	4	2	no	—
84	C₈×C₈₉	—	8	1	no	—
85	#10623494	—	8	2	no	—
86	C₂²×C₈₉	#397	1	1	yes	characteristic
87	C₂²×C₈₉	#464	1	2	yes	—
88	C₂²×C₈₉	#467	1	1	yes	characteristic
89	C₂²×C₈₉	#469	1	1	yes	characteristic
90	C₂²×C₈₉	#471	1	2	yes	—
91	C₄×C₈₉	—	2	2	no	—
92	C₄×C₈₉	—	4	2	no	—
93	C₄×C₈₉	—	8	1	no	—
94	C₄×C₈₉	#482	1	2	yes	—
95	C₄×C₈₉	#484	1	2	yes	—
96	C₄×C₈₉	#485	1	2	yes	—
97	C₄×C₈₉	#488	1	1	yes	characteristic
98	C₄×C₈₉	#491	1	1	yes	characteristic
99	Dic₈₉	—	16	2	no	—
100	#10624	—	89	1	no	maximal, Sylow
101	C₂×C₈₉	#1684	1	2	yes	—
102	C₂×C₈₉	#1702	1	2	yes	—
103	C₂×C₈₉	#2074	1	1	yes	characteristic
104	C₂×C₈₉	#2076	1	1	yes	characteristic
105	C₂×C₈₉	#2079	1	1	yes	characteristic
106	#1984	D₈₉	1	1	yes	characteristic
107	#1834	—	89	1	no	—
108	#1803	—	89	2	no	—
109	#1296	—	89	2	no	—
110	#1277	—	89	1	no	—
111	C₈₉	#10624	1	1	yes	characteristic, complemented, Sylow
112	C₂×C₄×Q₈	C₂×D₈₉	1	1	yes	characteristic
113	C₂×C₄⋊C₈	—	89	2	no	—
114	C₂×C₄⋊C₈	C₂×D₈₉	1	1	yes	characteristic
115	C₂×#103	—	178	2	no	—
116	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
117	#384	—	89	2	no	—
118	#340	—	178	2	no	—
119	#340	—	178	2	no	—
120	#336	—	2	2	no	—
121	C₂²×Q₈	D₃₅₆	1	1	yes	characteristic
122	C₂²×C₈	—	2	1	no	—
123	C₂²×C₈	—	178	2	no	—
124	C₂²×C₈	C₂²⋊D₈₉	1	2	yes	—
125	C₄×Q₈	—	2	2	no	—
126	C₄×Q₈	—	2	2	no	—
127	C₂×C₄⋊C₄	—	2	1	no	—
128	C₂×C₄⋊C₄	—	178	2	no	—
129	C₂×C₄⋊C₄	—	178	2	no	—
130	C₂×C₄⋊C₄	—	178	2	no	—
131	C₂×C₄⋊C₄	D₃₅₆	1	1	yes	characteristic
132	C₂×C₄²	—	2	1	no	—
133	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
134	C₄⋊C₈	—	2	2	no	—
135	C₄⋊C₈	—	178	2	no	—
136	C₄⋊C₈	—	178	2	no	—
137	#103	—	356	4	no	—
138	C₄×C₈	—	2	2	no	—
139	C₂×Q₈	—	2	1	no	—
140	C₂×Q₈	—	2	2	no	—
141	C₂×Q₈	—	2	2	no	—
142	C₂×Q₈	Dic₃₅₆	1	2	yes	—
143	C₂²×C₄	—	2	1	no	—
144	C₂²×C₄	—	2	1	no	—
145	C₂²×C₄	—	356	2	no	—
146	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
147	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
148	C₂²×C₄	#13083574	1	1	yes	characteristic
149	C₂×C₈	—	2	2	no	—
150	C₂×C₈	—	2	2	no	—
151	C₂×C₈	—	2	2	no	—
152	C₂×C₈	—	4	1	no	—
153	C₂×C₈	—	4	2	no	—
154	C₂×C₈	—	356	2	no	—
155	C₂×C₈	—	356	2	no	—
156	C₂×C₈	—	356	2	no	—
157	C₄⋊C₄	—	2	2	no	—
158	C₄⋊C₄	—	4	2	no	—
159	C₄⋊C₄	—	356	2	no	—
160	C₄⋊C₄	—	356	2	no	—
161	C₄⋊C₄	—	356	2	no	—
162	C₄⋊C₄	—	356	2	no	—
163	C₄⋊C₄	—	712	2	no	—
164	C₄²	—	4	1	no	—
165	C₄²	—	4	1	no	—
166	C₄²	#13083573	1	2	yes	—
167	C₄²	#13083573	1	2	yes	—
168	C₂³	#15667986	1	1	yes	characteristic
169	Q₈	—	2	4	no	—
170	Q₈	—	4	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	4	1	no	—
174	C₂×C₄	—	4	1	no	—
175	C₂×C₄	—	4	1	no	—
176	C₂×C₄	—	4	2	no	—
177	C₂×C₄	—	712	2	no	—
178	C₂×C₄	—	712	2	no	—
179	C₂×C₄	—	712	2	no	—
180	C₂×C₄	#15667993	1	1	yes	characteristic
181	C₂×C₄	#15667996	1	2	yes	—
182	C₂×C₄	#15667999	1	1	yes	characteristic
183	C₂×C₄	#15668005	1	2	yes	—
184	C₂×C₄	C₂×Dic₃₅₆	1	1	yes	characteristic
185	C₂×C₄	#15668032	1	2	yes	—
186	C₂×C₄	#15668034	1	1	yes	characteristic
187	C₂×C₄	#15668040	1	2	yes	—
188	C₂×C₄	#15668043	1	2	yes	—
189	C₈	—	4	2	no	—
190	C₈	—	8	1	no	—
191	C₈	—	712	2	no	—
192	C₂²	#19151523	1	1	yes	characteristic
193	C₂²	#19151820	1	2	yes	—
194	C₂²	#19151839	1	1	yes	characteristic
195	C₂²	#19151851	1	1	yes	characteristic
196	C₂²	#19151861	1	2	yes	—
197	C₄	—	2	2	no	—
198	C₄	—	4	2	no	—
199	C₄	—	8	1	no	—
200	C₄	—	1424	2	no	—
201	C₄	#19151923	1	2	yes	—
202	C₄	#19151934	1	2	yes	—
203	C₄	#19151936	1	2	yes	—
204	C₄	#19151952	1	1	yes	characteristic
205	C₄	#19151966	1	1	yes	characteristic
206	C₂	#26019710	1	2	yes	—
207	C₂	#26019820	1	2	yes	—
208	C₂	#26021670	1	1	yes	characteristic
209	C₂	#26021680	1	1	yes	characteristic
210	C₂	#26021697	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26020380

C₂

yes

maximal, characteristic

#26020257

C₂

yes

maximal

#26018136

C₂

yes

maximal

#26018087

C₂

yes

maximal, characteristic

C₈₉×#1984

C₂

yes

maximal, characteristic

C₂×C₄⋊#10623494

C₂²

yes

—

C₂×#15667936

—

#19151468

C₂²

yes

—

#19151361

—

#19151360

—

C₂×C₄×Q₈×C₈₉

C₂²

yes

characteristic

C₂×C₄⋊C₈×C₈₉

C₂²

yes

characteristic

C₂×C₄×C₈×C₈₉

C₂²

yes

characteristic

#336×C₈₉

—

C₂²×#10623494

—

C₂×C₄⋊Dic₈₉

—

C₂×C₄⋊Dic₈₉

—

C₂×C₄⋊Dic₈₉

—

C₄⋊#10623494

—

C₄⋊#10623494

—

#15667936

—

C₂²×Q₈×C₈₉

D₄

yes

characteristic

C₂²×C₈×C₈₉

—

C₂²×C₈×C₈₉

D₄

yes

—

C₄×Q₈×C₈₉

—

C₄×Q₈×C₈₉

—

C₂×C₄⋊C₄×C₈₉

—

C₂×C₄⋊C₄×C₈₉

D₄

yes

characteristic

C₂×C₄²×C₈₉

—

C₂×C₄²×C₈₉

C₂³

yes

characteristic

C₄⋊C₈×C₈₉

—

C₄×C₈×C₈₉

—

C₂²×Dic₈₉

—

C₂×#10623494

—

C₂×#10623494

—

C₂×#10623494

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₂×Q₈×C₈₉

—

C₂×Q₈×C₈₉

—

C₂×Q₈×C₈₉

—

C₂×Q₈×C₈₉

Q₁₆

yes

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₈₉

#41

yes

characteristic

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄²×C₈₉

—

C₄²×C₈₉

—

C₄²×C₈₉

#41

yes

—

C₄²×C₈₉

#41

yes

—

C₂³×C₈₉

#121

yes

characteristic

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

#123

yes

characteristic

C₂×C₄×C₈₉

#123

yes

—

C₂×C₄×C₈₉

#124

yes

characteristic

C₂×C₄×C₈₉

#126

yes

—

C₂×C₄×C₈₉

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₈₉

#135

yes

—

C₂×C₄×C₈₉

#135

yes

characteristic

C₂×C₄×C₈₉

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₈₉

#137

yes

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₈×C₈₉

—

C₈×C₈₉

—

#10623494

—

C₂²×C₈₉

#397

yes

characteristic

C₂²×C₈₉

#464

yes

—

C₂²×C₈₉

yes

C₂²×C₈₉

yes

C₂²×C₈₉

yes

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

#482

yes

—

C₄×C₈₉

#484

yes

—

C₄×C₈₉

#485

yes

—

C₄×C₈₉

yes

C₄×C₈₉

yes

Dic₈₉

—

100

#10624

—

maximal, Sylow

101

C₂×C₈₉

#1684

yes

—

102

C₂×C₈₉

#1702

yes

—

103

C₂×C₈₉

#2074

yes

characteristic

104

C₂×C₈₉

#2076

yes

characteristic

105

C₂×C₈₉

#2079

yes

characteristic

106

#1984

D₈₉

yes

characteristic

107

#1834

—

108

#1803

—

109

#1296

—

110

#1277

—

111

C₈₉

#10624

yes

characteristic, complemented, Sylow

112

C₂×C₄×Q₈

C₂×D₈₉

yes

characteristic

113

C₂×C₄⋊C₈

—

114

C₂×C₄⋊C₈

C₂×D₈₉

yes

characteristic

115

C₂×#103

—

178

—

116

C₂×C₄×C₈

C₂×D₈₉

yes

characteristic

117

#384

—

118

#340

—

178

—

119

#340

—

178

—

120

#336

—

121

C₂²×Q₈

D₃₅₆

yes

characteristic

122

C₂²×C₈

—

123

C₂²×C₈

—

178

—

124

C₂²×C₈

C₂²⋊D₈₉

yes

—

125

C₄×Q₈

—

126

C₄×Q₈

—

127

C₂×C₄⋊C₄

—

128

C₂×C₄⋊C₄

—

178

—

129

C₂×C₄⋊C₄

—

178

—

130

C₂×C₄⋊C₄

—

178

—

131

C₂×C₄⋊C₄

D₃₅₆

yes

characteristic

132

C₂×C₄²

—

133

C₂×C₄²

C₂²×D₈₉

yes

characteristic

134

C₄⋊C₈

—

135

C₄⋊C₈

—

178

—

136

C₄⋊C₈

—

178

—

137

#103

—

356

—

138

C₄×C₈

—

139

C₂×Q₈

—

140

C₂×Q₈

—

141

C₂×Q₈

—

142

C₂×Q₈

Dic₃₅₆

yes

—

143

C₂²×C₄

—

144

C₂²×C₄

—

145

C₂²×C₄

—

356

—

146

C₂²×C₄

C₂×C₂²⋊D₈₉

yes

characteristic

147

C₂²×C₄

C₂×D₃₅₆

yes

characteristic

148

C₂²×C₄

#13083574

yes

characteristic

149

C₂×C₈

—

150

C₂×C₈

—

151

C₂×C₈

—

152

C₂×C₈

—

153

C₂×C₈

—

154

C₂×C₈

—

356

—

155

C₂×C₈

—

356

—

156

C₂×C₈

—

356

—

157

C₄⋊C₄

—

158

C₄⋊C₄

—

159

C₄⋊C₄

—

356

—

160

C₄⋊C₄

—

356

—

161

C₄⋊C₄

—

356

—

162

C₄⋊C₄

—

356

—

163

C₄⋊C₄

—

712

—

164

C₄²

—

165

C₄²

—

166

C₄²

#13083573

yes

—

167

C₄²

#13083573

yes

—

168

C₂³

#15667986

yes

characteristic

169

Q₈

—

170

Q₈

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

712

—

178

C₂×C₄

—

712

—

179

C₂×C₄

—

712

—

180

C₂×C₄

#15667993

yes

characteristic

181

C₂×C₄

#15667996

yes

—

182

C₂×C₄

#15667999

yes

characteristic

183

C₂×C₄

#15668005

yes

—

184

C₂×C₄

C₂×Dic₃₅₆

yes

characteristic

185

C₂×C₄

#15668032

yes

—

186

C₂×C₄

#15668034

yes

characteristic

187

C₂×C₄

#15668040

yes

—

188

C₂×C₄

#15668043

yes

—

189

C₈

—

190

C₈

—

191

C₈

—

712

—

192

C₂²

#19151523

yes

characteristic

193

C₂²

#19151820

yes

—

194

C₂²

#19151839

yes

characteristic

195

C₂²

#19151851

yes

characteristic

196

C₂²

#19151861

yes

—

197

C₄

—

198

C₄

—

199

C₄

—

200

C₄

—

1424

—

201

C₄

#19151923

yes

—

202

C₄

#19151934

yes

—

203

C₄

#19151936

yes

—

204

C₄

#19151952

yes

characteristic

205

C₄

#19151966

yes

characteristic

206

C₂

#26019710

yes

—

207

C₂

#26019820

yes

—

208

C₂

#26021670

yes

characteristic

209

C₂

#26021680

yes

characteristic

210

C₂

#26021697

yes

characteristic

	subgroup	G/H
center	C₂³	#15667986
commutator, cosocle	C₂×C₄²×C₈₉	C₂³
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₈₉×#1984	C₂
socle, prime radical	C₂³×C₈₉	#121
hypercenter	#1984	D₈₉
hypocenter	C₈₉	#10624
LCS	G ▷ C₂×C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1984

subgroup

G/H

center

C₂³

#15667986

commutator, cosocle

C₂×C₄²×C₈₉

C₂³

Frattini

C₂×C₄²

C₂²×D₈₉

Fitting

C₈₉×#1984

C₂

C₂³×C₈₉

D₈₉

C₈₉

G ▷ C₂×C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1984

N	H ≅ G/N	type
111. C₈₉	#10624	semidirect

H ≅ G/N

type

111. C₈₉

#10624

semidirect

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

subgroup	index
#26020380	2
#26020257	2
#26018136	2
#26018087	2
C₈₉×#1984	2
#10624	89

subgroup

index

C₈₉×#1984

Subgroups — #36459177

Conjugacy classes of subgroups

Complements

Maximal subgroups