Group #36454578 — subgroups

Subgroups — #36454578

Order 22784.

438 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 438 conjugacy classes of subgroups — too many to draw. Only its 159 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	#26028615	C₂	1	1	yes	maximal, characteristic
2	#26020105	C₂	1	1	yes	maximal, characteristic
3	#26018136	C₂	1	4	yes	maximal
4	C₈₉×#1842	C₂	1	1	yes	maximal, characteristic
5	C₂×#15668004	—	2	1	no	—
6	C₂×C₄⋊#10623494	C₂²	1	2	yes	—
7	C₂×#15667923	—	2	1	no	—
8	#19151496	—	2	2	no	—
9	#19151496	C₄	1	4	yes	—
10	#19151468	C₂²	1	2	yes	—
11	#19151464	—	2	1	no	—
12	#19151353	—	2	1	no	—
13	C₂×C₄⋊C₈×C₈₉	—	2	2	no	—
14	C₂×C₄⋊C₈×C₈₉	C₂²	1	2	yes	—
15	C₄³×C₈₉	C₂²	1	1	yes	characteristic
16	C₂²×#10623494	—	2	2	no	—
17	#15668004	—	4	2	no	—
18	#15668004	—	4	2	no	—
19	C₂×C₄⋊Dic₈₉	—	2	2	no	—
20	C₂×C₄⋊Dic₈₉	—	2	2	no	—
21	C₂×C₄⋊Dic₈₉	C₂×C₄	1	4	yes	—
22	C₂×#13083553	—	2	4	no	—
23	C₄⋊#10623494	—	2	2	no	—
24	C₄⋊#10623494	Q₈	1	4	yes	—
25	#15667923	—	4	2	no	—
26	#15667919	—	2	4	no	—
27	C₂²×C₈×C₈₉	—	2	2	no	—
28	C₂×C₄²×C₈₉	C₂×C₄	1	2	yes	—
29	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
30	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
31	C₂×C₄²×C₈₉	D₄	1	2	yes	—
32	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
33	C₄⋊C₈×C₈₉	—	2	4	no	—
34	C₄⋊C₈×C₈₉	—	4	4	no	—
35	C₂²×Dic₈₉	—	2	4	no	—
36	C₂×#10623494	—	2	4	no	—
37	C₂×#10623494	—	4	2	no	—
38	C₂×#10623494	—	4	2	no	—
39	C₄⋊Dic₈₉	—	4	2	no	—
40	C₄⋊Dic₈₉	—	4	2	no	—
41	C₄⋊Dic₈₉	—	4	2	no	—
42	C₄⋊Dic₈₉	—	4	2	no	—
43	C₄⋊Dic₈₉	—	4	4	no	—
44	#13083553	—	4	8	no	—
45	C₂²×C₄×C₈₉	C₂²⋊C₄	1	2	yes	—
46	C₂²×C₄×C₈₉	C₂²⋊C₄	1	2	yes	—
47	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
48	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
50	C₂×C₈×C₈₉	—	4	2	no	—
51	C₂×C₈×C₈₉	—	4	4	no	—
52	C₄²×C₈₉	—	2	2	no	—
53	C₄²×C₈₉	—	2	2	no	—
54	C₄²×C₈₉	—	2	2	no	—
55	C₄²×C₈₉	—	4	2	no	—
56	C₄²×C₈₉	D₈	1	2	yes	—
57	C₄²×C₈₉	SD₁₆	1	2	yes	—
58	C₄²×C₈₉	C₂×Q₈	1	2	yes	—
59	C₄²×C₈₉	#41	1	2	yes	—
60	C₂³×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
61	C₂×C₄×C₈₉	—	2	2	no	—
62	C₂×C₄×C₈₉	—	2	2	no	—
63	C₂×C₄×C₈₉	—	2	2	no	—
64	C₂×C₄×C₈₉	—	2	2	no	—
65	C₂×C₄×C₈₉	—	2	2	no	—
66	C₂×C₄×C₈₉	—	2	2	no	—
67	C₂×C₄×C₈₉	#102	1	4	yes	—
68	C₂×C₄×C₈₉	C₄×Q₈	1	2	yes	—
69	C₂×C₄×C₈₉	C₂²⋊Q₈	1	2	yes	—
70	C₂×C₄×C₈₉	#124	1	1	yes	characteristic
71	C₂×C₄×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
72	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
73	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
74	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	2	yes	—
75	C₂×C₄×C₈₉	#137	1	2	yes	—
76	C₂×C₄×C₈₉	#137	1	2	yes	—
77	C₂×Dic₈₉	—	4	4	no	—
78	C₂×Dic₈₉	—	4	4	no	—
79	C₂×Dic₈₉	—	4	4	no	—
80	C₈×C₈₉	—	8	2	no	—
81	#10623494	—	4	4	no	—
82	C₂²×C₈₉	#391	1	1	yes	characteristic
83	C₂²×C₈₉	#413	1	2	yes	—
84	C₂²×C₈₉	C₂×#102	1	1	yes	characteristic
85	C₂²×C₈₉	#417	1	2	yes	—
86	C₂²×C₈₉	#419	1	1	yes	characteristic
87	C₄×C₈₉	—	2	4	no	—
88	C₄×C₈₉	—	2	2	no	—
89	C₄×C₈₉	—	4	2	no	—
90	C₄×C₈₉	#474	1	2	yes	—
91	C₄×C₈₉	#475	1	2	yes	—
92	C₄×C₈₉	#490	1	1	yes	characteristic
93	C₄×C₈₉	#491	1	1	yes	characteristic
94	C₄×C₈₉	#501	1	2	yes	—
95	Dic₈₉	—	8	4	no	—
96	#9904	—	89	1	no	maximal, Sylow
97	C₂×C₈₉	#1547	1	2	yes	—
98	C₂×C₈₉	#1559	1	2	yes	—
99	C₂×C₈₉	#1988	1	1	yes	characteristic
100	C₂×C₈₉	#1992	1	1	yes	characteristic
101	C₂×C₈₉	#1998	1	1	yes	characteristic
102	#2859	—	89	1	no	—
103	#1842	D₈₉	1	1	yes	characteristic
104	#1767	—	89	1	no	—
105	#1296	—	89	4	no	—
106	C₈₉	#9904	1	1	yes	characteristic, complemented, Sylow
107	C₂×#125	—	178	1	no	—
108	C₂×C₄⋊C₈	—	2	2	no	—
109	C₂×C₄⋊C₈	—	89	2	no	—
110	C₂×C₄⋊C₈	C₂×D₈₉	1	2	yes	—
111	C₂×#97	—	178	1	no	—
112	#389	—	89	4	no	—
113	#389	—	178	2	no	—
114	#384	—	89	2	no	—
115	#382	—	178	1	no	—
116	C₄³	C₂×D₈₉	1	1	yes	characteristic
117	#336	—	178	1	no	—
118	C₂²×C₈	—	2	2	no	—
119	C₂²×C₈	—	178	2	no	—
120	#125	—	356	2	no	—
121	#125	—	356	2	no	—
122	C₂×C₄⋊C₄	—	89	4	no	—
123	C₂×C₄⋊C₄	—	178	4	no	—
124	C₂×C₄⋊C₄	—	178	2	no	—
125	C₂×C₄⋊C₄	—	178	2	no	—
126	C₂×C₄²	C₄×D₈₉	1	2	yes	—
127	C₂×C₄²	D₃₅₆	1	1	yes	characteristic
128	C₂×C₄²	D₃₅₆	1	1	yes	characteristic
129	C₂×C₄²	C₂²⋊D₈₉	1	2	yes	—
130	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
131	C₄⋊C₈	—	2	4	no	—
132	C₄⋊C₈	—	4	4	no	—
133	C₄⋊C₈	—	89	4	no	—
134	C₄⋊C₈	—	178	2	no	—
135	#97	—	356	2	no	—
136	#95	—	178	4	no	—
137	C₂²×C₄	—	178	4	no	—
138	C₂²×C₄	#13083551	1	2	yes	—
139	C₂²×C₄	#13083551	1	2	yes	—
140	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
141	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
142	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
143	C₂×C₈	—	4	2	no	—
144	C₂×C₈	—	4	4	no	—
145	C₂×C₈	—	356	2	no	—
146	C₂×C₈	—	356	2	no	—
147	C₂×C₈	—	178	4	no	—
148	C₄⋊C₄	—	356	4	no	—
149	C₄⋊C₄	—	356	2	no	—
150	C₄⋊C₄	—	356	2	no	—
151	C₄⋊C₄	—	356	2	no	—
152	C₄⋊C₄	—	356	2	no	—
153	C₄⋊C₄	—	356	8	no	—
154	C₄²	—	2	2	no	—
155	C₄²	—	2	2	no	—
156	C₄²	—	2	2	no	—
157	C₄²	—	4	2	no	—
158	C₄²	D₇₁₂	1	2	yes	—
159	C₄²	#13083563	1	2	yes	—
160	C₄²	Q₈×D₈₉	1	2	yes	—
161	C₄²	#13083573	1	2	yes	—
162	C₂³	C₂×#13083551	1	1	yes	characteristic
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	356	4	no	—
170	C₂×C₄	—	356	4	no	—
171	C₂×C₄	—	356	4	no	—
172	C₂×C₄	#15667934	1	4	yes	—
173	C₂×C₄	#15667977	1	2	yes	—
174	C₂×C₄	#15667992	1	2	yes	—
175	C₂×C₄	#15667999	1	1	yes	characteristic
176	C₂×C₄	#15668010	1	1	yes	characteristic
177	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
178	C₂×C₄	C₂×#13083563	1	1	yes	characteristic
179	C₂×C₄	#15668038	1	2	yes	—
180	C₂×C₄	#15668043	1	2	yes	—
181	C₂×C₄	#15668045	1	2	yes	—
182	C₈	—	8	2	no	—
183	C₈	—	356	4	no	—
184	C₂²	#19151503	1	1	yes	characteristic
185	C₂²	#19151580	1	2	yes	—
186	C₂²	C₂×#15667934	1	1	yes	characteristic
187	C₂²	#19151597	1	2	yes	—
188	C₂²	#19151608	1	1	yes	characteristic
189	C₄	—	2	4	no	—
190	C₄	—	2	2	no	—
191	C₄	—	4	2	no	—
192	C₄	—	712	4	no	—
193	C₄	#19151879	1	2	yes	—
194	C₄	#19151884	1	2	yes	—
195	C₄	#19151962	1	1	yes	characteristic
196	C₄	#19151966	1	1	yes	characteristic
197	C₄	#19152004	1	2	yes	—
198	C₂	#26018927	1	2	yes	—
199	C₂	#26018980	1	2	yes	—
200	C₂	#26021203	1	1	yes	characteristic
201	C₂	#26021220	1	1	yes	characteristic
202	C₂	#26021244	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26028615

C₂

yes

maximal, characteristic

#26020105

C₂

yes

maximal, characteristic

#26018136

C₂

yes

maximal

C₈₉×#1842

C₂

yes

maximal, characteristic

C₂×#15668004

—

C₂×C₄⋊#10623494

C₂²

yes

—

C₂×#15667923

—

#19151496

—

#19151496

C₄

yes

—

#19151468

C₂²

yes

—

#19151464

—

#19151353

—

C₂×C₄⋊C₈×C₈₉

—

C₂×C₄⋊C₈×C₈₉

C₂²

yes

—

C₄³×C₈₉

C₂²

yes

—

—

—

C₂×C₄⋊Dic₈₉

—

C₂×C₄⋊Dic₈₉

—

C₂×C₄⋊Dic₈₉

C₂×C₄

yes

—

C₂×#13083553

—

C₄⋊#10623494

—

C₄⋊#10623494

Q₈

yes

—

#15667923

—

#15667919

—

C₂²×C₈×C₈₉

—

C₂×C₄²×C₈₉

C₂×C₄

yes

—

C₂×C₄²×C₈₉

D₄

yes

characteristic

C₂×C₄²×C₈₉

D₄

yes

characteristic

C₂×C₄²×C₈₉

D₄

yes

—

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

—

#13083553

—

C₂²×C₄×C₈₉

C₂²⋊C₄

yes

—

C₂²×C₄×C₈₉

C₂²⋊C₄

yes

—

C₂²×C₄×C₈₉

C₂²×C₄

yes

characteristic

C₂²×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₄²×C₈₉

—

C₄²×C₈₉

—

C₄²×C₈₉

—

C₄²×C₈₉

—

C₄²×C₈₉

D₈

yes

—

C₄²×C₈₉

SD₁₆

yes

—

C₄²×C₈₉

C₂×Q₈

yes

—

C₄²×C₈₉

#41

yes

—

C₂³×C₈₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

#102

yes

—

C₂×C₄×C₈₉

C₄×Q₈

yes

—

C₂×C₄×C₈₉

C₂²⋊Q₈

yes

—

C₂×C₄×C₈₉

#124

yes

characteristic

C₂×C₄×C₈₉

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₈₉

C₂×D₈

yes

characteristic

C₂×C₄×C₈₉

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₈₉

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₈₉

#137

yes

—

C₂×C₄×C₈₉

#137

yes

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₈×C₈₉

—

#10623494

—

C₂²×C₈₉

#391

yes

characteristic

C₂²×C₈₉

#413

yes

—

C₂²×C₈₉

C₂×#102

yes

characteristic

C₂²×C₈₉

#417

yes

—

C₂²×C₈₉

#419

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

#474

yes

—

C₄×C₈₉

#475

yes

—

C₄×C₈₉

yes

C₄×C₈₉

yes

C₄×C₈₉

yes

—

Dic₈₉

—

#9904

—

maximal, Sylow

C₂×C₈₉

#1547

yes

—

C₂×C₈₉

#1559

yes

—

C₂×C₈₉

#1988

yes

characteristic

100

C₂×C₈₉

#1992

yes

characteristic

101

C₂×C₈₉

#1998

yes

characteristic

102

#2859

—

103

#1842

D₈₉

yes

characteristic

104

#1767

—

105

#1296

—

106

C₈₉

#9904

yes

characteristic, complemented, Sylow

107

C₂×#125

—

178

—

108

C₂×C₄⋊C₈

—

109

C₂×C₄⋊C₈

—

110

C₂×C₄⋊C₈

C₂×D₈₉

yes

—

111

C₂×#97

—

178

—

112

#389

—

113

#389

—

178

—

114

#384

—

115

#382

—

178

—

116

C₄³

C₂×D₈₉

yes

characteristic

117

#336

—

178

—

118

C₂²×C₈

—

119

C₂²×C₈

—

178

—

120

#125

—

356

—

121

#125

—

356

—

122

C₂×C₄⋊C₄

—

123

C₂×C₄⋊C₄

—

178

—

124

C₂×C₄⋊C₄

—

178

—

125

C₂×C₄⋊C₄

—

178

—

126

C₂×C₄²

C₄×D₈₉

yes

—

127

C₂×C₄²

D₃₅₆

yes

characteristic

128

C₂×C₄²

D₃₅₆

yes

characteristic

129

C₂×C₄²

C₂²⋊D₈₉

yes

—

130

C₂×C₄²

C₂²×D₈₉

yes

characteristic

131

C₄⋊C₈

—

132

C₄⋊C₈

—

133

C₄⋊C₈

—

134

C₄⋊C₈

—

178

—

135

#97

—

356

—

136

#95

—

178

—

137

C₂²×C₄

—

178

—

138

C₂²×C₄

#13083551

yes

—

139

C₂²×C₄

#13083551

yes

—

140

C₂²×C₄

C₂×C₄×D₈₉

yes

characteristic

141

C₂²×C₄

C₂×C₂²⋊D₈₉

yes

characteristic

142

C₂²×C₄

C₂×D₃₅₆

yes

characteristic

143

C₂×C₈

—

144

C₂×C₈

—

145

C₂×C₈

—

356

—

146

C₂×C₈

—

356

—

147

C₂×C₈

—

178

—

148

C₄⋊C₄

—

356

—

149

C₄⋊C₄

—

356

—

150

C₄⋊C₄

—

356

—

151

C₄⋊C₄

—

356

—

152

C₄⋊C₄

—

356

—

153

C₄⋊C₄

—

356

—

154

C₄²

—

155

C₄²

—

156

C₄²

—

157

C₄²

—

158

C₄²

D₇₁₂

yes

—

159

C₄²

#13083563

yes

—

160

C₄²

Q₈×D₈₉

yes

—

161

C₄²

#13083573

yes

—

162

C₂³

C₂×#13083551

yes

characteristic

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

356

—

170

C₂×C₄

—

356

—

171

C₂×C₄

—

356

—

172

C₂×C₄

#15667934

yes

—

173

C₂×C₄

#15667977

yes

—

174

C₂×C₄

#15667992

yes

—

175

C₂×C₄

#15667999

yes

characteristic

176

C₂×C₄

#15668010

yes

characteristic

177

C₂×C₄

C₂×D₇₁₂

yes

characteristic

178

C₂×C₄

C₂×#13083563

yes

characteristic

179

C₂×C₄

#15668038

yes

—

180

C₂×C₄

#15668043

yes

—

181

C₂×C₄

#15668045

yes

—

182

C₈

—

183

C₈

—

356

—

184

C₂²

#19151503

yes

characteristic

185

C₂²

#19151580

yes

—

186

C₂²

C₂×#15667934

yes

characteristic

187

C₂²

#19151597

yes

—

188

C₂²

#19151608

yes

characteristic

189

C₄

—

190

C₄

—

191

C₄

—

192

C₄

—

712

—

193

C₄

#19151879

yes

—

194

C₄

#19151884

yes

—

195

C₄

#19151962

yes

characteristic

196

C₄

#19151966

yes

characteristic

197

C₄

#19152004

yes

—

198

C₂

#26018927

yes

—

199

C₂

#26018980

yes

—

200

C₂

#26021203

yes

characteristic

201

C₂

#26021220

yes

characteristic

202

C₂

#26021244

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×#13083551
commutator	C₂²×C₄×C₈₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₈₉×#1842	C₂
socle, prime radical	C₂³×C₈₉	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	#1842	D₈₉
hypocenter	C₈₉	#9904
LCS	G ▷ C₂²×C₄×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #1842

subgroup

G/H

center

C₂³

C₂×#13083551

commutator

C₂²×C₄×C₈₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₈₉

Fitting

C₈₉×#1842

C₂

socle, prime radical

C₂³×C₈₉

C₂×C₂²⋊C₄

cosocle

C₂×C₄²×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₂²×C₄×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂³ ◁ C₄³ ◁ #1842

N	H ≅ G/N	type
106. C₈₉	#9904	semidirect

H ≅ G/N

type

106. C₈₉

#9904

semidirect

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

subgroup	index
#26028615	2
#26020105	2
#26018136	2
C₈₉×#1842	2
#9904	89

subgroup

index

C₈₉×#1842

Subgroups — #36454578

Conjugacy classes of subgroups

Complements

Maximal subgroups