Group #36474036 — subgroups

Subgroups — #36474036

Order 22784.

410 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 109 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	C₂²×Dic₇₁₂	C₂	1	1	yes	maximal, characteristic
2	C₂×#19151416	C₂	1	1	yes	maximal, characteristic
3	C₂×#19151401	C₂	1	2	yes	maximal
4	#26021111	C₂	1	2	yes	maximal
5	C₈₉×#1380	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₃₅₆	C₂²	1	2	yes	—
7	C₂×Dic₇₁₂	—	2	2	no	—
8	C₂×Dic₇₁₂	—	2	4	no	—
9	C₂²×#13083549	C₂²	1	1	yes	characteristic
10	C₂×#15667935	—	2	2	no	—
11	C₂×C₈×Dic₈₉	C₂²	1	1	yes	characteristic
12	#19151503	—	2	2	no	—
13	#19151416	—	2	2	no	—
14	#19151401	—	2	4	no	—
15	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
16	C₂×#107×C₈₉	C₂²	1	2	yes	—
17	C₂²×Dic₁₇₈	—	2	2	no	—
18	C₂×Dic₃₅₆	—	2	2	no	—
19	C₂×Dic₃₅₆	—	2	2	no	—
20	C₂×Dic₃₅₆	—	2	2	no	—
21	C₂×Dic₃₅₆	—	2	2	no	—
22	C₂×Dic₃₅₆	—	2	2	no	—
23	C₂×Dic₃₅₆	D₄	1	4	yes	—
24	C₂²×#10623494	D₄	1	1	yes	characteristic
25	C₂×C₄×Dic₈₉	D₄	1	1	yes	characteristic
26	Dic₇₁₂	—	2	8	no	—
27	C₂×#13083549	—	2	2	no	—
28	C₂×#13083549	—	2	1	no	—
29	#15667935	—	4	4	no	—
30	C₈×Dic₈₉	—	2	1	no	—
31	C₈×Dic₈₉	—	2	1	no	—
32	#15667919	—	4	2	no	—
33	#15667919	—	4	2	no	—
34	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
35	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
36	C₂×C₁₆×C₈₉	—	2	2	no	—
37	C₂×C₁₆×C₈₉	D₄	1	2	yes	—
38	#107×C₈₉	—	2	2	no	—
39	#107×C₈₉	—	2	2	no	—
40	C₂×Dic₁₇₈	—	2	4	no	—
41	C₂×Dic₁₇₈	—	4	2	no	—
42	C₂×Dic₁₇₈	—	4	2	no	—
43	C₂×Dic₁₇₈	—	4	2	no	—
44	C₂×Dic₁₇₈	—	4	2	no	—
45	C₂×Dic₁₇₈	—	4	2	no	—
46	C₂²×Dic₈₉	—	4	2	no	—
47	C₂²×Dic₈₉	D₈	1	1	yes	characteristic
48	C₂²×Dic₈₉	D₈	1	1	yes	characteristic
49	Dic₃₅₆	—	2	4	no	—
50	Dic₃₅₆	—	2	4	no	—
51	Dic₃₅₆	—	4	2	no	—
52	Dic₃₅₆	—	4	2	no	—
53	C₂×#10623494	—	2	1	no	—
54	C₂×#10623494	—	2	2	no	—
55	C₄×Dic₈₉	—	2	1	no	—
56	C₄×Dic₈₉	—	2	1	no	—
57	#13083549	—	4	1	no	—
58	C₂²×C₄×C₈₉	—	4	2	no	—
59	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
60	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
61	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
62	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
63	C₂×C₈×C₈₉	#41	1	2	yes	—
64	C₄⋊C₄×C₈₉	—	4	2	no	—
65	C₄⋊C₄×C₈₉	—	4	2	no	—
66	C₁₆×C₈₉	—	2	2	no	—
67	C₂³×C₈₉	C₂×D₈	1	1	yes	characteristic
68	C₂×C₄×C₈₉	—	8	2	no	—
69	C₂×C₄×C₈₉	—	8	2	no	—
70	C₂×C₄×C₈₉	—	8	2	no	—
71	C₂×C₄×C₈₉	#121	1	2	yes	—
72	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
73	C₂×C₄×C₈₉	#135	1	1	yes	characteristic
74	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	2	yes	—
75	C₂×Dic₈₉	—	2	1	no	—
76	C₂×Dic₈₉	—	2	1	no	—
77	C₂×Dic₈₉	—	2	1	no	—
78	C₂×Dic₈₉	—	2	2	no	—
79	C₂×Dic₈₉	—	4	4	no	—
80	C₂×Dic₈₉	—	8	2	no	—
81	C₂×Dic₈₉	—	8	2	no	—
82	C₂×Dic₈₉	Q₃₂	1	2	yes	complemented
83	Dic₁₇₈	—	4	4	no	—
84	Dic₁₇₈	—	4	4	no	—
85	Dic₁₇₈	—	8	2	no	—
86	Dic₁₇₈	—	8	2	no	—
87	C₈×C₈₉	#120	1	1	yes	characteristic
88	C₈×C₈₉	#123	1	2	yes	—
89	C₈×C₈₉	#124	1	1	yes	characteristic
90	#10623494	—	4	1	no	—
91	C₂²×C₈₉	#459	1	2	yes	—
92	C₂²×C₈₉	#466	1	1	yes	characteristic
93	C₂²×C₈₉	C₂×Q₃₂	1	1	yes	characteristic
94	C₂²×C₈₉	#508	1	1	yes	characteristic
95	C₂²×C₈₉	#510	1	2	yes	—
96	C₄×C₈₉	—	16	2	no	—
97	C₄×C₈₉	#394	1	1	yes	characteristic
98	C₄×C₈₉	#463	1	2	yes	—
99	C₄×C₈₉	#469	1	1	yes	characteristic
100	Dic₈₉	—	2	2	no	—
101	Dic₈₉	—	4	1	no	—
102	Dic₈₉	—	8	4	no	—
103	#13450	—	89	1	no	maximal, Sylow
104	C₂×C₈₉	#2054	1	1	yes	characteristic
105	C₂×C₈₉	#2209	1	2	yes	—
106	C₂×C₈₉	#2211	1	2	yes	—
107	C₂×C₈₉	#2216	1	1	yes	characteristic
108	C₂×C₈₉	#2219	1	1	yes	characteristic
109	C₂²×Q₃₂	—	89	1	no	—
110	C₂×C₄⋊C₁₆	—	89	1	no	—
111	C₂×#358	—	89	2	no	—
112	#1969	—	89	2	no	—
113	#1380	D₈₉	1	1	yes	characteristic
114	C₈₉	#13450	1	1	yes	characteristic, complemented, Sylow
115	C₂²×Q₁₆	—	89	2	no	—
116	C₂×Q₃₂	—	178	4	no	—
117	C₂×Q₃₂	—	178	2	no	—
118	C₂²×C₁₆	—	89	1	no	—
119	C₂²×C₁₆	C₂×D₈₉	1	1	yes	characteristic
120	C₂×#107	C₂×D₈₉	1	2	yes	—
121	C₂×#103	—	178	2	no	—
122	C₂×C₄×C₈	—	89	1	no	—
123	#391	—	178	2	no	—
124	C₄⋊C₁₆	—	178	2	no	—
125	#358	—	178	4	no	—
126	C₂²×Q₈	—	178	2	no	—
127	C₂×Q₁₆	—	89	4	no	—
128	C₂×Q₁₆	—	178	2	no	—
129	C₂×Q₁₆	—	178	2	no	—
130	C₂×Q₁₆	—	178	2	no	—
131	C₂×Q₁₆	—	178	2	no	—
132	C₂×Q₁₆	—	178	2	no	—
133	C₂²×C₈	—	89	1	no	—
134	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
135	C₂×C₄⋊C₄	—	2	2	no	—
136	C₂×C₄²	—	89	1	no	—
137	Q₃₂	—	178	8	no	—
138	C₂×C₁₆	—	2	2	no	—
139	C₂×C₁₆	—	178	2	no	—
140	C₂×C₁₆	—	178	1	no	—
141	C₂×C₁₆	D₃₅₆	1	2	yes	—
142	#107	—	2	2	no	—
143	#107	—	2	2	no	—
144	#103	—	356	4	no	—
145	C₄×C₈	—	178	1	no	—
146	C₄×C₈	—	178	1	no	—
147	#95	—	356	2	no	—
148	#95	—	356	2	no	—
149	C₂×Q₈	—	356	2	no	—
150	C₂×Q₈	—	356	2	no	—
151	C₂×Q₈	—	356	2	no	—
152	C₂×Q₈	—	356	2	no	—
153	C₂×Q₈	—	356	2	no	—
154	C₂×Q₈	—	178	4	no	—
155	C₂²×C₄	—	4	2	no	—
156	C₂²×C₄	—	89	1	no	—
157	C₂²×C₄	—	89	1	no	—
158	C₂²×C₄	—	356	2	no	—
159	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
160	Q₁₆	—	356	2	no	—
161	Q₁₆	—	356	2	no	—
162	Q₁₆	—	178	4	no	—
163	Q₁₆	—	178	4	no	—
164	C₂×C₈	—	178	1	no	—
165	C₂×C₈	—	178	2	no	—
166	C₂×C₈	D₄×D₈₉	1	2	yes	—
167	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
168	C₂×C₈	#13083574	1	2	yes	—
169	C₂×C₈	#13083575	1	1	yes	characteristic
170	C₄⋊C₄	—	4	2	no	—
171	C₄⋊C₄	—	4	2	no	—
172	C₄²	—	178	1	no	—
173	C₄²	—	178	1	no	—
174	C₁₆	—	2	2	no	—
175	C₁₆	—	356	1	no	—
176	C₂³	D₈×D₈₉	1	1	yes	characteristic
177	Q₈	—	356	4	no	—
178	Q₈	—	356	4	no	—
179	Q₈	—	712	2	no	—
180	Q₈	—	712	2	no	—
181	C₂×C₄	—	8	2	no	—
182	C₂×C₄	—	8	2	no	—
183	C₂×C₄	—	8	2	no	—
184	C₂×C₄	—	89	2	no	—
185	C₂×C₄	—	356	4	no	—
186	C₂×C₄	—	178	2	no	—
187	C₂×C₄	—	178	1	no	—
188	C₂×C₄	—	178	1	no	—
189	C₂×C₄	—	178	1	no	—
190	C₂×C₄	—	712	2	no	—
191	C₂×C₄	—	712	2	no	—
192	C₂×C₄	#15667985	1	2	yes	—
193	C₂×C₄	#15667987	1	1	yes	characteristic
194	C₂×C₄	#15668035	1	1	yes	characteristic
195	C₂×C₄	#15668039	1	2	yes	—
196	C₈	—	356	1	no	—
197	C₈	#15667982	1	1	yes	characteristic
198	C₈	#15667997	1	2	yes	—
199	C₈	#15668000	1	1	yes	characteristic
200	C₂²	#19151792	1	2	yes	—
201	C₂²	#19151835	1	1	yes	characteristic
202	C₂²	Q₃₂×D₈₉	1	1	yes	characteristic
203	C₂²	#19152027	1	1	yes	characteristic
204	C₂²	#19152037	1	2	yes	—
205	C₄	—	16	2	no	—
206	C₄	—	356	1	no	—
207	C₄	—	178	2	no	—
208	C₄	—	712	4	no	—
209	C₄	#19151516	1	1	yes	characteristic
210	C₄	#19151816	1	2	yes	—
211	C₄	#19151852	1	1	yes	characteristic
212	C₂	#26021564	1	1	yes	characteristic
213	C₂	#26022301	1	2	yes	—
214	C₂	#26022313	1	2	yes	—
215	C₂	#26022344	1	1	yes	characteristic
216	C₂	#26022361	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₇₁₂

C₂

yes

maximal, characteristic

C₂×#19151416

C₂

yes

maximal, characteristic

C₂×#19151401

C₂

yes

maximal

#26021111

C₂

yes

maximal

C₈₉×#1380

C₂

yes

maximal, characteristic

C₂²×Dic₃₅₆

C₂²

yes

—

C₂×Dic₇₁₂

—

C₂×Dic₇₁₂

—

C₂²×#13083549

C₂²

yes

characteristic

C₂×#15667935

—

C₂×C₈×Dic₈₉

C₂²

yes

—

—

—

C₂²×C₁₆×C₈₉

C₂²

yes

characteristic

C₂×#107×C₈₉

C₂²

yes

—

C₂²×Dic₁₇₈

—

C₂×Dic₃₅₆

—

C₂×Dic₃₅₆

—

C₂×Dic₃₅₆

—

C₂×Dic₃₅₆

—

C₂×Dic₃₅₆

—

C₂×Dic₃₅₆

D₄

yes

—

C₂²×#10623494

D₄

yes

characteristic

C₂×C₄×Dic₈₉

D₄

yes

characteristic

Dic₇₁₂

—

C₂×#13083549

—

C₂×#13083549

—

#15667935

—

C₈×Dic₈₉

—

C₈×Dic₈₉

—

#15667919

—

#15667919

—

C₂²×C₈×C₈₉

C₂³

yes

characteristic

C₂×C₄⋊C₄×C₈₉

—

C₂×C₁₆×C₈₉

—

C₂×C₁₆×C₈₉

D₄

yes

—

#107×C₈₉

—

#107×C₈₉

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂²×Dic₈₉

—

C₂²×Dic₈₉

D₈

yes

characteristic

C₂²×Dic₈₉

D₈

yes

characteristic

Dic₃₅₆

—

Dic₃₅₆

—

Dic₃₅₆

—

Dic₃₅₆

—

C₂×#10623494

—

C₂×#10623494

—

C₄×Dic₈₉

—

C₄×Dic₈₉

—

#13083549

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₈×C₈₉

C₂×D₄

yes

—

C₂×C₈×C₈₉

C₂×D₄

yes

C₂×C₈×C₈₉

yes

C₂×C₈×C₈₉

yes

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₁₆×C₈₉

—

C₂³×C₈₉

C₂×D₈

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

#121

yes

—

C₂×C₄×C₈₉

yes

C₂×C₄×C₈₉

yes

C₂×C₄×C₈₉

yes

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

Q₃₂

yes

complemented

Dic₁₇₈

—

Dic₁₇₈

—

Dic₁₇₈

—

Dic₁₇₈

—

C₈×C₈₉

#120

yes

characteristic

C₈×C₈₉

#123

yes

—

C₈×C₈₉

#124

yes

characteristic

#10623494

—

C₂²×C₈₉

#459

yes

—

C₂²×C₈₉

#466

yes

characteristic

C₂²×C₈₉

C₂×Q₃₂

yes

C₂²×C₈₉

yes

C₂²×C₈₉

yes

—

C₄×C₈₉

—

C₄×C₈₉

#394

yes

characteristic

C₄×C₈₉

#463

yes

—

C₄×C₈₉

#469

yes

characteristic

100

Dic₈₉

—

101

Dic₈₉

—

102

Dic₈₉

—

103

#13450

—

maximal, Sylow

104

C₂×C₈₉

#2054

yes

characteristic

105

C₂×C₈₉

#2209

yes

—

106

C₂×C₈₉

#2211

yes

—

107

C₂×C₈₉

#2216

yes

characteristic

108

C₂×C₈₉

#2219

yes

characteristic

109

C₂²×Q₃₂

—

110

C₂×C₄⋊C₁₆

—

111

C₂×#358

—

112

#1969

—

113

#1380

D₈₉

yes

characteristic

114

C₈₉

#13450

yes

characteristic, complemented, Sylow

115

C₂²×Q₁₆

—

116

C₂×Q₃₂

—

178

—

117

C₂×Q₃₂

—

178

—

118

C₂²×C₁₆

—

119

C₂²×C₁₆

C₂×D₈₉

yes

characteristic

120

C₂×#107

C₂×D₈₉

yes

—

121

C₂×#103

—

178

—

122

C₂×C₄×C₈

—

123

#391

—

178

—

124

C₄⋊C₁₆

—

178

—

125

#358

—

178

—

126

C₂²×Q₈

—

178

—

127

C₂×Q₁₆

—

128

C₂×Q₁₆

—

178

—

129

C₂×Q₁₆

—

178

—

130

C₂×Q₁₆

—

178

—

131

C₂×Q₁₆

—

178

—

132

C₂×Q₁₆

—

178

—

133

C₂²×C₈

—

134

C₂²×C₈

C₂²×D₈₉

yes

characteristic

135

C₂×C₄⋊C₄

—

136

C₂×C₄²

—

137

Q₃₂

—

178

—

138

C₂×C₁₆

—

139

C₂×C₁₆

—

178

—

140

C₂×C₁₆

—

178

—

141

C₂×C₁₆

D₃₅₆

yes

—

142

#107

—

143

#107

—

144

#103

—

356

—

145

C₄×C₈

—

178

—

146

C₄×C₈

—

178

—

147

#95

—

356

—

148

#95

—

356

—

149

C₂×Q₈

—

356

—

150

C₂×Q₈

—

356

—

151

C₂×Q₈

—

356

—

152

C₂×Q₈

—

356

—

153

C₂×Q₈

—

356

—

154

C₂×Q₈

—

178

—

155

C₂²×C₄

—

156

C₂²×C₄

—

157

C₂²×C₄

—

158

C₂²×C₄

—

356

—

159

C₂²×C₄

D₄×D₈₉

yes

characteristic

160

Q₁₆

—

356

—

161

Q₁₆

—

356

—

162

Q₁₆

—

178

—

163

Q₁₆

—

178

—

164

C₂×C₈

—

178

—

165

C₂×C₈

—

178

—

166

C₂×C₈

D₄×D₈₉

yes

—

167

C₂×C₈

C₂×D₃₅₆

yes

characteristic

168

C₂×C₈

#13083574

yes

—

169

C₂×C₈

#13083575

yes

characteristic

170

C₄⋊C₄

—

171

C₄⋊C₄

—

172

C₄²

—

178

—

173

C₄²

—

178

—

174

C₁₆

—

175

C₁₆

—

356

—

176

C₂³

D₈×D₈₉

yes

characteristic

177

Q₈

—

356

—

178

Q₈

—

356

—

179

Q₈

—

712

—

180

Q₈

—

712

—

181

C₂×C₄

—

182

C₂×C₄

—

183

C₂×C₄

—

184

C₂×C₄

—

185

C₂×C₄

—

356

—

186

C₂×C₄

—

178

—

187

C₂×C₄

—

178

—

188

C₂×C₄

—

178

—

189

C₂×C₄

—

178

—

190

C₂×C₄

—

712

—

191

C₂×C₄

—

712

—

192

C₂×C₄

#15667985

yes

—

193

C₂×C₄

#15667987

yes

characteristic

194

C₂×C₄

#15668035

yes

characteristic

195

C₂×C₄

#15668039

yes

—

196

C₈

—

356

—

197

C₈

#15667982

yes

characteristic

198

C₈

#15667997

yes

—

199

C₈

#15668000

yes

characteristic

200

C₂²

#19151792

yes

—

201

C₂²

#19151835

yes

characteristic

202

C₂²

Q₃₂×D₈₉

yes

characteristic

203

C₂²

#19152027

yes

characteristic

204

C₂²

#19152037

yes

—

205

C₄

—

206

C₄

—

356

—

207

C₄

—

178

—

208

C₄

—

712

—

209

C₄

#19151516

yes

characteristic

210

C₄

#19151816

yes

—

211

C₄

#19151852

yes

characteristic

212

C₂

#26021564

yes

characteristic

213

C₂

#26022301

yes

—

214

C₂

#26022313

yes

—

215

C₂

#26022344

yes

characteristic

216

C₂

#26022361

yes

characteristic

	subgroup	G/H
center	C₂³	D₈×D₈₉
commutator, cosocle	C₂²×C₈×C₈₉	C₂³
Frattini	C₂²×C₈	C₂²×D₈₉
Fitting	C₈₉×#1380	C₂
socle, prime radical	C₂³×C₈₉	C₂×D₈
hypercenter	#1380	D₈₉
hypocenter	C₈₉	#13450
LCS	G ▷ C₂²×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1380

subgroup

G/H

center

C₂³

D₈×D₈₉

commutator, cosocle

C₂²×C₈×C₈₉

C₂³

Frattini

C₂²×C₈

C₂²×D₈₉

Fitting

C₈₉×#1380

C₂

socle, prime radical

C₂³×C₈₉

C₂×D₈

D₈₉

C₈₉

G ▷ C₂²×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1380

N	H ≅ G/N	type
82. C₂×Dic₈₉	Q₃₂	semidirect
114. C₈₉	#13450	semidirect

H ≅ G/N

type

82. C₂×Dic₈₉

Q₃₂

semidirect

114. C₈₉

#13450

semidirect

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

subgroup	index
C₂²×Dic₇₁₂	2
C₂×#19151416	2
C₂×#19151401	2
#26021111	2
C₈₉×#1380	2
#13450	89

subgroup

index

C₂²×Dic₇₁₂

C₈₉×#1380

Subgroups — #36474036

Conjugacy classes of subgroups

Complements

Maximal subgroups