Group #36448930 — subgroups

Subgroups — #36448930

Order 22784.

412 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 103 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	#26029319	C₂	1	1	yes	maximal, characteristic
2	#26019043	C₂	1	2	yes	maximal, complemented
3	#26018914	C₂	1	1	yes	maximal, characteristic
4	#26018085	C₂	1	2	yes	maximal, complemented
5	C₈₉×#1962	C₂	1	1	yes	maximal, characteristic
6	#19151676	—	2	2	no	—
7	D₄×#10623494	—	2	2	no	—
8	#19151651	C₂²	1	2	yes	—
9	#19151651	C₂²	1	2	yes	—
10	#19151565	—	2	2	no	—
11	#19151350	—	2	2	no	—
12	#19151348	—	2	2	no	—
13	#19151347	C₄	1	2	yes	complemented
14	#19151342	C₄	1	1	yes	characteristic, complemented
15	#19151342	C₄	1	1	yes	characteristic, complemented
16	#532×C₈₉	C₂²	1	1	yes	characteristic
17	#423×C₈₉	C₂²	1	1	yes	characteristic
18	C₂×#104×C₈₉	—	2	2	no	—
19	C₄³×C₈₉	C₂²	1	1	yes	characteristic
20	C₂×#13083558	—	2	2	no	—
21	C₂²×#10623494	—	2	2	no	—
22	C₄⋊#10623494	C₂×C₄	1	2	yes	—
23	C₄⋊#10623494	C₂×C₄	1	2	yes	—
24	C₄⋊#10623494	D₄	1	2	yes	—
25	C₄⋊#10623494	D₄	1	2	yes	—
26	C₂²⋊#10623494	—	2	4	no	—
27	C₂²⋊#10623494	—	2	2	no	—
28	C₂²⋊#10623494	—	2	2	no	—
29	C₄×#10623494	—	2	2	no	—
30	C₄×#10623494	—	2	2	no	—
31	C₂×#41×C₈₉	D₄	1	2	yes	—
32	C₂×M₁₆×C₈₉	—	2	1	no	—
33	C₄⋊Q₈×C₈₉	C₂×C₄	1	1	yes	characteristic
34	#127×C₈₉	C₂×C₄	1	1	yes	characteristic
35	#124×C₈₉	Q₈	1	2	yes	—
36	#121×C₈₉	—	2	2	no	—
37	C₄×D₄×C₈₉	—	2	2	no	—
38	C₂×C₄²×C₈₉	—	2	1	no	—
39	C₂×C₄²×C₈₉	—	2	2	no	—
40	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
41	C₄⋊C₈×C₈₉	—	2	2	no	—
42	#104×C₈₉	—	4	4	no	—
43	#13083558	—	8	2	no	—
44	C₂×#10623494	—	2	2	no	—
45	C₂×#10623494	—	2	2	no	—
46	C₂×#10623494	—	2	2	no	—
47	C₂×#10623494	—	2	2	no	—
48	C₂×#10623494	—	8	2	no	—
49	#41×C₈₉	—	4	2	no	—
50	#41×C₈₉	—	4	2	no	—
51	C₂×Q₈×C₈₉	C₄⋊C₄	1	2	yes	—
52	C₂×D₄×C₈₉	—	2	2	no	—
53	C₂×D₄×C₈₉	—	2	2	no	—
54	C₂×D₄×C₈₉	C₄⋊C₄	1	2	yes	—
55	C₂²×C₄×C₈₉	—	2	2	no	—
56	C₂²×C₄×C₈₉	—	4	1	no	—
57	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₈₉	#41	1	2	yes	—
59	M₁₆×C₈₉	—	4	2	no	—
60	C₂×C₈×C₈₉	—	4	1	no	—
61	C₄⋊C₄×C₈₉	—	2	2	no	—
62	C₂²⋊C₄×C₈₉	—	2	4	no	—
63	C₄²×C₈₉	—	2	2	no	—
64	C₄²×C₈₉	—	2	4	no	—
65	C₄²×C₈₉	—	4	2	no	—
66	C₄²×C₈₉	—	4	1	no	—
67	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
68	C₄²×C₈₉	C₂×D₄	1	2	yes	—
69	C₄²×C₈₉	C₂×Q₈	1	1	yes	characteristic
70	C₂³×C₈₉	—	2	2	no	—
71	C₂³×C₈₉	#123	1	1	yes	characteristic
72	Q₈×C₈₉	—	4	2	no	—
73	D₄×C₈₉	—	4	2	no	—
74	D₄×C₈₉	—	4	2	no	—
75	D₄×C₈₉	—	4	2	no	—
76	D₄×C₈₉	—	8	2	no	—
77	C₂×C₄×C₈₉	—	2	2	no	—
78	C₂×C₄×C₈₉	—	2	2	no	—
79	C₂×C₄×C₈₉	—	2	1	no	—
80	C₂×C₄×C₈₉	—	2	1	no	—
81	C₂×C₄×C₈₉	—	2	2	no	—
82	C₂×C₄×C₈₉	—	4	2	no	—
83	C₂×C₄×C₈₉	—	4	4	no	—
84	C₂×C₄×C₈₉	—	4	2	no	—
85	C₂×C₄×C₈₉	—	4	2	no	—
86	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
87	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
88	C₂×C₄×C₈₉	#120	1	1	yes	characteristic
89	C₂×C₄×C₈₉	C₂²⋊Q₈	1	2	yes	—
90	C₈×C₈₉	—	8	1	no	—
91	#10623494	—	4	2	no	—
92	#10623494	—	4	2	no	—
93	#10623494	—	4	2	no	—
94	#10623494	—	4	2	no	—
95	C₂²×C₈₉	—	2	1	no	—
96	C₂²×C₈₉	—	2	2	no	—
97	C₂²×C₈₉	—	4	2	no	—
98	C₂²×C₈₉	—	4	4	no	—
99	C₂²×C₈₉	#385	1	1	yes	characteristic
100	C₄×C₈₉	—	2	1	no	—
101	C₄×C₈₉	—	2	2	no	—
102	C₄×C₈₉	—	4	4	no	—
103	C₄×C₈₉	—	4	2	no	—
104	C₄×C₈₉	#443	1	2	yes	—
105	C₄×C₈₉	#444	1	2	yes	—
106	C₄×C₈₉	#453	1	1	yes	characteristic
107	C₄×C₈₉	#455	1	1	yes	characteristic
108	#8934	—	89	1	no	maximal, Sylow
109	C₂×C₈₉	—	4	1	no	—
110	C₂×C₈₉	—	8	2	no	—
111	C₂×C₈₉	#1856	1	1	yes	characteristic
112	C₂×C₈₉	#1866	1	2	yes	—
113	#2979	—	89	1	no	—
114	#1962	D₈₉	1	1	yes	characteristic
115	#1574	—	89	2	no	—
116	#1543	—	89	1	no	—
117	#1276	—	89	2	no	—
118	C₈₉	#8934	1	1	yes	characteristic, complemented, Sylow
119	#532	C₂×D₈₉	1	1	yes	characteristic
120	#436	—	178	2	no	—
121	C₈×D₄	—	178	2	no	—
122	#431	—	89	2	no	—
123	#431	—	89	2	no	—
124	#423	C₂×D₈₉	1	1	yes	characteristic
125	C₂×#104	—	2	2	no	—
126	#408	—	178	2	no	—
127	C₄³	C₂×D₈₉	1	1	yes	characteristic
128	#335	—	178	2	no	—
129	#334	—	178	2	no	—
130	#333	—	89	2	no	—
131	#331	—	89	1	no	—
132	#331	—	89	1	no	—
133	C₂×#41	C₂²⋊D₈₉	1	2	yes	—
134	C₂×M₁₆	—	2	1	no	—
135	C₂×M₁₆	—	178	2	no	—
136	C₂²×C₈	—	178	2	no	—
137	C₄⋊Q₈	C₄×D₈₉	1	1	yes	characteristic
138	#127	C₄×D₈₉	1	1	yes	characteristic
139	#124	Dic₁₇₈	1	2	yes	—
140	#121	—	2	2	no	—
141	C₄×D₄	—	2	2	no	—
142	C₂×C₄²	—	2	1	no	—
143	C₂×C₄²	—	2	2	no	—
144	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
145	C₄⋊C₈	—	2	2	no	—
146	C₄⋊C₈	—	89	2	no	—
147	C₄⋊C₈	—	89	2	no	—
148	C₄⋊C₈	—	89	2	no	—
149	C₄⋊C₈	—	89	2	no	—
150	#104	—	4	4	no	—
151	C₂²⋊C₈	—	178	4	no	—
152	C₂²⋊C₈	—	178	2	no	—
153	C₂²⋊C₈	—	178	2	no	—
154	C₄×C₈	—	178	2	no	—
155	C₄×C₈	—	178	2	no	—
156	#41	—	4	2	no	—
157	#41	—	4	2	no	—
158	C₂×Q₈	#13083553	1	2	yes	—
159	C₂×D₄	—	2	2	no	—
160	C₂×D₄	—	2	2	no	—
161	C₂×D₄	#13083553	1	2	yes	—
162	C₂²×C₄	—	2	2	no	—
163	C₂²×C₄	—	4	1	no	—
164	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
165	C₂²×C₄	#13083573	1	2	yes	—
166	M₁₆	—	4	2	no	—
167	M₁₆	—	712	2	no	—
168	C₂×C₈	—	4	1	no	—
169	C₂×C₈	—	178	2	no	—
170	C₂×C₈	—	178	2	no	—
171	C₂×C₈	—	178	2	no	—
172	C₂×C₈	—	178	2	no	—
173	C₂×C₈	—	712	2	no	—
174	C₄⋊C₄	—	2	2	no	—
175	C₂²⋊C₄	—	2	4	no	—
176	C₄²	—	2	2	no	—
177	C₄²	—	2	4	no	—
178	C₄²	—	4	2	no	—
179	C₄²	—	4	1	no	—
180	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
181	C₄²	D₄×D₈₉	1	2	yes	—
182	C₄²	C₂×Dic₁₇₈	1	1	yes	characteristic
183	C₂³	—	2	2	no	—
184	C₂³	#15667993	1	1	yes	characteristic
185	Q₈	—	4	2	no	—
186	D₄	—	4	2	no	—
187	D₄	—	4	2	no	—
188	D₄	—	4	2	no	—
189	D₄	—	8	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	1	no	—
193	C₂×C₄	—	2	1	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	4	2	no	—
196	C₂×C₄	—	4	4	no	—
197	C₂×C₄	—	4	2	no	—
198	C₂×C₄	—	4	2	no	—
199	C₂×C₄	C₂×#13083553	1	1	yes	characteristic
200	C₂×C₄	#15667972	1	2	yes	—
201	C₂×C₄	#15667980	1	1	yes	characteristic
202	C₂×C₄	#15667989	1	2	yes	—
203	C₈	—	8	1	no	—
204	C₈	—	356	2	no	—
205	C₈	—	356	2	no	—
206	C₈	—	356	2	no	—
207	C₈	—	356	2	no	—
208	C₂²	—	2	1	no	—
209	C₂²	—	2	2	no	—
210	C₂²	—	4	2	no	—
211	C₂²	—	4	4	no	—
212	C₂²	#19151470	1	1	yes	characteristic
213	C₄	—	2	1	no	—
214	C₄	—	2	2	no	—
215	C₄	—	4	4	no	—
216	C₄	—	4	2	no	—
217	C₄	#19151712	1	2	yes	—
218	C₄	#19151717	1	2	yes	—
219	C₄	#19151765	1	1	yes	characteristic
220	C₄	#19151775	1	1	yes	characteristic
221	C₂	—	4	1	no	—
222	C₂	—	8	2	no	—
223	C₂	#26020461	1	1	yes	characteristic
224	C₂	#26020525	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#26029319

C₂

yes

maximal, characteristic

#26019043

C₂

yes

maximal, complemented

#26018914

C₂

yes

maximal, characteristic

#26018085

C₂

yes

maximal, complemented

C₈₉×#1962

C₂

yes

maximal, characteristic

#19151676

—

D₄×#10623494

—

#19151651

C₂²

yes

—

#19151651

C₂²

yes

—

—

—

—

C₄

yes

complemented

#19151342

C₄

yes

characteristic, complemented

#19151342

C₄

yes

characteristic, complemented

#532×C₈₉

C₂²

yes

characteristic

#423×C₈₉

C₂²

yes

characteristic

C₂×#104×C₈₉

—

C₄³×C₈₉

C₂²

yes

—

—

C₂×C₄

yes

—

C₄⋊#10623494

C₂×C₄

yes

—

C₄⋊#10623494

D₄

yes

—

C₄⋊#10623494

D₄

yes

—

—

—

—

—

—

C₂×#41×C₈₉

D₄

yes

—

C₂×M₁₆×C₈₉

—

C₄⋊Q₈×C₈₉

C₂×C₄

yes

characteristic

#127×C₈₉

C₂×C₄

yes

characteristic

#124×C₈₉

Q₈

yes

—

#121×C₈₉

—

C₄×D₄×C₈₉

—

C₂×C₄²×C₈₉

—

C₂×C₄²×C₈₉

—

C₂×C₄²×C₈₉

C₂³

yes

characteristic

C₄⋊C₈×C₈₉

—

#104×C₈₉

—

—

—

—

—

—

—

#41×C₈₉

—

#41×C₈₉

—

C₂×Q₈×C₈₉

C₄⋊C₄

yes

—

C₂×D₄×C₈₉

—

C₂×D₄×C₈₉

—

C₂×D₄×C₈₉

C₄⋊C₄

yes

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₈₉

#41

yes

—

M₁₆×C₈₉

—

C₂×C₈×C₈₉

—

C₄⋊C₄×C₈₉

—

C₂²⋊C₄×C₈₉

—

C₄²×C₈₉

—

C₄²×C₈₉

—

C₄²×C₈₉

—

C₄²×C₈₉

—

C₄²×C₈₉

C₂²×C₄

yes

characteristic

C₄²×C₈₉

C₂×D₄

yes

—

C₄²×C₈₉

C₂×Q₈

yes

characteristic

C₂³×C₈₉

—

C₂³×C₈₉

#123

yes

characteristic

Q₈×C₈₉

—

D₄×C₈₉

—

D₄×C₈₉

—

D₄×C₈₉

—

D₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₈₉

C₄×D₄

yes

—

C₂×C₄×C₈₉

#120

yes

characteristic

C₂×C₄×C₈₉

C₂²⋊Q₈

yes

—

C₈×C₈₉

—

—

—

—

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

#385

yes

characteristic

100

C₄×C₈₉

—

101

C₄×C₈₉

—

102

C₄×C₈₉

—

103

C₄×C₈₉

—

104

C₄×C₈₉

#443

yes

—

105

C₄×C₈₉

#444

yes

—

106

C₄×C₈₉

#453

yes

characteristic

107

C₄×C₈₉

yes

108

—

109

C₂×C₈₉

—

110

C₂×C₈₉

—

111

C₂×C₈₉

#1856

yes

characteristic

112

C₂×C₈₉

#1866

yes

—

113

#2979

—

114

#1962

D₈₉

yes

characteristic

115

#1574

—

116

#1543

—

117

#1276

—

118

C₈₉

#8934

yes

characteristic, complemented, Sylow

119

#532

C₂×D₈₉

yes

characteristic

120

#436

—

178

—

121

C₈×D₄

—

178

—

122

#431

—

123

#431

—

124

#423

C₂×D₈₉

yes

characteristic

125

C₂×#104

—

126

#408

—

178

—

127

C₄³

C₂×D₈₉

yes

characteristic

128

#335

—

178

—

129

#334

—

178

—

130

#333

—

131

#331

—

132

#331

—

133

C₂×#41

C₂²⋊D₈₉

yes

—

134

C₂×M₁₆

—

135

C₂×M₁₆

—

178

—

136

C₂²×C₈

—

178

—

137

C₄⋊Q₈

C₄×D₈₉

yes

characteristic

138

#127

C₄×D₈₉

yes

characteristic

139

#124

Dic₁₇₈

yes

—

140

#121

—

141

C₄×D₄

—

142

C₂×C₄²

—

143

C₂×C₄²

—

144

C₂×C₄²

C₂²×D₈₉

yes

characteristic

145

C₄⋊C₈

—

146

C₄⋊C₈

—

147

C₄⋊C₈

—

148

C₄⋊C₈

—

149

C₄⋊C₈

—

150

#104

—

151

C₂²⋊C₈

—

178

—

152

C₂²⋊C₈

—

178

—

153

C₂²⋊C₈

—

178

—

154

C₄×C₈

—

178

—

155

C₄×C₈

—

178

—

156

#41

—

157

#41

—

158

C₂×Q₈

#13083553

yes

—

159

C₂×D₄

—

160

C₂×D₄

—

161

C₂×D₄

#13083553

yes

—

162

C₂²×C₄

—

163

C₂²×C₄

—

164

C₂²×C₄

C₂×C₂²⋊D₈₉

yes

characteristic

165

C₂²×C₄

#13083573

yes

—

166

M₁₆

—

167

M₁₆

—

712

—

168

C₂×C₈

—

169

C₂×C₈

—

178

—

170

C₂×C₈

—

178

—

171

C₂×C₈

—

178

—

172

C₂×C₈

—

178

—

173

C₂×C₈

—

712

—

174

C₄⋊C₄

—

175

C₂²⋊C₄

—

176

C₄²

—

177

C₄²

—

178

C₄²

—

179

C₄²

—

180

C₄²

C₂×C₄×D₈₉

yes

characteristic

181

C₄²

D₄×D₈₉

yes

—

182

C₄²

C₂×Dic₁₇₈

yes

characteristic

183

C₂³

—

184

C₂³

#15667993

yes

characteristic

185

Q₈

—

186

D₄

—

187

D₄

—

188

D₄

—

189

D₄

—

190

C₂×C₄

—

191

C₂×C₄

—

192

C₂×C₄

—

193

C₂×C₄

—

194

C₂×C₄

—

195

C₂×C₄

—

196

C₂×C₄

—

197

C₂×C₄

—

198

C₂×C₄

—

199

C₂×C₄

C₂×#13083553

yes

characteristic

200

C₂×C₄

#15667972

yes

—

201

C₂×C₄

#15667980

yes

characteristic

202

C₂×C₄

#15667989

yes

—

203

C₈

—

204

C₈

—

356

—

205

C₈

—

356

—

206

C₈

—

356

—

207

C₈

—

356

—

208

C₂²

—

209

C₂²

—

210

C₂²

—

211

C₂²

—

212

C₂²

#19151470

yes

characteristic

213

C₄

—

214

C₄

—

215

C₄

—

216

C₄

—

217

C₄

#19151712

yes

—

218

C₄

#19151717

yes

—

219

C₄

#19151765

yes

characteristic

220

C₄

#19151775

yes

characteristic

221

C₂

—

222

C₂

—

223

C₂

#26020461

yes

characteristic

224

C₂

#26020525

yes

—

	subgroup	G/H
center	C₂×C₄	#15667980
commutator	C₄²×C₈₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₈₉×#1962	C₂
socle	C₂²×C₈₉	#385
prime radical	#532×C₈₉	C₂²
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	#1962	D₈₉
hypocenter	C₈₉	#8934
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1962

subgroup

G/H

center

C₂×C₄

#15667980

commutator

C₄²×C₈₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₈₉

Fitting

C₈₉×#1962

C₂

socle

C₂²×C₈₉

#385

prime radical

#532×C₈₉

C₂²

cosocle

C₂×C₄²×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1962

N	H ≅ G/N	type
2. #26019043	C₂	semidirect
4. #26018085	C₂	semidirect
13. #19151347	C₄	semidirect
14. #19151342	C₄	semidirect
15. #19151342	C₄	semidirect
118. C₈₉	#8934	semidirect

H ≅ G/N

type

2. #26019043

C₂

semidirect

4. #26018085

C₂

semidirect

13. #19151347

C₄

semidirect

14. #19151342

C₄

semidirect

15. #19151342

C₄

semidirect

118. C₈₉

#8934

semidirect

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

subgroup	index
#26029319	2
#26019043	2
#26018914	2
#26018085	2
C₈₉×#1962	2
#8934	89

subgroup

index

C₈₉×#1962

Subgroups — #36448930

Conjugacy classes of subgroups

Complements

Maximal subgroups