Group #36449378 — subgroups

Subgroups — #36449378

Order 22784.

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

normal subgroups only

This group has 416 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	#26029323	C₂	1	1	yes	maximal, characteristic
2	#26018914	C₂	1	1	yes	maximal, characteristic
3	#26018725	C₂	1	2	yes	maximal, complemented
4	#26018151	C₂	1	2	yes	maximal, complemented
5	C₈₉×#3135	C₂	1	1	yes	maximal, characteristic
6	#19151670	—	2	4	no	—
7	#19151655	C₂²	1	2	yes	—
8	#19151651	C₂²	1	2	yes	—
9	#19151565	—	2	2	no	—
10	#19151347	—	2	1	no	—
11	#19151342	—	2	1	no	—
12	Q₈⋊#10623494	—	2	4	no	—
13	#533×C₈₉	C₂²	1	1	yes	characteristic
14	#532×C₈₉	C₂²	1	1	yes	characteristic
15	#491×C₈₉	—	2	1	no	—
16	#490×C₈₉	—	2	1	no	—
17	#489×C₈₉	C₄	1	4	yes	—
18	#416×C₈₉	—	2	2	no	—
19	C₂×#97×C₈₉	C₂²	1	1	yes	characteristic
20	C₂×#13083558	—	2	2	no	—
21	C₂²×#10623494	—	2	2	no	—
22	C₄⋊#10623494	—	2	2	no	—
23	C₄⋊#10623494	D₄	1	4	yes	—
24	C₂²⋊#10623494	—	2	4	no	—
25	C₂²⋊#10623494	—	2	2	no	—
26	C₂²⋊#10623494	—	2	2	no	—
27	#15667923	—	2	2	no	—
28	C₄×#10623494	—	2	2	no	—
29	C₂×#41×C₈₉	D₄	1	2	yes	—
30	C₂²×C₈×C₈₉	—	2	1	no	—
31	C₄⋊Q₈×C₈₉	C₂×C₄	1	2	yes	—
32	C₄⋊Q₈×C₈₉	D₄	1	1	yes	characteristic
33	#127×C₈₉	D₄	1	1	yes	characteristic
34	#125×C₈₉	—	2	1	no	—
35	#124×C₈₉	C₂×C₄	1	2	yes	—
36	C₂²⋊Q₈×C₈₉	—	2	2	no	—
37	#121×C₈₉	—	2	2	no	—
38	C₄×Q₈×C₈₉	—	2	2	no	—
39	C₄×D₄×C₈₉	—	2	2	no	—
40	#117×C₈₉	—	2	1	no	—
41	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
42	#103×C₈₉	—	2	4	no	—
43	#102×C₈₉	—	2	4	no	—
44	#97×C₈₉	—	2	1	no	—
45	#97×C₈₉	C₂×C₄	1	2	yes	—
46	#13083558	—	8	2	no	—
47	C₂×#10623494	—	2	4	no	—
48	C₂×#10623494	—	4	2	no	—
49	C₂×#10623494	—	8	2	no	—
50	#41×C₈₉	—	4	2	no	—
51	#41×C₈₉	—	4	2	no	—
52	C₂×Q₈×C₈₉	—	2	2	no	—
53	C₂×Q₈×C₈₉	C₂²⋊C₄	1	2	yes	—
54	C₂×D₄×C₈₉	—	2	2	no	—
55	C₂×D₄×C₈₉	—	2	2	no	—
56	C₂×D₄×C₈₉	C₂²⋊C₄	1	2	yes	—
57	C₂²×C₄×C₈₉	—	2	2	no	—
58	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₈₉	#41	1	2	yes	—
60	C₂×C₈×C₈₉	—	2	2	no	—
61	C₂×C₈×C₈₉	—	4	2	no	—
62	C₄⋊C₄×C₈₉	—	2	2	no	—
63	C₄⋊C₄×C₈₉	—	2	2	no	—
64	C₄⋊C₄×C₈₉	—	2	2	no	—
65	C₄⋊C₄×C₈₉	—	4	2	no	—
66	C₂²⋊C₄×C₈₉	—	2	4	no	—
67	C₂²⋊C₄×C₈₉	—	4	1	no	—
68	C₄²×C₈₉	—	4	1	no	—
69	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
70	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
71	C₄²×C₈₉	C₂×D₄	1	2	yes	—
72	C₂³×C₈₉	—	2	2	no	—
73	C₂³×C₈₉	#123	1	1	yes	characteristic
74	Q₈×C₈₉	—	4	2	no	—
75	Q₈×C₈₉	—	4	4	no	—
76	D₄×C₈₉	—	4	2	no	—
77	D₄×C₈₉	—	8	2	no	—
78	D₄×C₈₉	—	8	2	no	—
79	C₂×C₄×C₈₉	—	2	2	no	—
80	C₂×C₄×C₈₉	—	2	2	no	—
81	C₂×C₄×C₈₉	—	4	2	no	—
82	C₂×C₄×C₈₉	—	4	2	no	—
83	C₂×C₄×C₈₉	—	4	2	no	—
84	C₂×C₄×C₈₉	—	4	2	no	—
85	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
87	C₂×C₄×C₈₉	#120	1	1	yes	characteristic
88	C₂×C₄×C₈₉	#121	1	2	yes	—
89	C₈×C₈₉	—	4	2	no	—
90	#10623494	—	4	4	no	—
91	#10623494	—	8	2	no	—
92	C₂²×C₈₉	—	2	2	no	—
93	C₂²×C₈₉	—	2	1	no	—
94	C₂²×C₈₉	—	4	4	no	—
95	C₂²×C₈₉	—	4	2	no	—
96	C₂²×C₈₉	#386	1	1	yes	characteristic
97	C₄×C₈₉	—	2	2	no	—
98	C₄×C₈₉	—	2	2	no	—
99	C₄×C₈₉	—	2	1	no	—
100	C₄×C₈₉	—	4	2	no	—
101	C₄×C₈₉	—	8	2	no	—
102	C₄×C₈₉	#455	1	2	yes	—
103	#9005	—	89	1	no	maximal, Sylow
104	C₂×C₈₉	—	4	1	no	—
105	C₂×C₈₉	—	8	2	no	—
106	C₂×C₈₉	#1881	1	2	yes	—
107	C₂×C₈₉	#1885	1	1	yes	characteristic
108	#3135	D₈₉	1	1	yes	characteristic
109	#2980	—	89	1	no	—
110	#1543	—	89	1	no	—
111	#1497	—	89	2	no	—
112	#1302	—	89	2	no	—
113	C₈₉	#9005	1	1	yes	characteristic, complemented, Sylow
114	#533	C₂×D₈₉	1	1	yes	characteristic
115	#532	C₂×D₈₉	1	1	yes	characteristic
116	#491	—	2	1	no	—
117	#490	—	2	1	no	—
118	#489	Dic₈₉	1	4	yes	—
119	#435	—	178	4	no	—
120	#432	—	89	2	no	—
121	#431	—	89	2	no	—
122	#416	—	2	2	no	—
123	#408	—	178	2	no	—
124	C₂×#97	C₂×D₈₉	1	1	yes	characteristic
125	#333	—	178	1	no	—
126	#331	—	178	1	no	—
127	Q₈⋊C₈	—	178	4	no	—
128	C₂×#41	C₂²⋊D₈₉	1	2	yes	—
129	C₂×M₁₆	—	178	2	no	—
130	C₂²×C₈	—	2	1	no	—
131	C₂²×C₈	—	178	2	no	—
132	C₄⋊Q₈	C₂×Dic₈₉	1	2	yes	—
133	C₄⋊Q₈	C₂²⋊D₈₉	1	1	yes	characteristic
134	#127	C₂²⋊D₈₉	1	1	yes	characteristic
135	#125	—	2	1	no	—
136	#124	C₂×Dic₈₉	1	2	yes	—
137	C₂²⋊Q₈	—	2	2	no	—
138	#121	—	2	2	no	—
139	C₄×Q₈	—	2	2	no	—
140	C₄×D₄	—	2	2	no	—
141	#117	—	2	1	no	—
142	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
143	C₄⋊C₈	—	89	4	no	—
144	C₄⋊C₈	—	178	2	no	—
145	#103	—	2	4	no	—
146	#102	—	2	4	no	—
147	C₂²⋊C₈	—	178	2	no	—
148	C₂²⋊C₈	—	178	4	no	—
149	C₂²⋊C₈	—	178	2	no	—
150	#97	—	2	1	no	—
151	#97	—	178	2	no	—
152	#97	C₂×Dic₈₉	1	2	yes	—
153	C₄×C₈	—	178	2	no	—
154	#41	—	4	2	no	—
155	#41	—	4	2	no	—
156	C₂×Q₈	—	2	2	no	—
157	C₂×Q₈	C₂²⋊Dic₈₉	1	2	yes	—
158	C₂×D₄	—	2	2	no	—
159	C₂×D₄	—	2	2	no	—
160	C₂×D₄	C₂²⋊Dic₈₉	1	2	yes	—
161	C₂²×C₄	—	2	2	no	—
162	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
163	C₂²×C₄	#13083573	1	2	yes	—
164	M₁₆	—	712	2	no	—
165	C₂×C₈	—	2	2	no	—
166	C₂×C₈	—	4	2	no	—
167	C₂×C₈	—	356	2	no	—
168	C₂×C₈	—	178	4	no	—
169	C₂×C₈	—	712	2	no	—
170	C₄⋊C₄	—	2	2	no	—
171	C₄⋊C₄	—	2	2	no	—
172	C₄⋊C₄	—	2	2	no	—
173	C₄⋊C₄	—	4	2	no	—
174	C₂²⋊C₄	—	2	4	no	—
175	C₂²⋊C₄	—	4	1	no	—
176	C₄²	—	4	1	no	—
177	C₄²	C₂²×Dic₈₉	1	1	yes	characteristic
178	C₄²	D₄×D₈₉	1	2	yes	—
179	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
180	C₂³	—	2	2	no	—
181	C₂³	#15667993	1	1	yes	characteristic
182	Q₈	—	4	2	no	—
183	Q₈	—	4	4	no	—
184	D₄	—	4	2	no	—
185	D₄	—	8	2	no	—
186	D₄	—	8	2	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	4	2	no	—
190	C₂×C₄	—	4	2	no	—
191	C₂×C₄	—	4	2	no	—
192	C₂×C₄	—	4	2	no	—
193	C₂×C₄	C₂×C₂²⋊Dic₈₉	1	1	yes	characteristic
194	C₂×C₄	D₄×Dic₈₉	1	2	yes	—
195	C₂×C₄	#15667980	1	1	yes	characteristic
196	C₂×C₄	#15667984	1	2	yes	—
197	C₈	—	4	2	no	—
198	C₈	—	356	4	no	—
199	C₈	—	712	2	no	—
200	C₂²	—	2	2	no	—
201	C₂²	—	2	1	no	—
202	C₂²	—	4	4	no	—
203	C₂²	—	4	2	no	—
204	C₂²	#19151478	1	1	yes	characteristic
205	C₄	—	2	2	no	—
206	C₄	—	2	2	no	—
207	C₄	—	2	1	no	—
208	C₄	—	4	2	no	—
209	C₄	—	8	2	no	—
210	C₄	#19151775	1	2	yes	—
211	C₂	—	4	1	no	—
212	C₂	—	8	2	no	—
213	C₂	#26020616	1	2	yes	—
214	C₂	#26020641	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26029323

C₂

yes

maximal, characteristic

#26018914

C₂

yes

maximal, characteristic

#26018725

C₂

yes

maximal, complemented

#26018151

C₂

yes

maximal, complemented

C₈₉×#3135

C₂

yes

maximal, characteristic

#19151670

—

#19151655

C₂²

yes

—

#19151651

C₂²

yes

—

—

—

—

—

#533×C₈₉

C₂²

yes

characteristic

#532×C₈₉

C₂²

yes

characteristic

#491×C₈₉

—

#490×C₈₉

—

#489×C₈₉

C₄

yes

—

#416×C₈₉

—

C₂×#97×C₈₉

C₂²

yes

—

—

—

D₄

yes

—

—

—

—

—

—

C₂×#41×C₈₉

D₄

yes

—

C₂²×C₈×C₈₉

—

C₄⋊Q₈×C₈₉

C₂×C₄

yes

—

C₄⋊Q₈×C₈₉

D₄

yes

characteristic

#127×C₈₉

D₄

yes

characteristic

#125×C₈₉

—

#124×C₈₉

C₂×C₄

yes

—

C₂²⋊Q₈×C₈₉

—

#121×C₈₉

—

C₄×Q₈×C₈₉

—

C₄×D₄×C₈₉

—

#117×C₈₉

—

C₂×C₄²×C₈₉

C₂³

yes

characteristic

#103×C₈₉

—

#102×C₈₉

—

#97×C₈₉

—

#97×C₈₉

C₂×C₄

yes

—

—

—

—

—

#41×C₈₉

—

#41×C₈₉

—

C₂×Q₈×C₈₉

—

C₂×Q₈×C₈₉

C₂²⋊C₄

yes

—

C₂×D₄×C₈₉

—

C₂×D₄×C₈₉

—

C₂×D₄×C₈₉

C₂²⋊C₄

yes

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₈₉

#41

yes

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₂²⋊C₄×C₈₉

—

C₂²⋊C₄×C₈₉

—

C₄²×C₈₉

—

C₄²×C₈₉

C₂²×C₄

yes

characteristic

C₄²×C₈₉

C₂×D₄

yes

characteristic

C₄²×C₈₉

C₂×D₄

yes

—

C₂³×C₈₉

—

C₂³×C₈₉

#123

yes

characteristic

Q₈×C₈₉

—

Q₈×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×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₈₉

C₄×D₄

yes

—

C₂×C₄×C₈₉

#120

yes

characteristic

C₂×C₄×C₈₉

#121

yes

—

C₈×C₈₉

—

#10623494

—

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

#386

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

100

C₄×C₈₉

—

101

C₄×C₈₉

—

102

C₄×C₈₉

#455

yes

—

103

#9005

—

maximal, Sylow

104

C₂×C₈₉

—

105

C₂×C₈₉

—

106

C₂×C₈₉

#1881

yes

—

107

C₂×C₈₉

#1885

yes

characteristic

108

#3135

D₈₉

yes

characteristic

109

#2980

—

110

#1543

—

111

#1497

—

112

#1302

—

113

C₈₉

#9005

yes

characteristic, complemented, Sylow

114

#533

C₂×D₈₉

yes

characteristic

115

#532

C₂×D₈₉

yes

characteristic

116

#491

—

117

#490

—

118

#489

Dic₈₉

yes

—

119

#435

—

178

—

120

#432

—

121

#431

—

122

#416

—

123

#408

—

178

—

124

C₂×#97

C₂×D₈₉

yes

characteristic

125

#333

—

178

—

126

#331

—

178

—

127

Q₈⋊C₈

—

178

—

128

C₂×#41

C₂²⋊D₈₉

yes

—

129

C₂×M₁₆

—

178

—

130

C₂²×C₈

—

131

C₂²×C₈

—

178

—

132

C₄⋊Q₈

C₂×Dic₈₉

yes

—

133

C₄⋊Q₈

C₂²⋊D₈₉

yes

characteristic

134

#127

C₂²⋊D₈₉

yes

characteristic

135

#125

—

136

#124

C₂×Dic₈₉

yes

—

137

C₂²⋊Q₈

—

138

#121

—

139

C₄×Q₈

—

140

C₄×D₄

—

141

#117

—

142

C₂×C₄²

C₂²×D₈₉

yes

characteristic

143

C₄⋊C₈

—

144

C₄⋊C₈

—

178

—

145

#103

—

146

#102

—

147

C₂²⋊C₈

—

178

—

148

C₂²⋊C₈

—

178

—

149

C₂²⋊C₈

—

178

—

150

#97

—

151

#97

—

178

—

152

#97

C₂×Dic₈₉

yes

—

153

C₄×C₈

—

178

—

154

#41

—

155

#41

—

156

C₂×Q₈

—

157

C₂×Q₈

C₂²⋊Dic₈₉

yes

—

158

C₂×D₄

—

159

C₂×D₄

—

160

C₂×D₄

C₂²⋊Dic₈₉

yes

—

161

C₂²×C₄

—

162

C₂²×C₄

C₂×C₂²⋊D₈₉

yes

characteristic

163

C₂²×C₄

#13083573

yes

—

164

M₁₆

—

712

—

165

C₂×C₈

—

166

C₂×C₈

—

167

C₂×C₈

—

356

—

168

C₂×C₈

—

178

—

169

C₂×C₈

—

712

—

170

C₄⋊C₄

—

171

C₄⋊C₄

—

172

C₄⋊C₄

—

173

C₄⋊C₄

—

174

C₂²⋊C₄

—

175

C₂²⋊C₄

—

176

C₄²

—

177

C₄²

C₂²×Dic₈₉

yes

characteristic

178

C₄²

D₄×D₈₉

yes

—

179

C₄²

C₂×C₂²⋊D₈₉

yes

characteristic

180

C₂³

—

181

C₂³

#15667993

yes

characteristic

182

Q₈

—

183

Q₈

—

184

D₄

—

185

D₄

—

186

D₄

—

187

C₂×C₄

—

188

C₂×C₄

—

189

C₂×C₄

—

190

C₂×C₄

—

191

C₂×C₄

—

192

C₂×C₄

—

193

C₂×C₄

C₂×C₂²⋊Dic₈₉

yes

characteristic

194

C₂×C₄

D₄×Dic₈₉

yes

—

195

C₂×C₄

#15667980

yes

characteristic

196

C₂×C₄

#15667984

yes

—

197

C₈

—

198

C₈

—

356

—

199

C₈

—

712

—

200

C₂²

—

201

C₂²

—

202

C₂²

—

203

C₂²

—

204

C₂²

#19151478

yes

characteristic

205

C₄

—

206

C₄

—

207

C₄

—

208

C₄

—

209

C₄

—

210

C₄

#19151775

yes

—

211

C₂

—

212

C₂

—

213

C₂

#26020616

yes

—

214

C₂

#26020641

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#15667980
commutator	C₄²×C₈₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₈₉×#3135	C₂
socle	C₂²×C₈₉	#386
prime radical	#532×C₈₉	C₂²
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	#3135	D₈₉
hypocenter	C₈₉	#9005
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3135

subgroup

G/H

center

C₂×C₄

#15667980

commutator

C₄²×C₈₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₈₉

Fitting

C₈₉×#3135

C₂

socle

C₂²×C₈₉

#386

prime radical

#532×C₈₉

C₂²

cosocle

C₂×C₄²×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3135

N	H ≅ G/N	type
3. #26018725	C₂	semidirect
4. #26018151	C₂	semidirect
113. C₈₉	#9005	semidirect

H ≅ G/N

type

3. #26018725

C₂

semidirect

4. #26018151

C₂

semidirect

113. C₈₉

#9005

semidirect

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

subgroup	index
#26029323	2
#26018914	2
#26018725	2
#26018151	2
C₈₉×#3135	2
#9005	89

subgroup

index

C₈₉×#3135

Subgroups — #36449378

Conjugacy classes of subgroups

Complements

Maximal subgroups