Group #36474285 — subgroups

Subgroups — #36474285

Order 22784.

342 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 105 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	C₂×#19151377	C₂	1	1	yes	maximal, characteristic
2	#26020278	C₂	1	2	yes	maximal
3	#26018344	C₂	1	1	yes	maximal, characteristic, complemented
4	#26018343	C₂	1	2	yes	maximal, complemented
5	C₂×#366×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₈×D₈₉	C₂²	1	1	yes	characteristic
7	C₂²×#13083549	C₂²	1	1	yes	characteristic
8	C₂×#15667946	C₂²	1	2	yes	—
9	#19151455	—	2	2	no	—
10	#19151377	—	2	2	no	—
11	#19151360	—	2	2	no	—
12	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
13	C₂×#107×C₈₉	C₂²	1	2	yes	—
14	#366×C₈₉	—	2	2	no	—
15	C₂²×C₄×D₈₉	D₄	1	1	yes	characteristic
16	C₂²×#10623494	D₄	1	1	yes	characteristic
17	C₂×C₈×D₈₉	—	2	1	no	—
18	C₂×C₈×D₈₉	—	2	1	no	—
19	C₂×C₈×D₈₉	—	2	2	no	—
20	C₂×C₈×D₈₉	—	2	2	no	—
21	C₂×C₄⋊Dic₈₉	—	2	2	no	—
22	C₂×#13083551	—	4	2	no	—
23	C₂×#13083549	—	2	1	no	—
24	C₂×#13083549	—	2	2	no	—
25	#15667946	—	2	2	no	—
26	#15667946	—	2	2	no	—
27	#15667919	—	4	2	no	—
28	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
29	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
30	C₂×C₁₆×C₈₉	—	2	2	no	—
31	C₂×C₁₆×C₈₉	D₄	1	2	yes	—
32	#107×C₈₉	—	2	2	no	—
33	#107×C₈₉	—	2	2	no	—
34	C₂³×D₈₉	D₈	1	1	yes	characteristic
35	C₂²×Dic₈₉	—	4	2	no	—
36	C₂²×Dic₈₉	D₈	1	1	yes	characteristic
37	C₂×C₄×D₈₉	—	2	1	no	—
38	C₂×C₄×D₈₉	—	2	2	no	—
39	C₂×C₄×D₈₉	—	2	1	no	—
40	C₂×C₄×D₈₉	—	2	2	no	—
41	C₂×#10623494	—	2	1	no	—
42	C₂×#10623494	—	2	2	no	—
43	C₈×D₈₉	—	4	1	no	—
44	C₈×D₈₉	—	4	1	no	—
45	C₈×D₈₉	—	4	2	no	—
46	C₄⋊Dic₈₉	—	4	2	no	—
47	C₄⋊Dic₈₉	—	4	2	no	—
48	#13083551	—	8	4	no	—
49	#13083549	—	4	1	no	—
50	C₂²×C₄×C₈₉	—	4	2	no	—
51	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
52	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
53	C₂×C₈×C₈₉	#41	1	2	yes	—
54	C₂×C₈×C₈₉	#41	1	2	yes	—
55	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
56	C₄⋊C₄×C₈₉	—	4	2	no	—
57	C₄⋊C₄×C₈₉	—	4	2	no	—
58	C₁₆×C₈₉	—	2	2	no	—
59	C₂³×C₈₉	C₂×D₈	1	1	yes	characteristic
60	C₂²×D₈₉	—	2	1	no	—
61	C₂²×D₈₉	—	2	2	no	—
62	C₂²×D₈₉	—	2	1	no	—
63	C₂²×D₈₉	—	2	2	no	—
64	C₂²×D₈₉	D₁₆	1	2	yes	—
65	C₂×C₄×C₈₉	—	8	2	no	—
66	C₂×C₄×C₈₉	—	8	2	no	—
67	C₂×C₄×C₈₉	—	8	2	no	—
68	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
69	C₂×C₄×C₈₉	#123	1	2	yes	—
70	C₂×C₄×C₈₉	#135	1	1	yes	characteristic
71	C₂×C₄×C₈₉	#137	1	2	yes	—
72	C₂×Dic₈₉	—	2	1	no	—
73	C₂×Dic₈₉	—	2	2	no	—
74	C₂×Dic₈₉	—	8	2	no	—
75	C₂×Dic₈₉	—	8	2	no	—
76	C₂×Dic₈₉	—	8	2	no	—
77	C₄×D₈₉	—	4	1	no	—
78	C₄×D₈₉	—	4	1	no	—
79	C₄×D₈₉	—	4	2	no	—
80	C₈×C₈₉	#123	1	1	yes	characteristic
81	C₈×C₈₉	#124	1	1	yes	characteristic
82	C₈×C₈₉	#126	1	2	yes	—
83	#10623494	—	4	1	no	—
84	C₂²×C₈₉	#466	1	1	yes	characteristic
85	C₂²×C₈₉	#480	1	2	yes	—
86	C₂²×C₈₉	C₂×D₁₆	1	1	yes	characteristic
87	C₂²×C₈₉	#508	1	1	yes	characteristic
88	C₂²×C₈₉	#510	1	2	yes	—
89	C₂×D₈₉	—	2	2	no	—
90	C₂×D₈₉	—	2	4	no	—
91	C₂×D₈₉	—	4	1	no	—
92	C₂×D₈₉	—	4	1	no	—
93	C₂×D₈₉	—	4	2	no	—
94	C₄×C₈₉	—	16	2	no	—
95	C₄×C₈₉	#397	1	1	yes	characteristic
96	C₄×C₈₉	#470	1	1	yes	characteristic
97	C₄×C₈₉	#485	1	2	yes	—
98	Dic₈₉	—	4	1	no	—
99	Dic₈₉	—	16	2	no	—
100	#13492	—	89	1	no	maximal, Sylow
101	C₂×C₈₉	#2075	1	1	yes	characteristic
102	C₂×C₈₉	#2215	1	1	yes	characteristic
103	C₂×C₈₉	#2219	1	1	yes	characteristic
104	C₂×C₈₉	#2232	1	2	yes	—
105	C₂×C₈₉	#2237	1	2	yes	—
106	D₈₉	—	4	2	no	—
107	C₂×#366	D₈₉	1	1	yes	characteristic, complemented
108	C₂×C₂²⋊C₁₆	—	89	1	no	—
109	#1808	—	89	2	no	—
110	#1380	—	89	1	no	—
111	#1380	—	89	2	no	—
112	C₈₉	#13492	1	1	yes	characteristic, complemented, Sylow
113	C₂³×C₈	—	89	1	no	—
114	C₂²×C₁₆	—	89	1	no	—
115	C₂²×C₁₆	C₂×D₈₉	1	1	yes	characteristic
116	C₂×#107	—	89	2	no	—
117	C₂×#107	C₂×D₈₉	1	2	yes	—
118	#380	—	178	2	no	—
119	#366	—	2	2	no	—
120	C₂²⋊C₁₆	—	178	2	no	—
121	#340	—	178	2	no	—
122	C₂³×C₄	—	89	1	no	—
123	C₂²×C₈	—	89	1	no	—
124	C₂²×C₈	—	178	1	no	—
125	C₂²×C₈	—	178	1	no	—
126	C₂²×C₈	—	178	2	no	—
127	C₂²×C₈	—	178	2	no	—
128	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
129	C₂×C₄⋊C₄	—	2	2	no	—
130	C₂×C₄⋊C₄	—	178	2	no	—
131	C₂×C₂²⋊C₄	—	356	2	no	—
132	C₂×C₁₆	—	2	2	no	—
133	C₂×C₁₆	—	178	1	no	—
134	C₂×C₁₆	—	178	2	no	—
135	C₂×C₁₆	C₂²⋊D₈₉	1	2	yes	—
136	#107	—	2	2	no	—
137	#107	—	2	2	no	—
138	#107	—	178	2	no	—
139	#107	—	178	2	no	—
140	#95	—	356	2	no	—
141	C₂⁴	—	89	1	no	—
142	C₂²×C₄	—	4	2	no	—
143	C₂²×C₄	—	89	1	no	—
144	C₂²×C₄	—	356	2	no	—
145	C₂²×C₄	—	178	1	no	—
146	C₂²×C₄	—	178	2	no	—
147	C₂²×C₄	—	178	1	no	—
148	C₂²×C₄	—	178	2	no	—
149	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
150	C₂×C₈	—	356	1	no	—
151	C₂×C₈	—	356	1	no	—
152	C₂×C₈	—	356	2	no	—
153	C₂×C₈	—	178	1	no	—
154	C₂×C₈	—	178	2	no	—
155	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
156	C₂×C₈	#13083573	1	1	yes	characteristic
157	C₂×C₈	#13083574	1	2	yes	—
158	C₂×C₈	#13083575	1	2	yes	—
159	C₄⋊C₄	—	4	2	no	—
160	C₄⋊C₄	—	4	2	no	—
161	C₄⋊C₄	—	356	2	no	—
162	C₄⋊C₄	—	356	2	no	—
163	C₂²⋊C₄	—	712	4	no	—
164	C₁₆	—	2	2	no	—
165	C₁₆	—	356	1	no	—
166	C₂³	—	89	2	no	—
167	C₂³	—	178	2	no	—
168	C₂³	—	178	1	no	—
169	C₂³	—	178	2	no	—
170	C₂³	—	178	1	no	—
171	C₂³	D₈×D₈₉	1	1	yes	characteristic
172	C₂×C₄	—	8	2	no	—
173	C₂×C₄	—	8	2	no	—
174	C₂×C₄	—	8	2	no	—
175	C₂×C₄	—	356	2	no	—
176	C₂×C₄	—	356	1	no	—
177	C₂×C₄	—	356	1	no	—
178	C₂×C₄	—	178	2	no	—
179	C₂×C₄	—	178	1	no	—
180	C₂×C₄	—	712	2	no	—
181	C₂×C₄	—	712	2	no	—
182	C₂×C₄	—	712	2	no	—
183	C₂×C₄	#15667983	1	1	yes	characteristic
184	C₂×C₄	#15667997	1	2	yes	—
185	C₂×C₄	#15668031	1	1	yes	characteristic
186	C₂×C₄	#15668044	1	2	yes	—
187	C₈	—	356	1	no	—
188	C₈	#15667995	1	1	yes	characteristic
189	C₈	#15668001	1	1	yes	characteristic
190	C₈	#15668007	1	2	yes	—
191	C₂²	—	356	1	no	—
192	C₂²	—	356	2	no	—
193	C₂²	—	356	1	no	—
194	C₂²	—	178	4	no	—
195	C₂²	—	178	2	no	—
196	C₂²	#19151831	1	1	yes	characteristic
197	C₂²	#19151913	1	2	yes	—
198	C₂²	D₁₆×D₈₉	1	1	yes	characteristic
199	C₂²	#19152027	1	1	yes	characteristic
200	C₂²	#19152040	1	2	yes	—
201	C₄	—	16	2	no	—
202	C₄	—	356	1	no	—
203	C₄	—	1424	2	no	—
204	C₄	#19151524	1	1	yes	characteristic
205	C₄	#19151853	1	1	yes	characteristic
206	C₄	#19151941	1	2	yes	—
207	C₂	—	356	2	no	—
208	C₂	#26021676	1	1	yes	characteristic
209	C₂	#26022335	1	1	yes	characteristic
210	C₂	#26022357	1	1	yes	characteristic
211	C₂	#26022434	1	2	yes	—
212	C₂	#26022462	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19151377

C₂

yes

maximal, characteristic

#26020278

C₂

yes

maximal

#26018344

C₂

yes

maximal, characteristic, complemented

#26018343

C₂

yes

maximal, complemented

C₂×#366×C₈₉

C₂

yes

maximal, characteristic, complemented

C₂²×C₈×D₈₉

C₂²

yes

characteristic

C₂²×#13083549

C₂²

yes

characteristic

C₂×#15667946

C₂²

yes

—

#19151455

—

#19151377

—

#19151360

—

C₂²×C₁₆×C₈₉

C₂²

yes

characteristic

C₂×#107×C₈₉

C₂²

yes

—

#366×C₈₉

—

C₂²×C₄×D₈₉

D₄

yes

characteristic

C₂²×#10623494

D₄

yes

characteristic

C₂×C₈×D₈₉

—

C₂×C₈×D₈₉

—

C₂×C₈×D₈₉

—

C₂×C₈×D₈₉

—

C₂×C₄⋊Dic₈₉

—

—

—

—

—

—

—

C₂²×C₈×C₈₉

C₂³

yes

characteristic

C₂×C₄⋊C₄×C₈₉

—

C₂×C₁₆×C₈₉

—

C₂×C₁₆×C₈₉

D₄

yes

—

#107×C₈₉

—

#107×C₈₉

—

C₂³×D₈₉

D₈

yes

characteristic

C₂²×Dic₈₉

—

C₂²×Dic₈₉

D₈

yes

characteristic

C₂×C₄×D₈₉

—

C₂×C₄×D₈₉

—

C₂×C₄×D₈₉

—

C₂×C₄×D₈₉

—

C₂×#10623494

—

C₂×#10623494

—

C₈×D₈₉

—

C₈×D₈₉

—

C₈×D₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

#13083551

—

#13083549

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₈×C₈₉

C₂×D₄

yes

characteristic

C₂×C₈×C₈₉

#41

yes

—

C₂×C₈×C₈₉

#41

yes

—

C₂×C₈×C₈₉

#41

yes

characteristic

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₁₆×C₈₉

—

C₂³×C₈₉

C₂×D₈

yes

characteristic

C₂²×D₈₉

—

C₂²×D₈₉

—

C₂²×D₈₉

—

C₂²×D₈₉

—

C₂²×D₈₉

D₁₆

yes

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

#121

yes

characteristic

C₂×C₄×C₈₉

#123

yes

—

C₂×C₄×C₈₉

#135

yes

characteristic

C₂×C₄×C₈₉

#137

yes

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₄×D₈₉

—

C₄×D₈₉

—

C₄×D₈₉

—

C₈×C₈₉

yes

C₈×C₈₉

yes

C₈×C₈₉

yes

—

#10623494

—

C₂²×C₈₉

#466

yes

characteristic

C₂²×C₈₉

#480

yes

—

C₂²×C₈₉

C₂×D₁₆

yes

C₂²×C₈₉

yes

C₂²×C₈₉

yes

—

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₄×C₈₉

—

C₄×C₈₉

yes

C₄×C₈₉

yes

C₄×C₈₉

yes

—

Dic₈₉

—

Dic₈₉

—

100

#13492

—

maximal, Sylow

101

C₂×C₈₉

#2075

yes

characteristic

102

C₂×C₈₉

#2215

yes

characteristic

103

C₂×C₈₉

#2219

yes

characteristic

104

C₂×C₈₉

#2232

yes

—

105

C₂×C₈₉

#2237

yes

—

106

D₈₉

—

107

C₂×#366

D₈₉

yes

characteristic, complemented

108

C₂×C₂²⋊C₁₆

—

109

#1808

—

110

#1380

—

111

#1380

—

112

C₈₉

#13492

yes

characteristic, complemented, Sylow

113

C₂³×C₈

—

114

C₂²×C₁₆

—

115

C₂²×C₁₆

C₂×D₈₉

yes

characteristic

116

C₂×#107

—

117

C₂×#107

C₂×D₈₉

yes

—

118

#380

—

178

—

119

#366

—

120

C₂²⋊C₁₆

—

178

—

121

#340

—

178

—

122

C₂³×C₄

—

123

C₂²×C₈

—

124

C₂²×C₈

—

178

—

125

C₂²×C₈

—

178

—

126

C₂²×C₈

—

178

—

127

C₂²×C₈

—

178

—

128

C₂²×C₈

C₂²×D₈₉

yes

characteristic

129

C₂×C₄⋊C₄

—

130

C₂×C₄⋊C₄

—

178

—

131

C₂×C₂²⋊C₄

—

356

—

132

C₂×C₁₆

—

133

C₂×C₁₆

—

178

—

134

C₂×C₁₆

—

178

—

135

C₂×C₁₆

C₂²⋊D₈₉

yes

—

136

#107

—

137

#107

—

138

#107

—

178

—

139

#107

—

178

—

140

#95

—

356

—

141

C₂⁴

—

142

C₂²×C₄

—

143

C₂²×C₄

—

144

C₂²×C₄

—

356

—

145

C₂²×C₄

—

178

—

146

C₂²×C₄

—

178

—

147

C₂²×C₄

—

178

—

148

C₂²×C₄

—

178

—

149

C₂²×C₄

D₄×D₈₉

yes

characteristic

150

C₂×C₈

—

356

—

151

C₂×C₈

—

356

—

152

C₂×C₈

—

356

—

153

C₂×C₈

—

178

—

154

C₂×C₈

—

178

—

155

C₂×C₈

C₂×C₂²⋊D₈₉

yes

characteristic

156

C₂×C₈

#13083573

yes

characteristic

157

C₂×C₈

#13083574

yes

—

158

C₂×C₈

#13083575

yes

—

159

C₄⋊C₄

—

160

C₄⋊C₄

—

161

C₄⋊C₄

—

356

—

162

C₄⋊C₄

—

356

—

163

C₂²⋊C₄

—

712

—

164

C₁₆

—

165

C₁₆

—

356

—

166

C₂³

—

167

C₂³

—

178

—

168

C₂³

—

178

—

169

C₂³

—

178

—

170

C₂³

—

178

—

171

C₂³

D₈×D₈₉

yes

characteristic

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

—

356

—

176

C₂×C₄

—

356

—

177

C₂×C₄

—

356

—

178

C₂×C₄

—

178

—

179

C₂×C₄

—

178

—

180

C₂×C₄

—

712

—

181

C₂×C₄

—

712

—

182

C₂×C₄

—

712

—

183

C₂×C₄

#15667983

yes

characteristic

184

C₂×C₄

#15667997

yes

—

185

C₂×C₄

#15668031

yes

characteristic

186

C₂×C₄

#15668044

yes

—

187

C₈

—

356

—

188

C₈

#15667995

yes

characteristic

189

C₈

#15668001

yes

characteristic

190

C₈

#15668007

yes

—

191

C₂²

—

356

—

192

C₂²

—

356

—

193

C₂²

—

356

—

194

C₂²

—

178

—

195

C₂²

—

178

—

196

C₂²

#19151831

yes

characteristic

197

C₂²

#19151913

yes

—

198

C₂²

D₁₆×D₈₉

yes

characteristic

199

C₂²

#19152027

yes

characteristic

200

C₂²

#19152040

yes

—

201

C₄

—

202

C₄

—

356

—

203

C₄

—

1424

—

204

C₄

#19151524

yes

characteristic

205

C₄

#19151853

yes

characteristic

206

C₄

#19151941

yes

—

207

C₂

—

356

—

208

C₂

#26021676

yes

characteristic

209

C₂

#26022335

yes

characteristic

210

C₂

#26022357

yes

characteristic

211

C₂

#26022434

yes

—

212

C₂

#26022462

yes

—

	subgroup	G/H
center	C₂³	D₈×D₈₉
commutator, cosocle	C₂²×C₈×C₈₉	C₂³
Frattini	C₂²×C₈	C₂²×D₈₉
Fitting	C₂×#366×C₈₉	C₂
socle	C₂³×C₈₉	C₂×D₈
prime radical	C₂³×D₈₉	D₈
hypercenter	C₂×#366	D₈₉
hypocenter	C₈₉	#13492
LCS	G ▷ C₂²×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#366

subgroup

G/H

center

C₂³

D₈×D₈₉

commutator, cosocle

C₂²×C₈×C₈₉

C₂³

Frattini

C₂²×C₈

C₂²×D₈₉

Fitting

C₂×#366×C₈₉

C₂

socle

C₂³×C₈₉

C₂×D₈

prime radical

C₂³×D₈₉

D₈

D₈₉

C₈₉

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

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#366

N	H ≅ G/N	type
3. #26018344	C₂	semidirect
4. #26018343	C₂	semidirect
5. C₂×#366×C₈₉	C₂	semidirect
107. C₂×#366	D₈₉	semidirect
112. C₈₉	#13492	semidirect

H ≅ G/N

type

3. #26018344

C₂

semidirect

4. #26018343

C₂

semidirect

5. C₂×#366×C₈₉

C₂

semidirect

107. C₂×#366

D₈₉

semidirect

112. C₈₉

#13492

semidirect

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

subgroup	index
C₂×#19151377	2
#26020278	2
#26018344	2
#26018343	2
C₂×#366×C₈₉	2
#13492	89

subgroup

index

C₂×#366×C₈₉

Subgroups — #36474285

Conjugacy classes of subgroups

Complements

Maximal subgroups