Group #36455423 — subgroups

Subgroups — #36455423

Order 22784.

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

normal subgroups only

This group has 476 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	#26030776	C₂	1	1	yes	maximal, characteristic
2	#26030291	C₂	1	2	yes	maximal
3	#26018654	C₂	1	2	yes	maximal
4	#26018083	C₂	1	1	yes	maximal, characteristic
5	C₈₉×#1985	C₂	1	1	yes	maximal, characteristic
6	#19152155	C₂²	1	2	yes	—
7	#19152007	—	2	2	no	—
8	#19152007	—	2	2	no	—
9	#19151933	—	2	2	no	—
10	#19151893	—	2	2	no	—
11	#19151808	—	2	2	no	—
12	#19151754	—	2	2	no	—
13	C₄×#13083564	—	2	4	no	—
14	#19151651	C₂²	1	2	yes	—
15	#19151636	—	2	2	no	—
16	#19151328	—	2	4	no	—
17	C₂×C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
19	C₂×#97×C₈₉	C₂²	1	1	yes	characteristic
20	#336×C₈₉	—	2	2	no	—
21	C₂×#13083564	—	2	4	no	—
22	#15668012	D₄	1	2	yes	—
23	#15668012	D₄	1	2	yes	—
24	#15668004	—	2	2	no	—
25	#15667991	—	2	2	no	—
26	#15667991	—	2	2	no	—
27	C₄×Dic₁₇₈	—	2	2	no	—
28	C₄×Dic₁₇₈	—	2	2	no	—
29	#15667971	—	2	2	no	—
30	#15667946	—	2	2	no	—
31	#15667946	—	2	2	no	—
32	#15667945	—	2	4	no	—
33	#15667944	—	2	4	no	—
34	C₄⋊#10623494	—	2	2	no	—
35	#15667936	—	2	4	no	—
36	#15667935	—	2	4	no	—
37	C₂²⋊#10623494	—	2	2	no	—
38	C₄×#10623494	D₄	1	4	yes	—
39	C₂²×Q₈×C₈₉	D₄	1	1	yes	characteristic
40	C₂²×C₈×C₈₉	—	2	1	no	—
41	C₂²×C₈×C₈₉	D₄	1	2	yes	—
42	C₄×Q₈×C₈₉	—	2	2	no	—
43	C₄×Q₈×C₈₉	—	2	2	no	—
44	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
45	C₂×C₄⋊C₄×C₈₉	D₄	1	1	yes	characteristic
46	C₂×C₄²×C₈₉	—	2	1	no	—
47	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
48	C₄⋊C₈×C₈₉	—	2	2	no	—
49	#97×C₈₉	—	2	2	no	—
50	C₂×Dic₁₇₈	—	2	2	no	—
51	C₂×Dic₁₇₈	—	2	2	no	—
52	#13083564	—	4	8	no	—
53	C₂×#10623494	—	2	4	no	—
54	C₄⋊Dic₈₉	—	2	2	no	—
55	C₄⋊Dic₈₉	—	2	2	no	—
56	C₄⋊Dic₈₉	—	2	2	no	—
57	C₄⋊Dic₈₉	—	2	2	no	—
58	#13083553	—	4	2	no	—
59	#13083553	—	4	2	no	—
60	C₂²⋊Dic₈₉	—	4	2	no	—
61	C₄×Dic₈₉	—	4	2	no	—
62	C₂×Q₈×C₈₉	—	2	2	no	—
63	C₂×Q₈×C₈₉	—	4	2	no	—
64	C₂²×C₄×C₈₉	—	2	1	no	—
65	C₂²×C₄×C₈₉	—	2	1	no	—
66	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
69	C₂×C₈×C₈₉	—	2	2	no	—
70	C₂×C₈×C₈₉	—	2	2	no	—
71	C₂×C₈×C₈₉	—	4	1	no	—
72	C₂×C₈×C₈₉	—	4	2	no	—
73	C₂×C₈×C₈₉	Q₁₆	1	4	yes	complemented
74	C₄⋊C₄×C₈₉	—	2	2	no	—
75	C₄⋊C₄×C₈₉	—	4	2	no	—
76	C₄²×C₈₉	—	4	1	no	—
77	C₄²×C₈₉	—	4	1	no	—
78	C₄²×C₈₉	C₂×D₄	1	2	yes	—
79	C₄²×C₈₉	C₂×D₄	1	2	yes	—
80	C₂³×C₈₉	#121	1	1	yes	characteristic
81	Q₈×C₈₉	—	2	4	no	—
82	Q₈×C₈₉	—	4	2	no	—
83	C₂×C₄×C₈₉	—	2	2	no	—
84	C₂×C₄×C₈₉	—	4	1	no	—
85	C₂×C₄×C₈₉	—	4	1	no	—
86	C₂×C₄×C₈₉	—	4	1	no	—
87	C₂×C₄×C₈₉	—	4	1	no	—
88	C₂×C₄×C₈₉	—	4	2	no	—
89	C₂×C₄×C₈₉	—	4	1	no	—
90	C₂×C₄×C₈₉	#120	1	1	yes	characteristic
91	C₂×C₄×C₈₉	#120	1	2	yes	—
92	C₂×C₄×C₈₉	#121	1	2	yes	—
93	C₂×C₄×C₈₉	#127	1	1	yes	characteristic
94	C₂×C₄×C₈₉	C₂×Q₁₆	1	2	yes	—
95	C₂×C₄×C₈₉	#137	1	2	yes	—
96	C₂×Dic₈₉	—	4	2	no	—
97	C₂×Dic₈₉	—	4	2	no	—
98	Dic₁₇₈	—	4	4	no	—
99	Dic₁₇₈	—	4	4	no	—
100	C₈×C₈₉	—	2	4	no	—
101	C₈×C₈₉	—	8	1	no	—
102	#10623494	—	4	4	no	—
103	C₂²×C₈₉	—	2	1	no	—
104	C₂²×C₈₉	—	2	1	no	—
105	C₂²×C₈₉	#392	1	1	yes	characteristic
106	C₂²×C₈₉	#462	1	2	yes	—
107	C₄×C₈₉	—	2	2	no	—
108	C₄×C₈₉	—	2	2	no	—
109	C₄×C₈₉	—	4	2	no	—
110	C₄×C₈₉	—	8	1	no	—
111	C₄×C₈₉	#451	1	2	yes	—
112	C₄×C₈₉	#494	1	1	yes	characteristic
113	C₄×C₈₉	#497	1	1	yes	characteristic
114	Dic₈₉	—	8	2	no	—
115	Dic₈₉	—	8	2	no	—
116	#10037	—	89	1	no	maximal, Sylow
117	C₂×C₈₉	—	2	2	no	—
118	C₂×C₈₉	#2010	1	1	yes	characteristic
119	C₂×C₈₉	#2016	1	1	yes	characteristic
120	C₂×C₈₉	#2017	1	1	yes	characteristic
121	#3162	—	89	1	no	—
122	#3104	—	89	2	no	—
123	#1985	D₈₉	1	1	yes	characteristic
124	#1481	—	89	2	no	—
125	#1275	—	89	1	no	—
126	C₈₉	#10037	1	1	yes	characteristic, complemented, Sylow
127	#533	—	89	2	no	—
128	C₂×C₄×Q₈	C₂×D₈₉	1	1	yes	characteristic
129	#501	—	178	2	no	—
130	#501	—	178	2	no	—
131	#484	—	178	2	no	—
132	#477	—	178	2	no	—
133	#462	—	178	2	no	—
134	#451	—	178	2	no	—
135	C₄×Q₁₆	—	178	4	no	—
136	#431	—	89	2	no	—
137	#427	—	178	2	no	—
138	C₂×C₄⋊C₈	C₂×D₈₉	1	1	yes	characteristic
139	C₂×#97	C₂×D₈₉	1	1	yes	characteristic
140	#336	—	2	2	no	—
141	Q₈⋊C₈	—	178	4	no	—
142	C₂²×Q₈	D₃₅₆	1	1	yes	characteristic
143	C₂×Q₁₆	—	178	4	no	—
144	C₂²×C₈	—	2	1	no	—
145	C₂²×C₈	C₂²⋊D₈₉	1	2	yes	—
146	C₄⋊Q₈	—	89	2	no	—
147	C₄⋊Q₈	—	89	2	no	—
148	#125	—	178	2	no	—
149	C₂²⋊Q₈	—	178	2	no	—
150	C₂²⋊Q₈	—	178	2	no	—
151	C₄×Q₈	—	2	2	no	—
152	C₄×Q₈	—	2	2	no	—
153	C₄×Q₈	—	178	2	no	—
154	C₄×Q₈	—	178	2	no	—
155	#117	—	178	2	no	—
156	C₂×C₄⋊C₄	—	2	1	no	—
157	C₂×C₄⋊C₄	D₃₅₆	1	1	yes	characteristic
158	C₂×C₄²	—	2	1	no	—
159	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
160	#107	—	178	4	no	—
161	#107	—	178	2	no	—
162	#107	—	178	2	no	—
163	#106	—	178	4	no	—
164	C₄⋊C₈	—	2	2	no	—
165	C₄⋊C₈	—	178	2	no	—
166	#103	—	178	4	no	—
167	#103	—	178	4	no	—
168	C₂²⋊C₈	—	178	2	no	—
169	#97	—	2	2	no	—
170	C₄×C₈	—	89	4	no	—
171	C₂×Q₈	—	2	2	no	—
172	C₂×Q₈	—	4	2	no	—
173	C₂×Q₈	—	178	2	no	—
174	C₂×Q₈	—	178	2	no	—
175	C₂²×C₄	—	2	1	no	—
176	C₂²×C₄	—	2	1	no	—
177	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
178	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
179	C₂²×C₄	#13083574	1	1	yes	characteristic
180	Q₁₆	—	356	8	no	—
181	C₂×C₈	—	2	2	no	—
182	C₂×C₈	—	2	2	no	—
183	C₂×C₈	—	4	1	no	—
184	C₂×C₈	—	4	2	no	—
185	C₂×C₈	—	178	4	no	—
186	C₂×C₈	#13083564	1	4	yes	complemented
187	C₄⋊C₄	—	2	2	no	—
188	C₄⋊C₄	—	4	2	no	—
189	C₄⋊C₄	—	356	2	no	—
190	C₄⋊C₄	—	356	2	no	—
191	C₄⋊C₄	—	178	2	no	—
192	C₄⋊C₄	—	178	2	no	—
193	C₄⋊C₄	—	178	2	no	—
194	C₄⋊C₄	—	178	2	no	—
195	C₂²⋊C₄	—	356	2	no	—
196	C₄²	—	4	1	no	—
197	C₄²	—	4	1	no	—
198	C₄²	—	356	2	no	—
199	C₄²	D₄×D₈₉	1	2	yes	—
200	C₄²	D₄×D₈₉	1	2	yes	—
201	C₂³	#15667986	1	1	yes	characteristic
202	Q₈	—	2	4	no	—
203	Q₈	—	4	2	no	—
204	Q₈	—	356	4	no	—
205	Q₈	—	356	4	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	4	1	no	—
208	C₂×C₄	—	4	1	no	—
209	C₂×C₄	—	4	1	no	—
210	C₂×C₄	—	4	1	no	—
211	C₂×C₄	—	4	2	no	—
212	C₂×C₄	—	4	1	no	—
213	C₂×C₄	—	356	2	no	—
214	C₂×C₄	—	356	2	no	—
215	C₂×C₄	#15667980	1	1	yes	characteristic
216	C₂×C₄	#15667982	1	2	yes	—
217	C₂×C₄	#15667985	1	2	yes	—
218	C₂×C₄	#15668008	1	1	yes	characteristic
219	C₂×C₄	C₂×#13083564	1	2	yes	—
220	C₂×C₄	#15668042	1	2	yes	—
221	C₈	—	2	4	no	—
222	C₈	—	8	1	no	—
223	C₈	—	356	4	no	—
224	C₂²	—	2	1	no	—
225	C₂²	—	2	1	no	—
226	C₂²	#19151507	1	1	yes	characteristic
227	C₂²	#19151808	1	2	yes	—
228	C₄	—	2	2	no	—
229	C₄	—	2	2	no	—
230	C₄	—	4	2	no	—
231	C₄	—	8	1	no	—
232	C₄	—	712	2	no	—
233	C₄	—	712	2	no	—
234	C₄	#19151753	1	2	yes	—
235	C₄	#19151977	1	1	yes	characteristic
236	C₄	#19151993	1	1	yes	characteristic
237	C₂	—	2	2	no	—
238	C₂	#26021319	1	1	yes	characteristic
239	C₂	#26021351	1	1	yes	characteristic
240	C₂	#26021356	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26030776

C₂

yes

maximal, characteristic

#26030291

C₂

yes

maximal

#26018654

C₂

yes

maximal

#26018083

C₂

yes

maximal, characteristic

C₈₉×#1985

C₂

yes

maximal, characteristic

#19152155

C₂²

yes

—

—

—

—

—

—

—

—

C₂²

yes

—

#19151636

—

#19151328

—

C₂×C₄×Q₈×C₈₉

C₂²

yes

characteristic

C₂×C₄⋊C₈×C₈₉

C₂²

yes

characteristic

C₂×#97×C₈₉

C₂²

yes

characteristic

#336×C₈₉

—

C₂×#13083564

—

#15668012

D₄

yes

—

#15668012

D₄

yes

—

#15668004

—

#15667991

—

#15667991

—

C₄×Dic₁₇₈

—

C₄×Dic₁₇₈

—

—

—

—

—

—

—

—

—

—

D₄

yes

—

C₂²×Q₈×C₈₉

D₄

yes

characteristic

C₂²×C₈×C₈₉

—

C₂²×C₈×C₈₉

D₄

yes

—

C₄×Q₈×C₈₉

—

C₄×Q₈×C₈₉

—

C₂×C₄⋊C₄×C₈₉

—

C₂×C₄⋊C₄×C₈₉

D₄

yes

characteristic

C₂×C₄²×C₈₉

—

C₂×C₄²×C₈₉

C₂³

yes

characteristic

C₄⋊C₈×C₈₉

—

#97×C₈₉

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

#13083564

—

C₂×#10623494

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

#13083553

—

#13083553

—

C₂²⋊Dic₈₉

—

C₄×Dic₈₉

—

C₂×Q₈×C₈₉

—

C₂×Q₈×C₈₉

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₈₉

#41

yes

characteristic

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

Q₁₆

yes

complemented

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄²×C₈₉

—

C₄²×C₈₉

—

C₄²×C₈₉

C₂×D₄

yes

—

C₄²×C₈₉

C₂×D₄

yes

—

C₂³×C₈₉

#121

yes

characteristic

Q₈×C₈₉

—

Q₈×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

#120

yes

characteristic

C₂×C₄×C₈₉

#120

yes

—

C₂×C₄×C₈₉

#121

yes

—

C₂×C₄×C₈₉

#127

yes

characteristic

C₂×C₄×C₈₉

C₂×Q₁₆

yes

—

C₂×C₄×C₈₉

#137

yes

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

Dic₁₇₈

—

Dic₁₇₈

—

100

C₈×C₈₉

—

101

C₈×C₈₉

—

102

#10623494

—

103

C₂²×C₈₉

—

104

C₂²×C₈₉

—

105

C₂²×C₈₉

#392

yes

characteristic

106

C₂²×C₈₉

#462

yes

—

107

C₄×C₈₉

—

108

C₄×C₈₉

—

109

C₄×C₈₉

—

110

C₄×C₈₉

—

111

C₄×C₈₉

#451

yes

—

112

C₄×C₈₉

#494

yes

characteristic

113

C₄×C₈₉

#497

yes

characteristic

114

Dic₈₉

—

115

Dic₈₉

—

116

#10037

—

maximal, Sylow

117

C₂×C₈₉

—

118

C₂×C₈₉

#2010

yes

characteristic

119

C₂×C₈₉

#2016

yes

characteristic

120

C₂×C₈₉

#2017

yes

characteristic

121

#3162

—

122

#3104

—

123

#1985

D₈₉

yes

characteristic

124

#1481

—

125

#1275

—

126

C₈₉

#10037

yes

characteristic, complemented, Sylow

127

#533

—

128

C₂×C₄×Q₈

C₂×D₈₉

yes

characteristic

129

#501

—

178

—

130

#501

—

178

—

131

#484

—

178

—

132

#477

—

178

—

133

#462

—

178

—

134

#451

—

178

—

135

C₄×Q₁₆

—

178

—

136

#431

—

137

#427

—

178

—

138

C₂×C₄⋊C₈

C₂×D₈₉

yes

characteristic

139

C₂×#97

C₂×D₈₉

yes

characteristic

140

#336

—

141

Q₈⋊C₈

—

178

—

142

C₂²×Q₈

D₃₅₆

yes

characteristic

143

C₂×Q₁₆

—

178

—

144

C₂²×C₈

—

145

C₂²×C₈

C₂²⋊D₈₉

yes

—

146

C₄⋊Q₈

—

147

C₄⋊Q₈

—

148

#125

—

178

—

149

C₂²⋊Q₈

—

178

—

150

C₂²⋊Q₈

—

178

—

151

C₄×Q₈

—

152

C₄×Q₈

—

153

C₄×Q₈

—

178

—

154

C₄×Q₈

—

178

—

155

#117

—

178

—

156

C₂×C₄⋊C₄

—

157

C₂×C₄⋊C₄

D₃₅₆

yes

characteristic

158

C₂×C₄²

—

159

C₂×C₄²

C₂²×D₈₉

yes

characteristic

160

#107

—

178

—

161

#107

—

178

—

162

#107

—

178

—

163

#106

—

178

—

164

C₄⋊C₈

—

165

C₄⋊C₈

—

178

—

166

#103

—

178

—

167

#103

—

178

—

168

C₂²⋊C₈

—

178

—

169

#97

—

170

C₄×C₈

—

171

C₂×Q₈

—

172

C₂×Q₈

—

173

C₂×Q₈

—

178

—

174

C₂×Q₈

—

178

—

175

C₂²×C₄

—

176

C₂²×C₄

—

177

C₂²×C₄

C₂×C₂²⋊D₈₉

yes

characteristic

178

C₂²×C₄

C₂×D₃₅₆

yes

characteristic

179

C₂²×C₄

#13083574

yes

characteristic

180

Q₁₆

—

356

—

181

C₂×C₈

—

182

C₂×C₈

—

183

C₂×C₈

—

184

C₂×C₈

—

185

C₂×C₈

—

178

—

186

C₂×C₈

#13083564

yes

complemented

187

C₄⋊C₄

—

188

C₄⋊C₄

—

189

C₄⋊C₄

—

356

—

190

C₄⋊C₄

—

356

—

191

C₄⋊C₄

—

178

—

192

C₄⋊C₄

—

178

—

193

C₄⋊C₄

—

178

—

194

C₄⋊C₄

—

178

—

195

C₂²⋊C₄

—

356

—

196

C₄²

—

197

C₄²

—

198

C₄²

—

356

—

199

C₄²

D₄×D₈₉

yes

—

200

C₄²

D₄×D₈₉

yes

—

201

C₂³

#15667986

yes

characteristic

202

Q₈

—

203

Q₈

—

204

Q₈

—

356

—

205

Q₈

—

356

—

206

C₂×C₄

—

207

C₂×C₄

—

208

C₂×C₄

—

209

C₂×C₄

—

210

C₂×C₄

—

211

C₂×C₄

—

212

C₂×C₄

—

213

C₂×C₄

—

356

—

214

C₂×C₄

—

356

—

215

C₂×C₄

#15667980

yes

characteristic

216

C₂×C₄

#15667982

yes

—

217

C₂×C₄

#15667985

yes

—

218

C₂×C₄

#15668008

yes

characteristic

219

C₂×C₄

C₂×#13083564

yes

—

220

C₂×C₄

#15668042

yes

—

221

C₈

—

222

C₈

—

223

C₈

—

356

—

224

C₂²

—

225

C₂²

—

226

C₂²

#19151507

yes

characteristic

227

C₂²

#19151808

yes

—

228

C₄

—

229

C₄

—

230

C₄

—

231

C₄

—

232

C₄

—

712

—

233

C₄

—

712

—

234

C₄

#19151753

yes

—

235

C₄

#19151977

yes

characteristic

236

C₄

#19151993

yes

characteristic

237

C₂

—

238

C₂

#26021319

yes

characteristic

239

C₂

#26021351

yes

characteristic

240

C₂

#26021356

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#15668008
commutator, cosocle	C₂×C₄²×C₈₉	C₂³
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₈₉×#1985	C₂
socle	C₂²×C₈₉	#392
prime radical	C₂³×C₈₉	#121
hypercenter	#1985	D₈₉
hypocenter	C₈₉	#10037
LCS	G ▷ C₂×C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1985

subgroup

G/H

center

C₂×C₄

#15668008

commutator, cosocle

C₂×C₄²×C₈₉

C₂³

Frattini

C₂×C₄²

C₂²×D₈₉

Fitting

C₈₉×#1985

C₂

C₂²×C₈₉

C₂³×C₈₉

D₈₉

C₈₉

G ▷ C₂×C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1985

N	H ≅ G/N	type
73. C₂×C₈×C₈₉	Q₁₆	semidirect
126. C₈₉	#10037	semidirect
186. C₂×C₈	#13083564	semidirect

H ≅ G/N

type

73. C₂×C₈×C₈₉

Q₁₆

semidirect

126. C₈₉

#10037

semidirect

186. C₂×C₈

#13083564

semidirect

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

subgroup	index
#26030776	2
#26030291	2
#26018654	2
#26018083	2
C₈₉×#1985	2
#10037	89

subgroup

index

C₈₉×#1985

Subgroups — #36455423

Conjugacy classes of subgroups

Complements

Maximal subgroups