Group #26031369 — subgroups

Subgroups — #26031369

Order 11392.

404 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 404 conjugacy classes of subgroups — too many to draw. Only its 125 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	#19152146	C₂	1	1	yes	maximal, characteristic
2	C₄⋊Q₈×D₈₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#19152003	C₂	1	1	yes	maximal, characteristic, complemented
4	#19152001	C₂	1	1	yes	maximal, characteristic, complemented
5	#19151944	C₂	1	1	yes	maximal, characteristic, complemented
6	#19151910	C₂	1	4	yes	maximal, complemented
7	#19151876	C₂	1	4	yes	maximal, complemented
8	#19151649	C₂	1	1	yes	maximal, characteristic, complemented
9	#486×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#13083573	—	2	1	no	—
11	C₂×Q₈×D₈₉	—	2	1	no	—
12	#15668012	C₂²	1	1	yes	characteristic
13	#15668012	C₂²	1	1	yes	characteristic
14	#15668011	—	2	1	no	—
15	#15668011	C₂²	1	2	yes	—
16	#15668010	—	2	1	no	—
17	#15667999	—	2	1	no	—
18	#15667983	—	2	1	no	—
19	#15667983	C₂²	1	2	yes	—
20	D₄×Dic₈₉	—	2	1	no	—
21	D₄×Dic₈₉	C₂²	1	2	yes	—
22	C₄⋊C₄×D₈₉	—	2	1	no	—
23	C₄⋊C₄×D₈₉	C₂²	1	2	yes	—
24	C₄²×D₈₉	C₂²	1	1	yes	characteristic
25	#15667946	C₂²	1	4	yes	—
26	#15667945	C₂²	1	4	yes	—
27	#15667941	C₂²	1	2	yes	—
28	#15667933	C₂²	1	4	yes	—
29	#15667924	C₂²	1	2	yes	—
30	C₄×#10623494	C₂²	1	1	yes	characteristic
31	C₄⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
32	#127×C₈₉	C₂²	1	1	yes	characteristic
33	#102×C₈₉	C₂²	1	4	yes	—
34	C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
35	#13083573	—	4	4	no	—
36	C₂×Dic₁₇₈	—	2	1	no	—
37	C₂×Dic₁₇₈	—	2	1	no	—
38	C₂×Dic₁₇₈	—	2	2	no	—
39	Q₈×D₈₉	—	4	4	no	—
40	C₂×C₂²⋊D₈₉	—	2	2	no	—
41	C₂²×Dic₈₉	—	2	2	no	—
42	C₂×C₄×D₈₉	—	2	2	no	—
43	C₂×C₄×D₈₉	D₄	1	1	yes	characteristic
44	C₂×C₄×D₈₉	D₄	1	1	yes	characteristic
45	C₂×C₄×D₈₉	C₂³	1	1	yes	characteristic
46	C₂×#10623494	C₂³	1	2	yes	—
47	C₄⋊Dic₈₉	—	2	1	no	—
48	C₄⋊Dic₈₉	—	2	1	no	—
49	C₄⋊Dic₈₉	C₂³	1	2	yes	—
50	C₄⋊Dic₈₉	C₂³	1	2	yes	—
51	#13083553	—	2	2	no	—
52	#13083553	—	2	2	no	—
53	C₂²⋊Dic₈₉	—	2	2	no	—
54	C₂²⋊Dic₈₉	—	2	2	no	—
55	C₄×Dic₈₉	D₄	1	1	yes	characteristic
56	C₄×Dic₈₉	D₄	1	1	yes	characteristic
57	C₄×Dic₈₉	C₂³	1	1	yes	characteristic
58	C₂×Q₈×C₈₉	—	2	1	no	—
59	C₂×D₄×C₈₉	—	2	1	no	—
60	C₂×D₄×C₈₉	—	2	1	no	—
61	C₂×D₄×C₈₉	C₂³	1	2	yes	—
62	C₂×C₈×C₈₉	C₂³	1	2	yes	—
63	C₄⋊C₄×C₈₉	—	2	1	no	—
64	C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
65	C₄²×C₈₉	C₂³	1	1	yes	characteristic
66	C₂³×C₈₉	—	2	2	no	—
67	C₂²×D₈₉	C₂×D₄	1	1	yes	characteristic
68	Q₈×C₈₉	—	4	2	no	—
69	D₄×C₈₉	—	2	4	no	—
70	D₄×C₈₉	—	4	2	no	—
71	D₄×C₈₉	—	4	2	no	—
72	C₂×C₄×C₈₉	—	2	2	no	—
73	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
74	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
75	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
76	C₂²⋊D₈₉	—	4	4	no	—
77	C₂×Dic₈₉	—	2	2	no	—
78	C₂×Dic₈₉	—	2	2	no	—
79	C₂×Dic₈₉	—	4	4	no	—
80	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
81	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
82	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
83	C₄×D₈₉	—	2	2	no	—
84	C₄×D₈₉	—	2	2	no	—
85	C₄×D₈₉	—	4	4	no	—
86	C₄×D₈₉	D₈	1	4	yes	—
87	Dic₁₇₈	—	4	2	no	—
88	Dic₁₇₈	—	4	2	no	—
89	Dic₁₇₈	—	4	4	no	—
90	C₈×C₈₉	—	2	2	no	—
91	#10623494	—	2	2	no	—
92	C₂²×C₈₉	—	4	2	no	—
93	C₂²×C₈₉	—	4	2	no	—
94	C₂²×C₈₉	—	4	2	no	—
95	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
96	C₂×D₈₉	—	2	1	no	—
97	C₂×D₈₉	—	2	1	no	—
98	C₂×D₈₉	C₂×D₈	1	2	yes	—
99	C₄×C₈₉	—	4	2	no	—
100	C₄×C₈₉	C₂×D₈	1	2	yes	—
101	C₄×C₈₉	#137	1	2	yes	—
102	C₄×C₈₉	2₋¹⁺⁴	1	2	yes	—
103	Dic₈₉	—	2	1	no	—
104	Dic₈₉	—	2	1	no	—
105	Dic₈₉	—	4	2	no	—
106	Dic₈₉	—	4	2	no	—
107	Dic₈₉	C₂×D₈	1	2	yes	—
108	C₂×C₈₉	—	8	2	no	—
109	C₂×C₈₉	#536	1	1	yes	characteristic
110	C₂×C₈₉	C₂²×D₈	1	1	yes	characteristic
111	C₂×C₈₉	C₂×#137	1	1	yes	characteristic
112	D₈₉	—	2	2	no	—
113	#3226	—	89	1	no	maximal, Sylow
114	C₈₉	#3226	1	1	yes	characteristic, complemented, Sylow
115	#532	—	89	1	no	—
116	C₂×C₄⋊Q₈	—	89	1	no	—
117	#500	—	89	1	no	—
118	#500	—	89	1	no	—
119	#486	—	89	1	no	—
120	#486	D₈₉	1	1	yes	characteristic, complemented
121	#480	—	89	4	no	—
122	#474	—	89	4	no	—
123	#431	—	89	1	no	—
124	C₂×#41	—	178	1	no	—
125	C₂²×Q₈	—	178	1	no	—
126	C₄⋊Q₈	—	89	2	no	—
127	C₄⋊Q₈	—	89	1	no	—
128	C₄⋊Q₈	—	89	1	no	—
129	C₄⋊Q₈	—	178	1	no	—
130	C₄⋊Q₈	—	178	1	no	—
131	C₄⋊Q₈	C₂×D₈₉	1	1	yes	characteristic
132	#127	C₂×D₈₉	1	1	yes	characteristic
133	#124	—	178	1	no	—
134	#121	—	89	2	no	—
135	#121	—	178	1	no	—
136	C₄×D₄	—	89	2	no	—
137	C₄×D₄	—	178	1	no	—
138	C₂×C₄⋊C₄	—	89	2	no	—
139	C₂×C₄⋊C₄	—	178	1	no	—
140	C₂×C₄²	—	89	1	no	—
141	#107	—	89	4	no	—
142	#107	—	89	4	no	—
143	C₄⋊C₈	—	89	2	no	—
144	#102	—	89	4	no	—
145	#102	C₂×D₈₉	1	4	yes	—
146	C₂²⋊C₈	—	89	2	no	—
147	C₄×C₈	—	89	1	no	—
148	C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
149	#41	—	356	4	no	—
150	C₂×Q₈	—	2	1	no	—
151	C₂×Q₈	—	356	4	no	—
152	C₂×Q₈	—	178	2	no	—
153	C₂×Q₈	—	178	1	no	—
154	C₂×Q₈	—	178	1	no	—
155	C₂×D₄	—	2	1	no	—
156	C₂×D₄	—	2	1	no	—
157	C₂×D₄	—	178	2	no	—
158	C₂×D₄	C₂²×D₈₉	1	2	yes	—
159	C₂²×C₄	—	89	1	no	—
160	C₂²×C₄	—	89	1	no	—
161	C₂²×C₄	—	89	1	no	—
162	C₂²×C₄	—	178	2	no	—
163	C₂²×C₄	—	178	2	no	—
164	C₂×C₈	—	89	2	no	—
165	C₂×C₈	C₂²×D₈₉	1	2	yes	—
166	C₄⋊C₄	—	2	1	no	—
167	C₄⋊C₄	—	89	2	no	—
168	C₄⋊C₄	—	89	2	no	—
169	C₄⋊C₄	—	178	2	no	—
170	C₄⋊C₄	—	178	2	no	—
171	C₄⋊C₄	—	178	1	no	—
172	C₄⋊C₄	—	178	1	no	—
173	C₄⋊C₄	C₂²×D₈₉	1	2	yes	—
174	C₂²⋊C₄	—	178	2	no	—
175	C₂²⋊C₄	—	178	2	no	—
176	C₄²	—	89	1	no	—
177	C₄²	—	89	1	no	—
178	C₄²	—	89	1	no	—
179	C₄²	C₂²×D₈₉	1	1	yes	characteristic
180	C₂³	—	2	2	no	—
181	C₂³	—	89	1	no	—
182	Q₈	—	4	2	no	—
183	Q₈	—	356	4	no	—
184	Q₈	—	356	2	no	—
185	Q₈	—	356	2	no	—
186	D₄	—	2	4	no	—
187	D₄	—	4	2	no	—
188	D₄	—	4	2	no	—
189	D₄	—	356	4	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	89	1	no	—
192	C₂×C₄	—	89	4	no	—
193	C₂×C₄	—	89	1	no	—
194	C₂×C₄	—	89	1	no	—
195	C₂×C₄	—	356	4	no	—
196	C₂×C₄	—	356	4	no	—
197	C₂×C₄	—	178	2	no	—
198	C₂×C₄	—	178	2	no	—
199	C₂×C₄	—	178	2	no	—
200	C₂×C₄	—	178	2	no	—
201	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
202	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
203	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
204	C₈	—	2	2	no	—
205	C₈	—	178	2	no	—
206	C₂²	—	4	2	no	—
207	C₂²	—	4	2	no	—
208	C₂²	—	4	2	no	—
209	C₂²	—	89	2	no	—
210	C₂²	—	178	1	no	—
211	C₂²	—	178	1	no	—
212	C₂²	C₂×D₄×D₈₉	1	1	yes	characteristic
213	C₄	—	4	2	no	—
214	C₄	—	89	2	no	—
215	C₄	—	356	2	no	—
216	C₄	—	356	2	no	—
217	C₄	—	178	1	no	—
218	C₄	—	178	1	no	—
219	C₄	D₈×D₈₉	1	2	yes	—
220	C₄	#15668041	1	2	yes	—
221	C₄	#15668059	1	2	yes	—
222	C₂	—	8	2	no	—
223	C₂	—	178	2	no	—
224	C₂	#19152168	1	1	yes	characteristic
225	C₂	C₂×D₈×D₈₉	1	1	yes	characteristic
226	C₂	C₂×#15668041	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19152146

C₂

yes

maximal, characteristic

C₄⋊Q₈×D₈₉

C₂

yes

maximal, characteristic, complemented

#19152003

C₂

yes

maximal, characteristic, complemented

#19152001

C₂

yes

maximal, characteristic, complemented

#19151944

C₂

yes

maximal, characteristic, complemented

#19151910

C₂

yes

maximal, complemented

#19151876

C₂

yes

maximal, complemented

#19151649

C₂

yes

maximal, characteristic, complemented

#486×C₈₉

C₂

yes

maximal, characteristic, complemented

C₂×#13083573

—

C₂×Q₈×D₈₉

—

#15668012

C₂²

yes

characteristic

#15668012

C₂²

yes

characteristic

#15668011

—

#15668011

C₂²

yes

—

—

—

—

C₂²

yes

—

D₄×Dic₈₉

—

D₄×Dic₈₉

C₂²

yes

—

C₄⋊C₄×D₈₉

—

C₄⋊C₄×D₈₉

C₂²

yes

—

C₄²×D₈₉

C₂²

yes

characteristic

#15667946

C₂²

yes

—

#15667945

C₂²

yes

—

#15667941

C₂²

yes

—

#15667933

C₂²

yes

—

#15667924

C₂²

yes

—

C₄×#10623494

C₂²

yes

characteristic

C₄⋊Q₈×C₈₉

C₂²

yes

characteristic

#127×C₈₉

C₂²

yes

characteristic

#102×C₈₉

C₂²

yes

—

C₄×C₈×C₈₉

C₂²

yes

characteristic

#13083573

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

Q₈×D₈₉

—

C₂×C₂²⋊D₈₉

—

C₂²×Dic₈₉

—

C₂×C₄×D₈₉

—

C₂×C₄×D₈₉

D₄

yes

characteristic

C₂×C₄×D₈₉

D₄

yes

characteristic

C₂×C₄×D₈₉

C₂³

yes

characteristic

C₂×#10623494

C₂³

yes

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

C₂³

yes

—

C₄⋊Dic₈₉

C₂³

yes

—

#13083553

—

#13083553

—

C₂²⋊Dic₈₉

—

C₂²⋊Dic₈₉

—

C₄×Dic₈₉

D₄

yes

characteristic

C₄×Dic₈₉

D₄

yes

characteristic

C₄×Dic₈₉

C₂³

yes

characteristic

C₂×Q₈×C₈₉

—

C₂×D₄×C₈₉

—

C₂×D₄×C₈₉

—

C₂×D₄×C₈₉

C₂³

yes

—

C₂×C₈×C₈₉

C₂³

yes

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

C₂³

yes

—

C₄²×C₈₉

C₂³

yes

characteristic

C₂³×C₈₉

—

C₂²×D₈₉

C₂×D₄

yes

characteristic

Q₈×C₈₉

—

D₄×C₈₉

—

D₄×C₈₉

—

D₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

C₂⁴

yes

characteristic

C₂²⋊D₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

C₂×D₄

yes

characteristic

C₂×Dic₈₉

C₂×D₄

yes

characteristic

C₂×Dic₈₉

C₂×D₄

yes

characteristic

C₄×D₈₉

—

C₄×D₈₉

—

C₄×D₈₉

—

C₄×D₈₉

D₈

yes

—

Dic₁₇₈

—

Dic₁₇₈

—

Dic₁₇₈

—

C₈×C₈₉

—

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

C₂²×D₄

yes

characteristic

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

C₂×D₈

yes

—

C₄×C₈₉

—

100

C₄×C₈₉

C₂×D₈

yes

—

101

C₄×C₈₉

#137

yes

—

102

C₄×C₈₉

2₋¹⁺⁴

yes

—

103

Dic₈₉

—

104

Dic₈₉

—

105

Dic₈₉

—

106

Dic₈₉

—

107

Dic₈₉

C₂×D₈

yes

—

108

C₂×C₈₉

—

109

C₂×C₈₉

#536

yes

characteristic

110

C₂×C₈₉

C₂²×D₈

yes

characteristic

111

C₂×C₈₉

C₂×#137

yes

characteristic

112

D₈₉

—

113

#3226

—

maximal, Sylow

114

C₈₉

#3226

yes

characteristic, complemented, Sylow

115

#532

—

116

C₂×C₄⋊Q₈

—

117

#500

—

118

#500

—

119

#486

—

120

#486

D₈₉

yes

characteristic, complemented

121

#480

—

122

#474

—

123

#431

—

124

C₂×#41

—

178

—

125

C₂²×Q₈

—

178

—

126

C₄⋊Q₈

—

127

C₄⋊Q₈

—

128

C₄⋊Q₈

—

129

C₄⋊Q₈

—

178

—

130

C₄⋊Q₈

—

178

—

131

C₄⋊Q₈

C₂×D₈₉

yes

characteristic

132

#127

C₂×D₈₉

yes

characteristic

133

#124

—

178

—

134

#121

—

135

#121

—

178

—

136

C₄×D₄

—

137

C₄×D₄

—

178

—

138

C₂×C₄⋊C₄

—

139

C₂×C₄⋊C₄

—

178

—

140

C₂×C₄²

—

141

#107

—

142

#107

—

143

C₄⋊C₈

—

144

#102

—

145

#102

C₂×D₈₉

yes

—

146

C₂²⋊C₈

—

147

C₄×C₈

—

148

C₄×C₈

C₂×D₈₉

yes

characteristic

149

#41

—

356

—

150

C₂×Q₈

—

151

C₂×Q₈

—

356

—

152

C₂×Q₈

—

178

—

153

C₂×Q₈

—

178

—

154

C₂×Q₈

—

178

—

155

C₂×D₄

—

156

C₂×D₄

—

157

C₂×D₄

—

178

—

158

C₂×D₄

C₂²×D₈₉

yes

—

159

C₂²×C₄

—

160

C₂²×C₄

—

161

C₂²×C₄

—

162

C₂²×C₄

—

178

—

163

C₂²×C₄

—

178

—

164

C₂×C₈

—

165

C₂×C₈

C₂²×D₈₉

yes

—

166

C₄⋊C₄

—

167

C₄⋊C₄

—

168

C₄⋊C₄

—

169

C₄⋊C₄

—

178

—

170

C₄⋊C₄

—

178

—

171

C₄⋊C₄

—

178

—

172

C₄⋊C₄

—

178

—

173

C₄⋊C₄

C₂²×D₈₉

yes

—

174

C₂²⋊C₄

—

178

—

175

C₂²⋊C₄

—

178

—

176

C₄²

—

177

C₄²

—

178

C₄²

—

179

C₄²

C₂²×D₈₉

yes

characteristic

180

C₂³

—

181

C₂³

—

182

Q₈

—

183

Q₈

—

356

—

184

Q₈

—

356

—

185

Q₈

—

356

—

186

D₄

—

187

D₄

—

188

D₄

—

189

D₄

—

356

—

190

C₂×C₄

—

191

C₂×C₄

—

192

C₂×C₄

—

193

C₂×C₄

—

194

C₂×C₄

—

195

C₂×C₄

—

356

—

196

C₂×C₄

—

356

—

197

C₂×C₄

—

178

—

198

C₂×C₄

—

178

—

199

C₂×C₄

—

178

—

200

C₂×C₄

—

178

—

201

C₂×C₄

D₄×D₈₉

yes

characteristic

202

C₂×C₄

D₄×D₈₉

yes

characteristic

203

C₂×C₄

C₂³×D₈₉

yes

characteristic

204

C₈

—

205

C₈

—

178

—

206

C₂²

—

207

C₂²

—

208

C₂²

—

209

C₂²

—

210

C₂²

—

178

—

211

C₂²

—

178

—

212

C₂²

C₂×D₄×D₈₉

yes

characteristic

213

C₄

—

214

C₄

—

215

C₄

—

356

—

216

C₄

—

356

—

217

C₄

—

178

—

218

C₄

—

178

—

219

C₄

D₈×D₈₉

yes

—

220

C₄

#15668041

yes

—

221

C₄

#15668059

yes

—

222

C₂

—

223

C₂

—

178

—

224

C₂

#19152168

yes

characteristic

225

C₂

C₂×D₈×D₈₉

yes

characteristic

226

C₂

C₂×#15668041

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₈₉
commutator, cosocle	C₂×C₄×C₈₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₉
Fitting	#486×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₄
prime radical	#19152146	C₂
hypercenter	#486	D₈₉
hypocenter	C₈₉	#3226
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #486

subgroup

G/H

center

C₂²

C₂×D₄×D₈₉

commutator, cosocle

C₂×C₄×C₈₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₈₉

Fitting

#486×C₈₉

C₂

socle

C₂²×C₈₉

C₂²×D₄

C₂

D₈₉

C₈₉

G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #486

N	H ≅ G/N	type
2. C₄⋊Q₈×D₈₉	C₂	semidirect
3. #19152003	C₂	semidirect
4. #19152001	C₂	semidirect
5. #19151944	C₂	semidirect
6. #19151910	C₂	semidirect
7. #19151876	C₂	semidirect
8. #19151649	C₂	semidirect
9. #486×C₈₉	C₂	semidirect
114. C₈₉	#3226	semidirect
120. #486	D₈₉	semidirect

H ≅ G/N

type

2. C₄⋊Q₈×D₈₉

C₂

semidirect

3. #19152003

C₂

semidirect

4. #19152001

C₂

semidirect

5. #19151944

C₂

semidirect

6. #19151910

C₂

semidirect

7. #19151876

C₂

semidirect

8. #19151649

C₂

semidirect

9. #486×C₈₉

C₂

semidirect

114. C₈₉

#3226

semidirect

120. #486

D₈₉

semidirect

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

subgroup	index
#19152146	2
C₄⋊Q₈×D₈₉	2
#19152003	2
#19152001	2
#19151944	2
#19151910	2
#19151876	2
#19151649	2
#486×C₈₉	2
#3226	89

subgroup

index

C₄⋊Q₈×D₈₉

#486×C₈₉

Subgroups — #26031369

Conjugacy classes of subgroups

Complements

Maximal subgroups