Group #26030647 — subgroups

Subgroups — #26030647

Order 11392.

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

normal subgroups only

This group has 424 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	#19152149	C₂	1	1	yes	maximal, characteristic
2	C₄⋊Q₈×D₈₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#19151959	C₂	1	4	yes	maximal, complemented
4	#19151948	C₂	1	1	yes	maximal, characteristic, complemented
5	#19151946	C₂	1	1	yes	maximal, characteristic, complemented
6	#19151945	C₂	1	1	yes	maximal, characteristic, complemented
7	#19151598	C₂	1	4	yes	maximal, complemented
8	C₄×#13083557	C₂	1	1	yes	maximal, characteristic, complemented
9	#487×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#13083575	C₂²	1	2	yes	—
11	C₂×Q₈×D₈₉	—	2	1	no	—
12	C₂×#13083557	C₂²	1	2	yes	—
13	#15668012	C₂²	1	1	yes	characteristic
14	#15668011	—	2	1	no	—
15	#15668011	C₂²	1	2	yes	—
16	#15668010	—	2	1	no	—
17	#15668009	C₂²	1	1	yes	characteristic
18	#15668001	C₂²	1	2	yes	—
19	#15667987	—	2	1	no	—
20	#15667987	—	2	1	no	—
21	#15667974	—	2	1	no	—
22	#15667974	—	2	1	no	—
23	C₄⋊C₄×D₈₉	—	2	1	no	—
24	C₄⋊C₄×D₈₉	C₂²	1	2	yes	—
25	C₄²×D₈₉	C₂²	1	1	yes	characteristic
26	#15667936	C₂²	1	4	yes	—
27	#15667934	C₂²	1	4	yes	—
28	#15667932	C₂²	1	4	yes	—
29	#15667922	C₂²	1	2	yes	—
30	C₄×#10623494	C₂²	1	1	yes	characteristic
31	C₄⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
32	C₄⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
33	#103×C₈₉	C₂²	1	4	yes	—
34	C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
35	#13083575	—	2	8	no	—
36	C₂×Dic₁₇₈	—	2	1	no	—
37	C₂×Dic₁₇₈	—	2	2	no	—
38	Q₈×D₈₉	—	4	4	no	—
39	C₂×D₃₅₆	—	2	2	no	—
40	C₂×D₃₅₆	—	2	1	no	—
41	C₂×D₃₅₆	—	2	1	no	—
42	C₂×D₃₅₆	C₂³	1	2	yes	—
43	C₂×C₄×D₈₉	—	2	2	no	—
44	C₂×C₄×D₈₉	—	2	2	no	—
45	C₂×C₄×D₈₉	D₄	1	1	yes	characteristic
46	C₂×C₄×D₈₉	D₄	1	1	yes	characteristic
47	C₂×C₄×D₈₉	C₂³	1	1	yes	characteristic
48	#13083557	—	2	4	no	—
49	C₂×#10623494	C₂³	1	2	yes	—
50	C₄⋊Dic₈₉	—	2	1	no	—
51	C₄⋊Dic₈₉	C₂³	1	2	yes	—
52	#13083553	—	2	2	no	—
53	#13083553	—	2	2	no	—
54	#13083551	—	2	2	no	—
55	#13083551	—	2	2	no	—
56	C₄×Dic₈₉	D₄	1	1	yes	characteristic
57	C₄×Dic₈₉	D₄	1	1	yes	characteristic
58	C₄×Dic₈₉	C₂³	1	1	yes	characteristic
59	C₂×Q₈×C₈₉	—	2	1	no	—
60	C₂×Q₈×C₈₉	C₂³	1	2	yes	—
61	C₂×C₈×C₈₉	C₂³	1	2	yes	—
62	C₄⋊C₄×C₈₉	—	2	1	no	—
63	C₄⋊C₄×C₈₉	—	2	1	no	—
64	C₄⋊C₄×C₈₉	—	2	1	no	—
65	C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
66	C₄²×C₈₉	C₂³	1	1	yes	characteristic
67	C₂²×D₈₉	—	2	2	no	—
68	C₂²×D₈₉	C₂×D₄	1	1	yes	characteristic
69	Q₈×C₈₉	—	2	4	no	—
70	Q₈×C₈₉	—	4	2	no	—
71	C₂×C₄×C₈₉	—	2	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	C₂×Dic₈₉	—	2	2	no	—
77	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
78	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
79	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
80	D₃₅₆	—	2	4	no	—
81	D₃₅₆	—	4	4	no	—
82	D₃₅₆	—	4	2	no	—
83	D₃₅₆	—	4	2	no	—
84	C₄×D₈₉	—	2	2	no	—
85	C₄×D₈₉	—	2	2	no	—
86	C₄×D₈₉	—	4	4	no	—
87	C₄×D₈₉	—	4	4	no	—
88	C₄×D₈₉	#41	1	4	yes	—
89	Dic₁₇₈	—	4	4	no	—
90	Dic₁₇₈	—	4	2	no	—
91	C₈×C₈₉	—	2	2	no	—
92	#10623494	—	2	2	no	—
93	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
94	C₂×D₈₉	—	2	1	no	—
95	C₂×D₈₉	—	2	1	no	—
96	C₂×D₈₉	—	4	2	no	—
97	C₂×D₈₉	—	4	2	no	—
98	C₂×D₈₉	—	4	2	no	—
99	C₂×D₈₉	#124	1	2	yes	—
100	C₄×C₈₉	—	4	2	no	—
101	C₄×C₈₉	—	4	2	no	—
102	C₄×C₈₉	AGL(1,ℤ/8)	1	2	yes	—
103	C₄×C₈₉	#137	1	2	yes	—
104	C₄×C₈₉	C₂×#41	1	2	yes	—
105	Dic₈₉	—	2	1	no	—
106	Dic₈₉	—	2	1	no	—
107	Dic₈₉	—	4	2	no	—
108	Dic₈₉	#124	1	2	yes	—
109	C₂×C₈₉	C₂×#124	1	1	yes	characteristic
110	C₂×C₈₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
111	C₂×C₈₉	C₂×#137	1	1	yes	characteristic
112	D₈₉	—	2	2	no	—
113	D₈₉	—	8	2	no	—
114	#3141	—	89	1	no	maximal, Sylow
115	C₈₉	#3141	1	1	yes	characteristic, complemented, Sylow
116	#532	—	89	1	no	—
117	C₂×C₄⋊Q₈	—	89	1	no	—
118	#489	—	89	4	no	—
119	#487	—	89	1	no	—
120	#487	D₈₉	1	1	yes	characteristic, complemented
121	#486	—	89	1	no	—
122	#486	—	89	1	no	—
123	#417	—	89	4	no	—
124	C₄×M₁₆	—	89	1	no	—
125	C₂×#41	—	89	2	no	—
126	C₂²×Q₈	—	178	1	no	—
127	C₂×M₁₆	—	89	2	no	—
128	C₄⋊Q₈	—	89	2	no	—
129	C₄⋊Q₈	—	89	1	no	—
130	C₄⋊Q₈	—	178	1	no	—
131	C₄⋊Q₈	—	178	1	no	—
132	C₄⋊Q₈	C₂×D₈₉	1	1	yes	characteristic
133	C₄⋊Q₈	C₂×D₈₉	1	1	yes	characteristic
134	#127	—	89	1	no	—
135	#124	—	89	2	no	—
136	#121	—	178	1	no	—
137	#121	—	178	1	no	—
138	C₄×D₄	—	178	1	no	—
139	C₄×D₄	—	178	1	no	—
140	C₂×C₄⋊C₄	—	89	2	no	—
141	C₂×C₄⋊C₄	—	178	1	no	—
142	C₂×C₄²	—	89	1	no	—
143	#103	—	89	4	no	—
144	#103	C₂×D₈₉	1	4	yes	—
145	#102	—	89	4	no	—
146	#102	—	89	4	no	—
147	#97	—	89	2	no	—
148	C₄×C₈	—	89	1	no	—
149	C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
150	#41	—	178	8	no	—
151	C₂×Q₈	—	2	1	no	—
152	C₂×Q₈	—	356	4	no	—
153	C₂×Q₈	—	178	2	no	—
154	C₂×Q₈	—	178	1	no	—
155	C₂×Q₈	C₂²×D₈₉	1	2	yes	—
156	C₂×D₄	—	89	2	no	—
157	C₂×D₄	—	178	2	no	—
158	C₂×D₄	—	178	1	no	—
159	C₂×D₄	—	178	1	no	—
160	C₂²×C₄	—	89	1	no	—
161	C₂²×C₄	—	89	1	no	—
162	C₂²×C₄	—	89	1	no	—
163	C₂²×C₄	—	178	2	no	—
164	C₂²×C₄	—	178	2	no	—
165	M₁₆	—	178	4	no	—
166	C₂×C₈	—	89	2	no	—
167	C₂×C₈	C₂²×D₈₉	1	2	yes	—
168	C₄⋊C₄	—	2	1	no	—
169	C₄⋊C₄	—	2	1	no	—
170	C₄⋊C₄	—	2	1	no	—
171	C₄⋊C₄	—	89	2	no	—
172	C₄⋊C₄	—	178	2	no	—
173	C₄⋊C₄	—	178	2	no	—
174	C₄⋊C₄	—	178	1	no	—
175	C₄⋊C₄	C₂²×D₈₉	1	2	yes	—
176	C₂²⋊C₄	—	178	2	no	—
177	C₂²⋊C₄	—	178	2	no	—
178	C₄²	—	89	1	no	—
179	C₄²	—	89	1	no	—
180	C₄²	—	89	1	no	—
181	C₄²	C₂²×D₈₉	1	1	yes	characteristic
182	C₂³	—	89	1	no	—
183	C₂³	—	178	2	no	—
184	Q₈	—	2	4	no	—
185	Q₈	—	4	2	no	—
186	Q₈	—	356	4	no	—
187	Q₈	—	356	2	no	—
188	D₄	—	356	4	no	—
189	D₄	—	356	2	no	—
190	D₄	—	356	2	no	—
191	D₄	—	178	4	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	89	1	no	—
195	C₂×C₄	—	89	4	no	—
196	C₂×C₄	—	89	1	no	—
197	C₂×C₄	—	89	1	no	—
198	C₂×C₄	—	356	4	no	—
199	C₂×C₄	—	356	4	no	—
200	C₂×C₄	—	178	2	no	—
201	C₂×C₄	—	178	2	no	—
202	C₂×C₄	—	178	2	no	—
203	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
204	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
205	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
206	C₈	—	2	2	no	—
207	C₈	—	178	2	no	—
208	C₂²	—	89	2	no	—
209	C₂²	—	356	2	no	—
210	C₂²	—	356	2	no	—
211	C₂²	—	356	2	no	—
212	C₂²	—	178	1	no	—
213	C₂²	—	178	1	no	—
214	C₂²	C₂×D₄×D₈₉	1	1	yes	characteristic
215	C₄	—	4	2	no	—
216	C₄	—	4	2	no	—
217	C₄	—	89	2	no	—
218	C₄	—	356	2	no	—
219	C₄	—	178	1	no	—
220	C₄	—	178	1	no	—
221	C₄	#15668039	1	2	yes	—
222	C₄	#15668044	1	2	yes	—
223	C₄	#41×D₈₉	1	2	yes	—
224	C₂	—	178	2	no	—
225	C₂	—	712	2	no	—
226	C₂	#124×D₈₉	1	1	yes	characteristic
227	C₂	C₂×#15668039	1	1	yes	characteristic
228	C₂	C₂×#15668044	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19152149

C₂

yes

maximal, characteristic

C₄⋊Q₈×D₈₉

C₂

yes

maximal, characteristic, complemented

#19151959

C₂

yes

maximal, complemented

#19151948

C₂

yes

maximal, characteristic, complemented

#19151946

C₂

yes

maximal, characteristic, complemented

#19151945

C₂

yes

maximal, characteristic, complemented

#19151598

C₂

yes

maximal, complemented

C₄×#13083557

C₂

yes

maximal, characteristic, complemented

#487×C₈₉

C₂

yes

maximal, characteristic, complemented

C₂×#13083575

C₂²

yes

—

C₂×Q₈×D₈₉

—

C₂×#13083557

C₂²

yes

—

#15668012

C₂²

yes

characteristic

#15668011

—

#15668011

C₂²

yes

—

#15668010

—

#15668009

C₂²

yes

characteristic

#15668001

C₂²

yes

—

—

—

—

—

C₄⋊C₄×D₈₉

—

C₄⋊C₄×D₈₉

C₂²

yes

—

C₄²×D₈₉

C₂²

yes

characteristic

#15667936

C₂²

yes

—

#15667934

C₂²

yes

—

#15667932

C₂²

yes

—

#15667922

C₂²

yes

—

C₄×#10623494

C₂²

yes

characteristic

C₄⋊Q₈×C₈₉

C₂²

yes

characteristic

C₄⋊Q₈×C₈₉

C₂²

yes

characteristic

#103×C₈₉

C₂²

yes

—

C₄×C₈×C₈₉

C₂²

yes

characteristic

#13083575

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

Q₈×D₈₉

—

C₂×D₃₅₆

—

C₂×D₃₅₆

—

C₂×D₃₅₆

—

C₂×D₃₅₆

C₂³

yes

—

C₂×C₄×D₈₉

—

C₂×C₄×D₈₉

—

C₂×C₄×D₈₉

D₄

yes

characteristic

C₂×C₄×D₈₉

D₄

yes

characteristic

C₂×C₄×D₈₉

C₂³

yes

characteristic

#13083557

—

C₂×#10623494

C₂³

yes

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

C₂³

yes

—

—

—

—

—

C₄×Dic₈₉

D₄

yes

characteristic

C₄×Dic₈₉

D₄

yes

characteristic

C₄×Dic₈₉

C₂³

yes

characteristic

C₂×Q₈×C₈₉

—

C₂×Q₈×C₈₉

C₂³

yes

—

C₂×C₈×C₈₉

C₂³

yes

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

C₂³

yes

—

C₄²×C₈₉

C₂³

yes

characteristic

C₂²×D₈₉

—

C₂²×D₈₉

C₂×D₄

yes

characteristic

Q₈×C₈₉

—

Q₈×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

C₂⁴

yes

characteristic

C₂×Dic₈₉

—

C₂×Dic₈₉

C₂×D₄

yes

characteristic

C₂×Dic₈₉

C₂×D₄

yes

characteristic

C₂×Dic₈₉

C₂×D₄

yes

characteristic

D₃₅₆

—

D₃₅₆

—

D₃₅₆

—

D₃₅₆

—

C₄×D₈₉

—

C₄×D₈₉

—

C₄×D₈₉

—

C₄×D₈₉

—

C₄×D₈₉

#41

yes

—

Dic₁₇₈

—

Dic₁₇₈

—

C₈×C₈₉

—

#10623494

—

C₂²×C₈₉

C₂²×D₄

yes

characteristic

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

#124

yes

—

100

C₄×C₈₉

—

101

C₄×C₈₉

—

102

C₄×C₈₉

AGL(1,ℤ/8)

yes

—

103

C₄×C₈₉

#137

yes

—

104

C₄×C₈₉

C₂×#41

yes

—

105

Dic₈₉

—

106

Dic₈₉

—

107

Dic₈₉

—

108

Dic₈₉

#124

yes

—

109

C₂×C₈₉

C₂×#124

yes

characteristic

110

C₂×C₈₉

C₂×AGL(1,ℤ/8)

yes

characteristic

111

C₂×C₈₉

C₂×#137

yes

characteristic

112

D₈₉

—

113

D₈₉

—

114

#3141

—

maximal, Sylow

115

C₈₉

#3141

yes

characteristic, complemented, Sylow

116

#532

—

117

C₂×C₄⋊Q₈

—

118

#489

—

119

#487

—

120

#487

D₈₉

yes

characteristic, complemented

121

#486

—

122

#486

—

123

#417

—

124

C₄×M₁₆

—

125

C₂×#41

—

126

C₂²×Q₈

—

178

—

127

C₂×M₁₆

—

128

C₄⋊Q₈

—

129

C₄⋊Q₈

—

130

C₄⋊Q₈

—

178

—

131

C₄⋊Q₈

—

178

—

132

C₄⋊Q₈

C₂×D₈₉

yes

characteristic

133

C₄⋊Q₈

C₂×D₈₉

yes

characteristic

134

#127

—

135

#124

—

136

#121

—

178

—

137

#121

—

178

—

138

C₄×D₄

—

178

—

139

C₄×D₄

—

178

—

140

C₂×C₄⋊C₄

—

141

C₂×C₄⋊C₄

—

178

—

142

C₂×C₄²

—

143

#103

—

144

#103

C₂×D₈₉

yes

—

145

#102

—

146

#102

—

147

#97

—

148

C₄×C₈

—

149

C₄×C₈

C₂×D₈₉

yes

characteristic

150

#41

—

178

—

151

C₂×Q₈

—

152

C₂×Q₈

—

356

—

153

C₂×Q₈

—

178

—

154

C₂×Q₈

—

178

—

155

C₂×Q₈

C₂²×D₈₉

yes

—

156

C₂×D₄

—

157

C₂×D₄

—

178

—

158

C₂×D₄

—

178

—

159

C₂×D₄

—

178

—

160

C₂²×C₄

—

161

C₂²×C₄

—

162

C₂²×C₄

—

163

C₂²×C₄

—

178

—

164

C₂²×C₄

—

178

—

165

M₁₆

—

178

—

166

C₂×C₈

—

167

C₂×C₈

C₂²×D₈₉

yes

—

168

C₄⋊C₄

—

169

C₄⋊C₄

—

170

C₄⋊C₄

—

171

C₄⋊C₄

—

172

C₄⋊C₄

—

178

—

173

C₄⋊C₄

—

178

—

174

C₄⋊C₄

—

178

—

175

C₄⋊C₄

C₂²×D₈₉

yes

—

176

C₂²⋊C₄

—

178

—

177

C₂²⋊C₄

—

178

—

178

C₄²

—

179

C₄²

—

180

C₄²

—

181

C₄²

C₂²×D₈₉

yes

characteristic

182

C₂³

—

183

C₂³

—

178

—

184

Q₈

—

185

Q₈

—

186

Q₈

—

356

—

187

Q₈

—

356

—

188

D₄

—

356

—

189

D₄

—

356

—

190

D₄

—

356

—

191

D₄

—

178

—

192

C₂×C₄

—

193

C₂×C₄

—

194

C₂×C₄

—

195

C₂×C₄

—

196

C₂×C₄

—

197

C₂×C₄

—

198

C₂×C₄

—

356

—

199

C₂×C₄

—

356

—

200

C₂×C₄

—

178

—

201

C₂×C₄

—

178

—

202

C₂×C₄

—

178

—

203

C₂×C₄

D₄×D₈₉

yes

characteristic

204

C₂×C₄

D₄×D₈₉

yes

characteristic

205

C₂×C₄

C₂³×D₈₉

yes

characteristic

206

C₈

—

207

C₈

—

178

—

208

C₂²

—

209

C₂²

—

356

—

210

C₂²

—

356

—

211

C₂²

—

356

—

212

C₂²

—

178

—

213

C₂²

—

178

—

214

C₂²

C₂×D₄×D₈₉

yes

characteristic

215

C₄

—

216

C₄

—

217

C₄

—

218

C₄

—

356

—

219

C₄

—

178

—

220

C₄

—

178

—

221

C₄

#15668039

yes

—

222

C₄

#15668044

yes

—

223

C₄

#41×D₈₉

yes

—

224

C₂

—

178

—

225

C₂

—

712

—

226

C₂

#124×D₈₉

yes

characteristic

227

C₂

C₂×#15668039

yes

characteristic

228

C₂

C₂×#15668044

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₈₉
commutator, cosocle	C₂×C₄×C₈₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₉
Fitting	#487×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₄
prime radical	#19152149	C₂
hypercenter	#487	D₈₉
hypocenter	C₈₉	#3141
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #487

subgroup

G/H

center

C₂²

C₂×D₄×D₈₉

commutator, cosocle

C₂×C₄×C₈₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₈₉

Fitting

#487×C₈₉

C₂

socle

C₂²×C₈₉

C₂²×D₄

C₂

D₈₉

C₈₉

G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #487

N	H ≅ G/N	type
2. C₄⋊Q₈×D₈₉	C₂	semidirect
3. #19151959	C₂	semidirect
4. #19151948	C₂	semidirect
5. #19151946	C₂	semidirect
6. #19151945	C₂	semidirect
7. #19151598	C₂	semidirect
8. C₄×#13083557	C₂	semidirect
9. #487×C₈₉	C₂	semidirect
115. C₈₉	#3141	semidirect
120. #487	D₈₉	semidirect

H ≅ G/N

type

2. C₄⋊Q₈×D₈₉

C₂

semidirect

3. #19151959

C₂

semidirect

4. #19151948

C₂

semidirect

5. #19151946

C₂

semidirect

6. #19151945

C₂

semidirect

7. #19151598

C₂

semidirect

8. C₄×#13083557

C₂

semidirect

9. #487×C₈₉

C₂

semidirect

115. C₈₉

#3141

semidirect

120. #487

D₈₉

semidirect

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

subgroup	index
#19152149	2
C₄⋊Q₈×D₈₉	2
#19151959	2
#19151948	2
#19151946	2
#19151945	2
#19151598	2
C₄×#13083557	2
#487×C₈₉	2
#3141	89

subgroup

index

C₄⋊Q₈×D₈₉

#487×C₈₉

Subgroups — #26030647

Conjugacy classes of subgroups

Complements

Maximal subgroups