Group #26031329 — subgroups

Subgroups — #26031329

Order 11392.

364 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 117 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	#19152174	C₂	1	1	yes	maximal, characteristic
2	#19151990	C₂	1	1	yes	maximal, characteristic, complemented
3	#19151977	C₂	1	1	yes	maximal, characteristic, complemented
4	#19151968	C₂	1	1	yes	maximal, characteristic, complemented
5	#19151955	C₂	1	1	yes	maximal, characteristic, complemented
6	#19151820	C₂	1	2	yes	maximal, complemented
7	#19151808	C₂	1	2	yes	maximal, complemented
8	#19151758	C₂	1	2	yes	maximal
9	#19151753	C₂	1	2	yes	maximal, complemented
10	#19151659	C₂	1	1	yes	maximal, characteristic, complemented
11	#541×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13083574	C₂²	1	1	yes	characteristic
13	C₂²×Dic₁₇₈	C₂²	1	1	yes	characteristic
14	C₂×#13083564	C₂²	1	2	yes	—
15	C₂×#13083564	C₂²	1	2	yes	—
16	C₂×#13083564	C₂²	1	2	yes	—
17	C₂×#13083561	C₂²	1	2	yes	—
18	#15668012	C₂²	1	1	yes	characteristic
19	#15668012	C₂²	1	1	yes	characteristic
20	#15668000	C₂²	1	1	yes	characteristic
21	#15668000	C₂²	1	1	yes	characteristic
22	#15667996	—	2	2	no	—
23	#15667992	—	2	2	no	—
24	C₄⋊#10623494	C₂²	1	2	yes	—
25	#15667935	C₂²	1	2	yes	—
26	#15667935	C₂²	1	2	yes	—
27	#15667935	C₂²	1	2	yes	—
28	#15667932	C₂²	1	2	yes	—
29	C₂²⋊#10623494	C₂²	1	2	yes	—
30	#15667923	C₂²	1	1	yes	characteristic
31	C₄×#10623494	C₂²	1	1	yes	characteristic
32	C₄⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
33	#126×C₈₉	—	2	2	no	—
34	#125×C₈₉	—	2	2	no	—
35	#125×C₈₉	C₂²	1	1	yes	characteristic
36	C₂²⋊Q₈×C₈₉	C₂²	1	2	yes	—
37	C₄×Q₈×C₈₉	C₂²	1	2	yes	—
38	#117×C₈₉	C₂²	1	1	yes	characteristic
39	#13083574	—	4	2	no	—
40	C₂×Dic₁₇₈	—	2	1	no	—
41	C₂×Dic₁₇₈	—	4	2	no	—
42	C₂×Dic₁₇₈	C₂³	1	1	yes	characteristic
43	C₂×Dic₁₇₈	C₂³	1	1	yes	characteristic
44	C₂×Dic₁₇₈	C₂³	1	1	yes	characteristic
45	C₂×D₃₅₆	C₂³	1	1	yes	characteristic
46	C₂×C₂²⋊D₈₉	—	2	1	no	—
47	C₂²×Dic₈₉	—	2	1	no	—
48	C₂×C₄×D₈₉	—	2	1	no	—
49	#13083564	—	4	2	no	—
50	#13083564	—	4	2	no	—
51	#13083564	—	4	2	no	—
52	#13083561	—	4	2	no	—
53	C₂×#10623494	C₂³	1	2	yes	—
54	C₂×#10623494	C₂³	1	2	yes	—
55	C₄⋊Dic₈₉	—	2	2	no	—
56	C₄⋊Dic₈₉	—	2	2	no	—
57	#13083551	—	2	2	no	—
58	#13083551	—	2	2	no	—
59	C₂×Q₈×C₈₉	C₂³	1	2	yes	—
60	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
61	C₄⋊C₄×C₈₉	—	2	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	D₄	1	2	yes	—
66	C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
67	C₂²⋊C₄×C₈₉	—	2	2	no	—
68	C₂²⋊C₄×C₈₉	D₄	1	2	yes	—
69	C₄²×C₈₉	—	2	2	no	—
70	C₄²×C₈₉	C₂³	1	1	yes	characteristic
71	C₄²×C₈₉	C₂³	1	1	yes	characteristic
72	C₂³×C₈₉	C₂×D₄	1	1	yes	characteristic
73	C₂²×D₈₉	—	2	1	no	—
74	Q₈×C₈₉	—	2	2	no	—
75	Q₈×C₈₉	—	2	2	no	—
76	C₂×C₄×C₈₉	—	2	2	no	—
77	C₂×C₄×C₈₉	—	2	2	no	—
78	C₂×C₄×C₈₉	—	2	2	no	—
79	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
80	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
81	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
82	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
83	C₂²⋊D₈₉	—	4	2	no	—
84	C₂×Dic₈₉	—	2	1	no	—
85	C₂×Dic₈₉	—	2	1	no	—
86	C₂×Dic₈₉	—	2	1	no	—
87	C₂×Dic₈₉	—	4	2	no	—
88	D₃₅₆	—	2	2	no	—
89	C₄×D₈₉	—	4	2	no	—
90	Dic₁₇₈	—	2	2	no	—
91	Dic₁₇₈	—	2	2	no	—
92	Dic₁₇₈	—	2	2	no	—
93	Dic₁₇₈	—	4	2	no	—
94	#10623494	—	2	2	no	—
95	#10623494	—	2	2	no	—
96	C₂²×C₈₉	—	2	1	no	—
97	C₂²×C₈₉	—	2	2	no	—
98	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
99	C₂×D₈₉	—	4	1	no	—
100	C₂×D₈₉	—	4	2	no	—
101	C₄×C₈₉	—	2	1	no	—
102	C₄×C₈₉	—	2	2	no	—
103	C₄×C₈₉	—	2	2	no	—
104	C₄×C₈₉	—	4	2	no	—
105	C₄×C₈₉	—	4	2	no	—
106	C₄×C₈₉	H(5,𝔽₂)	1	2	yes	—
107	Dic₈₉	—	4	1	no	—
108	Dic₈₉	—	4	1	no	—
109	Dic₈₉	—	4	1	no	—
110	C₂×C₈₉	—	4	1	no	—
111	C₂×C₈₉	#535	1	1	yes	characteristic
112	C₂×C₈₉	#578	1	2	yes	—
113	D₈₉	—	8	1	no	—
114	#3223	—	89	1	no	maximal, Sylow
115	C₈₉	#3223	1	1	yes	characteristic, complemented, Sylow
116	#541	D₈₉	1	1	yes	characteristic, complemented
117	#536	—	89	1	no	—
118	#497	—	89	1	no	—
119	#494	—	89	1	no	—
120	#491	—	89	1	no	—
121	#488	—	89	1	no	—
122	#464	—	89	2	no	—
123	#462	—	89	2	no	—
124	#452	—	89	2	no	—
125	#451	—	89	2	no	—
126	#433	—	89	1	no	—
127	C₂×#41	—	89	1	no	—
128	C₂²×Q₈	—	89	1	no	—
129	C₂×Q₁₆	—	89	2	no	—
130	C₂×Q₁₆	—	89	2	no	—
131	C₂×Q₁₆	—	89	2	no	—
132	C₂×SD₁₆	—	89	2	no	—
133	C₄⋊Q₈	—	89	1	no	—
134	C₄⋊Q₈	—	89	1	no	—
135	C₄⋊Q₈	C₂×D₈₉	1	1	yes	characteristic
136	#126	—	2	2	no	—
137	#125	—	2	2	no	—
138	#125	C₂×D₈₉	1	1	yes	characteristic
139	#124	—	89	1	no	—
140	#124	—	89	1	no	—
141	#123	—	178	2	no	—
142	C₂²⋊Q₈	—	178	2	no	—
143	C₂²⋊Q₈	C₂×D₈₉	1	2	yes	—
144	C₄×Q₈	C₂×D₈₉	1	2	yes	—
145	#117	C₂×D₈₉	1	1	yes	characteristic
146	C₄⋊C₈	—	89	2	no	—
147	#103	—	89	2	no	—
148	#103	—	89	2	no	—
149	#103	—	89	2	no	—
150	#102	—	89	2	no	—
151	C₂²⋊C₈	—	89	2	no	—
152	#97	—	89	1	no	—
153	C₄×C₈	—	89	1	no	—
154	#41	—	356	2	no	—
155	C₂×Q₈	—	89	1	no	—
156	C₂×Q₈	—	89	1	no	—
157	C₂×Q₈	—	89	1	no	—
158	C₂×Q₈	—	356	2	no	—
159	C₂×Q₈	—	178	1	no	—
160	C₂×Q₈	C₂²×D₈₉	1	2	yes	—
161	C₂×D₄	—	89	1	no	—
162	C₂×D₄	—	178	1	no	—
163	C₂²×C₄	—	178	1	no	—
164	C₂²×C₄	—	178	1	no	—
165	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
166	Q₁₆	—	356	2	no	—
167	Q₁₆	—	356	2	no	—
168	Q₁₆	—	356	2	no	—
169	SD₁₆	—	356	2	no	—
170	C₂×C₈	—	89	2	no	—
171	C₂×C₈	—	89	2	no	—
172	C₄⋊C₄	—	2	2	no	—
173	C₄⋊C₄	—	2	2	no	—
174	C₄⋊C₄	—	2	2	no	—
175	C₄⋊C₄	—	2	2	no	—
176	C₄⋊C₄	—	178	2	no	—
177	C₄⋊C₄	—	178	2	no	—
178	C₄⋊C₄	C₂²⋊D₈₉	1	2	yes	—
179	C₄⋊C₄	C₂²×D₈₉	1	2	yes	—
180	C₂²⋊C₄	—	2	2	no	—
181	C₂²⋊C₄	—	178	2	no	—
182	C₂²⋊C₄	—	178	2	no	—
183	C₂²⋊C₄	C₂²⋊D₈₉	1	2	yes	—
184	C₄²	—	2	2	no	—
185	C₄²	C₂²×D₈₉	1	1	yes	characteristic
186	C₄²	C₂²×D₈₉	1	1	yes	characteristic
187	C₂³	—	178	1	no	—
188	C₂³	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
189	Q₈	—	2	2	no	—
190	Q₈	—	2	2	no	—
191	Q₈	—	356	2	no	—
192	Q₈	—	178	2	no	—
193	Q₈	—	178	2	no	—
194	Q₈	—	178	2	no	—
195	D₄	—	356	2	no	—
196	D₄	—	178	2	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	356	2	no	—
201	C₂×C₄	—	356	2	no	—
202	C₂×C₄	—	178	1	no	—
203	C₂×C₄	—	178	1	no	—
204	C₂×C₄	—	178	1	no	—
205	C₂×C₄	C₂×C₂²⋊D₈₉	1	2	yes	—
206	C₂×C₄	C₂×C₂²⋊D₈₉	1	2	yes	—
207	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
208	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
209	C₈	—	178	2	no	—
210	C₈	—	178	2	no	—
211	C₂²	—	2	2	no	—
212	C₂²	—	2	1	no	—
213	C₂²	—	356	2	no	—
214	C₂²	—	356	1	no	—
215	C₂²	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
216	C₄	—	2	2	no	—
217	C₄	—	2	2	no	—
218	C₄	—	2	1	no	—
219	C₄	—	4	2	no	—
220	C₄	—	4	2	no	—
221	C₄	—	356	1	no	—
222	C₄	—	356	1	no	—
223	C₄	—	356	1	no	—
224	C₄	#15668058	1	2	yes	—
225	C₂	—	4	1	no	—
226	C₂	—	712	1	no	—
227	C₂	#19152162	1	1	yes	characteristic
228	C₂	#19152413	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#19152174

C₂

yes

maximal, characteristic

#19151990

C₂

yes

maximal, characteristic, complemented

#19151977

C₂

yes

maximal, characteristic, complemented

#19151968

C₂

yes

maximal, characteristic, complemented

#19151955

C₂

yes

maximal, characteristic, complemented

#19151820

C₂

yes

maximal, complemented

#19151808

C₂

yes

maximal, complemented

#19151758

C₂

yes

maximal

#19151753

C₂

yes

maximal, complemented

#19151659

C₂

yes

maximal, characteristic, complemented

#541×C₈₉

C₂

yes

maximal, characteristic, complemented

C₂×#13083574

C₂²

yes

characteristic

C₂²×Dic₁₇₈

C₂²

yes

characteristic

C₂×#13083564

C₂²

yes

—

C₂×#13083564

C₂²

yes

—

C₂×#13083564

C₂²

yes

—

C₂×#13083561

C₂²

yes

—

#15668012

C₂²

yes

characteristic

#15668012

C₂²

yes

characteristic

#15668000

C₂²

yes

characteristic

#15668000

C₂²

yes

—

—

C₂²

yes

—

#15667935

C₂²

yes

—

#15667935

C₂²

yes

—

#15667935

C₂²

yes

—

#15667932

C₂²

yes

—

C₂²⋊#10623494

C₂²

yes

—

#15667923

C₂²

yes

characteristic

C₄×#10623494

C₂²

yes

characteristic

C₄⋊Q₈×C₈₉

C₂²

yes

characteristic

#126×C₈₉

—

#125×C₈₉

—

#125×C₈₉

C₂²

yes

characteristic

C₂²⋊Q₈×C₈₉

C₂²

yes

—

C₄×Q₈×C₈₉

C₂²

yes

—

#117×C₈₉

C₂²

yes

characteristic

#13083574

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

C₂³

yes

characteristic

C₂×Dic₁₇₈

C₂³

yes

characteristic

C₂×Dic₁₇₈

C₂³

yes

characteristic

C₂×D₃₅₆

C₂³

yes

characteristic

C₂×C₂²⋊D₈₉

—

C₂²×Dic₈₉

—

C₂×C₄×D₈₉

—

—

—

—

—

C₂³

yes

—

C₂×#10623494

C₂³

yes

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

#13083551

—

#13083551

—

C₂×Q₈×C₈₉

C₂³

yes

—

C₂²×C₄×C₈₉

C₂³

yes

characteristic

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

D₄

yes

—

C₄⋊C₄×C₈₉

C₂³

yes

—

C₂²⋊C₄×C₈₉

—

C₂²⋊C₄×C₈₉

D₄

yes

—

C₄²×C₈₉

—

C₄²×C₈₉

C₂³

yes

characteristic

C₄²×C₈₉

C₂³

yes

characteristic

C₂³×C₈₉

C₂×D₄

yes

characteristic

C₂²×D₈₉

—

Q₈×C₈₉

—

Q₈×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

C₂⁴

yes

characteristic

C₂²⋊D₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

D₃₅₆

—

C₄×D₈₉

—

Dic₁₇₈

—

Dic₁₇₈

—

Dic₁₇₈

—

Dic₁₇₈

—

#10623494

—

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

C₂²×D₄

yes

characteristic

C₂×D₈₉

—

100

C₂×D₈₉

—

101

C₄×C₈₉

—

102

C₄×C₈₉

—

103

C₄×C₈₉

—

104

C₄×C₈₉

—

105

C₄×C₈₉

—

106

C₄×C₈₉

H(5,𝔽₂)

yes

—

107

Dic₈₉

—

108

Dic₈₉

—

109

Dic₈₉

—

110

C₂×C₈₉

—

111

C₂×C₈₉

#535

yes

characteristic

112

C₂×C₈₉

#578

yes

—

113

D₈₉

—

114

#3223

—

maximal, Sylow

115

C₈₉

#3223

yes

characteristic, complemented, Sylow

116

#541

D₈₉

yes

characteristic, complemented

117

#536

—

118

#497

—

119

#494

—

120

#491

—

121

#488

—

122

#464

—

123

#462

—

124

#452

—

125

#451

—

126

#433

—

127

C₂×#41

—

128

C₂²×Q₈

—

129

C₂×Q₁₆

—

130

C₂×Q₁₆

—

131

C₂×Q₁₆

—

132

C₂×SD₁₆

—

133

C₄⋊Q₈

—

134

C₄⋊Q₈

—

135

C₄⋊Q₈

C₂×D₈₉

yes

characteristic

136

#126

—

137

#125

—

138

#125

C₂×D₈₉

yes

characteristic

139

#124

—

140

#124

—

141

#123

—

178

—

142

C₂²⋊Q₈

—

178

—

143

C₂²⋊Q₈

C₂×D₈₉

yes

—

144

C₄×Q₈

C₂×D₈₉

yes

—

145

#117

C₂×D₈₉

yes

characteristic

146

C₄⋊C₈

—

147

#103

—

148

#103

—

149

#103

—

150

#102

—

151

C₂²⋊C₈

—

152

#97

—

153

C₄×C₈

—

154

#41

—

356

—

155

C₂×Q₈

—

156

C₂×Q₈

—

157

C₂×Q₈

—

158

C₂×Q₈

—

356

—

159

C₂×Q₈

—

178

—

160

C₂×Q₈

C₂²×D₈₉

yes

—

161

C₂×D₄

—

162

C₂×D₄

—

178

—

163

C₂²×C₄

—

178

—

164

C₂²×C₄

—

178

—

165

C₂²×C₄

C₂²×D₈₉

yes

characteristic

166

Q₁₆

—

356

—

167

Q₁₆

—

356

—

168

Q₁₆

—

356

—

169

SD₁₆

—

356

—

170

C₂×C₈

—

171

C₂×C₈

—

172

C₄⋊C₄

—

173

C₄⋊C₄

—

174

C₄⋊C₄

—

175

C₄⋊C₄

—

176

C₄⋊C₄

—

178

—

177

C₄⋊C₄

—

178

—

178

C₄⋊C₄

C₂²⋊D₈₉

yes

—

179

C₄⋊C₄

C₂²×D₈₉

yes

—

180

C₂²⋊C₄

—

181

C₂²⋊C₄

—

178

—

182

C₂²⋊C₄

—

178

—

183

C₂²⋊C₄

C₂²⋊D₈₉

yes

—

184

C₄²

—

185

C₄²

C₂²×D₈₉

yes

characteristic

186

C₄²

C₂²×D₈₉

yes

characteristic

187

C₂³

—

178

—

188

C₂³

C₂×C₂²⋊D₈₉

yes

characteristic

189

Q₈

—

190

Q₈

—

191

Q₈

—

356

—

192

Q₈

—

178

—

193

Q₈

—

178

—

194

Q₈

—

178

—

195

D₄

—

356

—

196

D₄

—

178

—

197

C₂×C₄

—

198

C₂×C₄

—

199

C₂×C₄

—

200

C₂×C₄

—

356

—

201

C₂×C₄

—

356

—

202

C₂×C₄

—

178

—

203

C₂×C₄

—

178

—

204

C₂×C₄

—

178

—

205

C₂×C₄

C₂×C₂²⋊D₈₉

yes

—

206

C₂×C₄

C₂×C₂²⋊D₈₉

yes

—

207

C₂×C₄

C₂×C₂²⋊D₈₉

yes

characteristic

208

C₂×C₄

C₂³×D₈₉

yes

characteristic

209

C₈

—

178

—

210

C₈

—

178

—

211

C₂²

—

212

C₂²

—

213

C₂²

—

356

—

214

C₂²

—

356

—

215

C₂²

C₂²×C₂²⋊D₈₉

yes

characteristic

216

C₄

—

217

C₄

—

218

C₄

—

219

C₄

—

220

C₄

—

221

C₄

—

356

—

222

C₄

—

356

—

223

C₄

—

356

—

224

C₄

#15668058

yes

—

225

C₂

—

226

C₂

—

712

—

227

C₂

#19152162

yes

characteristic

228

C₂

#19152413

yes

—

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₈₉
commutator, cosocle	C₂×C₄×C₈₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₉
Fitting	#541×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₄
prime radical	C₂×#13083574	C₂²
hypercenter	#541	D₈₉
hypocenter	C₈₉	#3223
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ #117 ◁ #541

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₈₉

commutator, cosocle

C₂×C₄×C₈₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₈₉

Fitting

#541×C₈₉

C₂

socle

C₂²×C₈₉

C₂²×D₄

C₂²

D₈₉

C₈₉

G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ #117 ◁ #541

N	H ≅ G/N	type
2. #19151990	C₂	semidirect
3. #19151977	C₂	semidirect
4. #19151968	C₂	semidirect
5. #19151955	C₂	semidirect
6. #19151820	C₂	semidirect
7. #19151808	C₂	semidirect
9. #19151753	C₂	semidirect
10. #19151659	C₂	semidirect
11. #541×C₈₉	C₂	semidirect
115. C₈₉	#3223	semidirect
116. #541	D₈₉	semidirect

H ≅ G/N

type

2. #19151990

C₂

semidirect

3. #19151977

C₂

semidirect

4. #19151968

C₂

semidirect

5. #19151955

C₂

semidirect

6. #19151820

C₂

semidirect

7. #19151808

C₂

semidirect

9. #19151753

C₂

semidirect

10. #19151659

C₂

semidirect

11. #541×C₈₉

C₂

semidirect

115. C₈₉

#3223

semidirect

116. #541

D₈₉

semidirect

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

subgroup	index
#19152174	2
#19151990	2
#19151977	2
#19151968	2
#19151955	2
#19151820	2
#19151808	2
#19151758	2
#19151753	2
#19151659	2
#541×C₈₉	2
#3223	89

subgroup

index

#541×C₈₉

Subgroups — #26031329

Conjugacy classes of subgroups

Complements

Maximal subgroups