Group #26031718 — subgroups

Subgroups — #26031718

Order 11392.

346 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 109 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	#19152221	C₂	1	2	yes	maximal, complemented
2	#19151887	C₂	1	2	yes	maximal, complemented
3	#19151881	C₂	1	2	yes	maximal, complemented
4	#19151851	C₂	1	1	yes	maximal, characteristic
5	#19151829	C₂	1	2	yes	maximal, complemented
6	#19151822	C₂	1	2	yes	maximal, complemented
7	#19151816	C₂	1	2	yes	maximal, complemented
8	#19151626	C₂	1	1	yes	maximal, characteristic, complemented
9	#470×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#13083561	C₂²	1	2	yes	—
11	C₂×D₇₁₂	C₂²	1	1	yes	characteristic
12	C₂²×#10623494	C₂²	1	1	yes	characteristic
13	#15668011	C₂²	1	2	yes	—
14	#15668007	—	2	2	no	—
15	#15668001	C₂²	1	2	yes	—
16	#15667998	—	2	2	no	—
17	#15667994	—	2	2	no	—
18	#15667992	—	2	2	no	—
19	#15667986	C₂²	1	2	yes	—
20	Q₈×Dic₈₉	C₂²	1	2	yes	—
21	#15667974	C₂²	1	2	yes	—
22	#15667971	C₂²	1	1	yes	characteristic
23	#15667944	C₂²	1	1	yes	characteristic
24	#15667943	C₂²	1	2	yes	—
25	#15667941	C₂²	1	2	yes	—
26	#15667941	C₂²	1	2	yes	—
27	#15667936	C₂²	1	2	yes	—
28	#15667934	C₂²	1	2	yes	—
29	#15667932	C₂²	1	2	yes	—
30	C₂×Q₁₆×C₈₉	C₂²	1	1	yes	characteristic
31	C₂×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
32	C₂²⋊Q₈×C₈₉	C₂²	1	2	yes	—
33	#106×C₈₉	C₂²	1	1	yes	characteristic
34	#103×C₈₉	C₂²	1	2	yes	—
35	C₂×Dic₁₇₈	—	2	2	no	—
36	C₂×D₃₅₆	C₂³	1	2	yes	—
37	C₂×C₂²⋊D₈₉	—	2	2	no	—
38	C₂×C₄×D₈₉	—	2	2	no	—
39	#13083561	—	4	2	no	—
40	D₇₁₂	—	4	1	no	—
41	C₂×#10623494	—	4	1	no	—
42	C₂×#10623494	C₂³	1	2	yes	—
43	C₄⋊Dic₈₉	—	2	2	no	—
44	C₄⋊Dic₈₉	D₄	1	2	yes	—
45	C₄⋊Dic₈₉	C₂³	1	2	yes	—
46	#13083553	—	2	2	no	—
47	C₂²⋊Dic₈₉	D₄	1	1	yes	characteristic
48	C₂²⋊Dic₈₉	D₄	1	1	yes	characteristic
49	#13083551	—	2	2	no	—
50	#13083551	—	2	2	no	—
51	#13083551	—	2	2	no	—
52	#13083551	—	2	2	no	—
53	C₄×Dic₈₉	—	2	2	no	—
54	C₄×Dic₈₉	C₂³	1	2	yes	—
55	C₂×Q₈×C₈₉	C₂³	1	2	yes	—
56	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
57	Q₁₆×C₈₉	—	4	1	no	—
58	M₁₆×C₈₉	—	4	1	no	—
59	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
60	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
61	C₄⋊C₄×C₈₉	—	2	2	no	—
62	C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
63	C₂²⋊C₄×C₈₉	—	2	2	no	—
64	C₂³×C₈₉	C₂×D₄	1	1	yes	characteristic
65	C₂²×D₈₉	—	2	2	no	—
66	Q₈×C₈₉	—	2	2	no	—
67	Q₈×C₈₉	—	2	2	no	—
68	C₂×C₄×C₈₉	—	2	1	no	—
69	C₂×C₄×C₈₉	—	2	1	no	—
70	C₂×C₄×C₈₉	—	2	2	no	—
71	C₂×C₄×C₈₉	—	2	2	no	—
72	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
73	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
74	C₂²⋊D₈₉	—	8	2	no	—
75	C₂×Dic₈₉	—	2	2	no	—
76	C₂×Dic₈₉	C₂×D₄	1	2	yes	—
77	C₂×Dic₈₉	C₂×D₄	1	2	yes	—
78	D₃₅₆	—	2	2	no	—
79	D₃₅₆	—	2	2	no	—
80	C₄×D₈₉	—	8	2	no	—
81	Dic₁₇₈	—	8	2	no	—
82	C₈×C₈₉	—	2	1	no	—
83	C₈×C₈₉	—	2	1	no	—
84	#10623494	—	2	2	no	—
85	C₂²×C₈₉	—	2	1	no	—
86	C₂²×C₈₉	—	2	2	no	—
87	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
88	C₂×D₈₉	—	4	2	no	—
89	C₂×D₈₉	—	4	2	no	—
90	C₂×D₈₉	—	4	2	no	—
91	C₄×C₈₉	—	2	1	no	—
92	C₄×C₈₉	—	4	2	no	—
93	C₄×C₈₉	—	4	2	no	—
94	C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
95	C₄×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
96	Dic₈₉	—	2	2	no	—
97	Dic₈₉	—	2	2	no	—
98	Dic₈₉	—	4	2	no	—
99	C₂×C₈₉	—	4	1	no	—
100	C₂×C₈₉	#537	1	1	yes	characteristic
101	C₂×C₈₉	#577	1	2	yes	—
102	D₈₉	—	8	2	no	—
103	#3263	—	89	1	no	maximal, Sylow
104	C₈₉	#3263	1	1	yes	characteristic, complemented, Sylow
105	#542	—	89	2	no	—
106	#476	—	89	2	no	—
107	#475	—	89	2	no	—
108	#470	D₈₉	1	1	yes	characteristic, complemented
109	#469	—	89	1	no	—
110	#465	—	89	2	no	—
111	#464	—	89	2	no	—
112	#463	—	89	2	no	—
113	#424	—	89	1	no	—
114	C₂×Q₁₆	C₂×D₈₉	1	1	yes	characteristic
115	C₂×SD₁₆	—	89	2	no	—
116	C₂×D₈	—	89	1	no	—
117	C₂×M₁₆	C₂×D₈₉	1	1	yes	characteristic
118	C₂²×C₈	—	89	1	no	—
119	C₄⋊Q₈	—	89	2	no	—
120	#126	—	178	2	no	—
121	#124	—	89	2	no	—
122	#124	—	178	2	no	—
123	#123	—	178	2	no	—
124	C₂²⋊Q₈	—	178	2	no	—
125	C₂²⋊Q₈	C₂×D₈₉	1	2	yes	—
126	#121	—	89	2	no	—
127	C₄×Q₈	—	89	2	no	—
128	C₄×D₄	—	89	2	no	—
129	#117	—	89	1	no	—
130	#106	—	89	1	no	—
131	#106	—	89	2	no	—
132	#106	C₂×D₈₉	1	1	yes	characteristic
133	C₄⋊C₈	—	89	2	no	—
134	C₄⋊C₈	—	89	2	no	—
135	#103	—	89	2	no	—
136	#103	C₂×D₈₉	1	2	yes	—
137	#102	—	89	2	no	—
138	#102	—	89	2	no	—
139	C₂×Q₈	—	178	2	no	—
140	C₂×Q₈	C₂²×D₈₉	1	2	yes	—
141	C₂×D₄	—	89	2	no	—
142	C₂×D₄	—	178	2	no	—
143	C₂²×C₄	—	178	2	no	—
144	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
145	Q₁₆	—	4	1	no	—
146	SD₁₆	—	356	2	no	—
147	D₈	—	356	1	no	—
148	M₁₆	—	4	1	no	—
149	C₂×C₈	—	89	2	no	—
150	C₂×C₈	—	356	1	no	—
151	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
152	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
153	C₄⋊C₄	—	2	2	no	—
154	C₄⋊C₄	—	89	2	no	—
155	C₄⋊C₄	—	89	2	no	—
156	C₄⋊C₄	—	178	2	no	—
157	C₄⋊C₄	—	178	2	no	—
158	C₄⋊C₄	C₂²×D₈₉	1	2	yes	—
159	C₂²⋊C₄	—	2	2	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	C₂²⋊C₄	—	178	2	no	—
165	C₂²⋊C₄	—	178	2	no	—
166	C₄²	—	89	2	no	—
167	C₄²	—	178	2	no	—
168	C₂³	—	178	2	no	—
169	C₂³	D₄×D₈₉	1	1	yes	characteristic
170	Q₈	—	2	2	no	—
171	Q₈	—	2	2	no	—
172	Q₈	—	712	2	no	—
173	D₄	—	178	2	no	—
174	D₄	—	178	2	no	—
175	D₄	—	712	2	no	—
176	C₂×C₄	—	2	1	no	—
177	C₂×C₄	—	2	1	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	89	2	no	—
181	C₂×C₄	—	89	2	no	—
182	C₂×C₄	—	178	2	no	—
183	C₂×C₄	—	712	2	no	—
184	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
185	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
186	C₈	—	2	1	no	—
187	C₈	—	2	1	no	—
188	C₈	—	178	2	no	—
189	C₂²	—	2	2	no	—
190	C₂²	—	2	1	no	—
191	C₂²	—	356	2	no	—
192	C₂²	—	356	2	no	—
193	C₂²	—	356	2	no	—
194	C₂²	C₂×D₄×D₈₉	1	1	yes	characteristic
195	C₄	—	2	1	no	—
196	C₄	—	4	2	no	—
197	C₄	—	4	2	no	—
198	C₄	—	356	2	no	—
199	C₄	—	178	2	no	—
200	C₄	—	178	2	no	—
201	C₄	#15668057	1	1	yes	characteristic
202	C₄	#15668059	1	1	yes	characteristic
203	C₂	—	4	1	no	—
204	C₂	—	712	2	no	—
205	C₂	#19152175	1	1	yes	characteristic
206	C₂	#19152407	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#19152221

C₂

yes

maximal, complemented

#19151887

C₂

yes

maximal, complemented

#19151881

C₂

yes

maximal, complemented

#19151851

C₂

yes

maximal, characteristic

#19151829

C₂

yes

maximal, complemented

#19151822

C₂

yes

maximal, complemented

#19151816

C₂

yes

maximal, complemented

#19151626

C₂

yes

maximal, characteristic, complemented

#470×C₈₉

C₂

yes

maximal, characteristic, complemented

C₂×#13083561

C₂²

yes

—

C₂×D₇₁₂

C₂²

yes

characteristic

C₂²×#10623494

C₂²

yes

characteristic

#15668011

C₂²

yes

—

#15668007

—

#15668001

C₂²

yes

—

—

—

—

C₂²

yes

—

Q₈×Dic₈₉

C₂²

yes

—

#15667974

C₂²

yes

—

#15667971

C₂²

yes

characteristic

#15667944

C₂²

yes

characteristic

#15667943

C₂²

yes

—

#15667941

C₂²

yes

—

#15667941

C₂²

yes

—

#15667936

C₂²

yes

—

#15667934

C₂²

yes

—

#15667932

C₂²

yes

—

C₂×Q₁₆×C₈₉

C₂²

yes

characteristic

C₂×M₁₆×C₈₉

C₂²

yes

characteristic

C₂²⋊Q₈×C₈₉

C₂²

yes

—

#106×C₈₉

C₂²

yes

characteristic

#103×C₈₉

C₂²

yes

—

C₂×Dic₁₇₈

—

C₂×D₃₅₆

C₂³

yes

—

C₂×C₂²⋊D₈₉

—

C₂×C₄×D₈₉

—

#13083561

—

D₇₁₂

—

C₂×#10623494

—

C₂×#10623494

C₂³

yes

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

D₄

yes

—

C₄⋊Dic₈₉

C₂³

yes

—

#13083553

—

C₂²⋊Dic₈₉

D₄

yes

characteristic

C₂²⋊Dic₈₉

D₄

yes

—

—

—

—

C₄×Dic₈₉

—

C₄×Dic₈₉

C₂³

yes

—

C₂×Q₈×C₈₉

C₂³

yes

—

C₂²×C₄×C₈₉

C₂³

yes

characteristic

Q₁₆×C₈₉

—

M₁₆×C₈₉

—

C₂×C₈×C₈₉

C₂³

yes

characteristic

C₂×C₈×C₈₉

C₂³

yes

characteristic

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

C₂³

yes

—

C₂²⋊C₄×C₈₉

—

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×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

C₂⁴

yes

characteristic

C₂²⋊D₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

C₂×D₄

yes

—

C₂×Dic₈₉

C₂×D₄

yes

—

D₃₅₆

—

D₃₅₆

—

C₄×D₈₉

—

Dic₁₇₈

—

C₈×C₈₉

—

C₈×C₈₉

—

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

C₂²×D₄

yes

characteristic

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

H(5,𝔽₂)

yes

characteristic

C₄×C₈₉

2₋¹⁺⁴

yes

characteristic

Dic₈₉

—

Dic₈₉

—

Dic₈₉

—

C₂×C₈₉

—

100

C₂×C₈₉

#537

yes

characteristic

101

C₂×C₈₉

#577

yes

—

102

D₈₉

—

103

#3263

—

maximal, Sylow

104

C₈₉

#3263

yes

characteristic, complemented, Sylow

105

#542

—

106

#476

—

107

#475

—

108

#470

D₈₉

yes

characteristic, complemented

109

#469

—

110

#465

—

111

#464

—

112

#463

—

113

#424

—

114

C₂×Q₁₆

C₂×D₈₉

yes

characteristic

115

C₂×SD₁₆

—

116

C₂×D₈

—

117

C₂×M₁₆

C₂×D₈₉

yes

characteristic

118

C₂²×C₈

—

119

C₄⋊Q₈

—

120

#126

—

178

—

121

#124

—

122

#124

—

178

—

123

#123

—

178

—

124

C₂²⋊Q₈

—

178

—

125

C₂²⋊Q₈

C₂×D₈₉

yes

—

126

#121

—

127

C₄×Q₈

—

128

C₄×D₄

—

129

#117

—

130

#106

—

131

#106

—

132

#106

C₂×D₈₉

yes

characteristic

133

C₄⋊C₈

—

134

C₄⋊C₈

—

135

#103

—

136

#103

C₂×D₈₉

yes

—

137

#102

—

138

#102

—

139

C₂×Q₈

—

178

—

140

C₂×Q₈

C₂²×D₈₉

yes

—

141

C₂×D₄

—

142

C₂×D₄

—

178

—

143

C₂²×C₄

—

178

—

144

C₂²×C₄

C₂²×D₈₉

yes

characteristic

145

Q₁₆

—

146

SD₁₆

—

356

—

147

D₈

—

356

—

148

M₁₆

—

149

C₂×C₈

—

150

C₂×C₈

—

356

—

151

C₂×C₈

C₂²×D₈₉

yes

characteristic

152

C₂×C₈

C₂²×D₈₉

yes

characteristic

153

C₄⋊C₄

—

154

C₄⋊C₄

—

155

C₄⋊C₄

—

156

C₄⋊C₄

—

178

—

157

C₄⋊C₄

—

178

—

158

C₄⋊C₄

C₂²×D₈₉

yes

—

159

C₂²⋊C₄

—

160

C₂²⋊C₄

—

161

C₂²⋊C₄

—

162

C₂²⋊C₄

—

178

—

163

C₂²⋊C₄

—

178

—

164

C₂²⋊C₄

—

178

—

165

C₂²⋊C₄

—

178

—

166

C₄²

—

167

C₄²

—

178

—

168

C₂³

—

178

—

169

C₂³

D₄×D₈₉

yes

characteristic

170

Q₈

—

171

Q₈

—

172

Q₈

—

712

—

173

D₄

—

178

—

174

D₄

—

178

—

175

D₄

—

712

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

—

181

C₂×C₄

—

182

C₂×C₄

—

178

—

183

C₂×C₄

—

712

—

184

C₂×C₄

D₄×D₈₉

yes

characteristic

185

C₂×C₄

C₂³×D₈₉

yes

characteristic

186

C₈

—

187

C₈

—

188

C₈

—

178

—

189

C₂²

—

190

C₂²

—

191

C₂²

—

356

—

192

C₂²

—

356

—

193

C₂²

—

356

—

194

C₂²

C₂×D₄×D₈₉

yes

characteristic

195

C₄

—

196

C₄

—

197

C₄

—

198

C₄

—

356

—

199

C₄

—

178

—

200

C₄

—

178

—

201

C₄

#15668057

yes

characteristic

202

C₄

#15668059

yes

characteristic

203

C₂

—

204

C₂

—

712

—

205

C₂

#19152175

yes

characteristic

206

C₂

#19152407

yes

—

	subgroup	G/H
center	C₂²	C₂×D₄×D₈₉
commutator, cosocle	C₂×C₄×C₈₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₉
Fitting	#470×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₄
prime radical	#19151851	C₂
hypercenter	#470	D₈₉
hypocenter	C₈₉	#3263
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #470

subgroup

G/H

center

C₂²

C₂×D₄×D₈₉

commutator, cosocle

C₂×C₄×C₈₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₈₉

Fitting

#470×C₈₉

C₂

socle

C₂²×C₈₉

C₂²×D₄

C₂

D₈₉

C₈₉

G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #470

N	H ≅ G/N	type
1. #19152221	C₂	semidirect
2. #19151887	C₂	semidirect
3. #19151881	C₂	semidirect
5. #19151829	C₂	semidirect
6. #19151822	C₂	semidirect
7. #19151816	C₂	semidirect
8. #19151626	C₂	semidirect
9. #470×C₈₉	C₂	semidirect
104. C₈₉	#3263	semidirect
108. #470	D₈₉	semidirect

H ≅ G/N

type

1. #19152221

C₂

semidirect

2. #19151887

C₂

semidirect

3. #19151881

C₂

semidirect

5. #19151829

C₂

semidirect

6. #19151822

C₂

semidirect

7. #19151816

C₂

semidirect

8. #19151626

C₂

semidirect

9. #470×C₈₉

C₂

semidirect

104. C₈₉

#3263

semidirect

108. #470

D₈₉

semidirect

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

subgroup	index
#19152221	2
#19151887	2
#19151881	2
#19151851	2
#19151829	2
#19151822	2
#19151816	2
#19151626	2
#470×C₈₉	2
#3263	89

subgroup

index

#470×C₈₉

Subgroups — #26031718

Conjugacy classes of subgroups

Complements

Maximal subgroups