Group #26031723 — subgroups

Subgroups — #26031723

Order 11392.

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

normal subgroups only

This group has 346 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	#19152219	C₂	1	2	yes	maximal, complemented
2	#19151889	C₂	1	2	yes	maximal, complemented
3	#19151884	C₂	1	2	yes	maximal, complemented
4	#19151856	C₂	1	1	yes	maximal, characteristic, complemented
5	#19151851	C₂	1	1	yes	maximal, characteristic
6	#19151827	C₂	1	2	yes	maximal, complemented
7	#19151821	C₂	1	2	yes	maximal, complemented
8	#19151814	C₂	1	2	yes	maximal, complemented
9	#424×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×Dic₃₅₆	C₂²	1	1	yes	characteristic
11	C₂×#13083563	C₂²	1	2	yes	—
12	C₂×D₇₁₂	C₂²	1	1	yes	characteristic
13	#15668012	C₂²	1	2	yes	—
14	#15668007	—	2	2	no	—
15	#15668000	C₂²	1	2	yes	—
16	#15667998	—	2	2	no	—
17	#15667994	—	2	2	no	—
18	#15667991	C₂²	1	2	yes	—
19	#15667990	—	2	2	no	—
20	#15667986	C₂²	1	2	yes	—
21	C₄×Dic₁₇₈	C₂²	1	2	yes	—
22	C₄×D₃₅₆	C₂²	1	2	yes	—
23	#15667944	C₂²	1	1	yes	characteristic
24	#15667944	C₂²	1	2	yes	—
25	#15667944	C₂²	1	1	yes	characteristic
26	#15667937	C₂²	1	2	yes	—
27	#15667937	C₂²	1	2	yes	—
28	#15667934	C₂²	1	2	yes	—
29	#15667934	C₂²	1	2	yes	—
30	C₂×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
31	C₂²×C₈×C₈₉	C₂²	1	1	yes	characteristic
32	#117×C₈₉	C₂²	1	1	yes	characteristic
33	C₄⋊C₈×C₈₉	C₂²	1	2	yes	—
34	C₄⋊C₈×C₈₉	C₂²	1	2	yes	—
35	C₂×Dic₁₇₈	—	2	2	no	—
36	C₂×Dic₁₇₈	C₂³	1	2	yes	—
37	C₂×D₃₅₆	C₂³	1	2	yes	—
38	C₂×C₂²⋊D₈₉	—	2	2	no	—
39	C₂×C₄×D₈₉	—	2	2	no	—
40	Dic₃₅₆	—	4	1	no	—
41	#13083563	—	4	2	no	—
42	D₇₁₂	—	4	1	no	—
43	C₄⋊Dic₈₉	—	2	2	no	—
44	C₄⋊Dic₈₉	C₂³	1	2	yes	—
45	C₄⋊Dic₈₉	C₂³	1	2	yes	—
46	#13083553	—	2	2	no	—
47	#13083553	—	2	2	no	—
48	C₂²⋊Dic₈₉	—	2	2	no	—
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	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
55	M₁₆×C₈₉	—	4	1	no	—
56	C₂×C₈×C₈₉	—	4	1	no	—
57	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
58	C₂×C₈×C₈₉	C₂³	1	2	yes	—
59	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
60	C₄⋊C₄×C₈₉	D₄	1	2	yes	—
61	C₂²⋊C₄×C₈₉	D₄	1	1	yes	characteristic
62	C₂²⋊C₄×C₈₉	D₄	1	1	yes	characteristic
63	C₄²×C₈₉	C₂³	1	2	yes	—
64	C₂³×C₈₉	C₂×D₄	1	1	yes	characteristic
65	C₂²×D₈₉	—	2	2	no	—
66	C₂×C₄×C₈₉	—	2	1	no	—
67	C₂×C₄×C₈₉	—	2	1	no	—
68	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
69	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
70	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
71	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
72	C₂²⋊D₈₉	—	8	2	no	—
73	C₂×Dic₈₉	—	2	2	no	—
74	C₂×Dic₈₉	—	2	2	no	—
75	C₂×Dic₈₉	—	2	2	no	—
76	D₃₅₆	—	2	2	no	—
77	D₃₅₆	—	2	2	no	—
78	C₄×D₈₉	—	8	2	no	—
79	Dic₁₇₈	—	2	2	no	—
80	Dic₁₇₈	—	2	2	no	—
81	Dic₁₇₈	—	8	2	no	—
82	C₈×C₈₉	—	2	2	no	—
83	C₈×C₈₉	—	2	1	no	—
84	C₈×C₈₉	—	2	1	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	—	2	2	no	—
93	C₄×C₈₉	—	2	2	no	—
94	C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
95	C₄×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
96	Dic₈₉	—	4	2	no	—
97	Dic₈₉	—	4	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	—	89	1	no	—
109	#469	—	89	1	no	—
110	#465	—	89	2	no	—
111	#464	—	89	2	no	—
112	#463	—	89	2	no	—
113	#424	D₈₉	1	1	yes	characteristic, complemented
114	C₂×Q₁₆	—	89	1	no	—
115	C₂×SD₁₆	—	89	2	no	—
116	C₂×D₈	—	89	1	no	—
117	C₂×M₁₆	C₂×D₈₉	1	1	yes	characteristic
118	C₂²×C₈	C₂×D₈₉	1	1	yes	characteristic
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	—	89	2	no	—
125	C₂²⋊Q₈	—	178	2	no	—
126	#121	—	89	2	no	—
127	C₄×Q₈	—	89	2	no	—
128	C₄×D₄	—	89	2	no	—
129	#117	C₂×D₈₉	1	1	yes	characteristic
130	#106	—	89	2	no	—
131	#106	—	89	1	no	—
132	#106	—	89	1	no	—
133	C₄⋊C₈	C₂×D₈₉	1	2	yes	—
134	C₄⋊C₈	C₂×D₈₉	1	2	yes	—
135	#103	—	89	2	no	—
136	#103	—	89	2	no	—
137	#102	—	89	2	no	—
138	#102	—	89	2	no	—
139	C₂×Q₈	—	89	2	no	—
140	C₂×Q₈	—	178	2	no	—
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	—	356	1	no	—
146	SD₁₆	—	356	2	no	—
147	D₈	—	356	1	no	—
148	M₁₆	—	4	1	no	—
149	C₂×C₈	—	4	1	no	—
150	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
151	C₂×C₈	C₂²×D₈₉	1	2	yes	—
152	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
153	C₄⋊C₄	—	89	2	no	—
154	C₄⋊C₄	—	89	2	no	—
155	C₄⋊C₄	—	178	2	no	—
156	C₄⋊C₄	—	178	2	no	—
157	C₄⋊C₄	—	178	2	no	—
158	C₄⋊C₄	D₃₅₆	1	2	yes	—
159	C₂²⋊C₄	—	178	2	no	—
160	C₂²⋊C₄	—	178	2	no	—
161	C₂²⋊C₄	—	178	2	no	—
162	C₂²⋊C₄	—	178	2	no	—
163	C₂²⋊C₄	—	178	2	no	—
164	C₂²⋊C₄	D₃₅₆	1	1	yes	characteristic
165	C₂²⋊C₄	D₃₅₆	1	1	yes	characteristic
166	C₄²	—	178	2	no	—
167	C₄²	C₂²×D₈₉	1	2	yes	—
168	C₂³	—	178	2	no	—
169	C₂³	C₂×D₃₅₆	1	1	yes	characteristic
170	Q₈	—	178	2	no	—
171	Q₈	—	178	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	—	178	2	no	—
179	C₂×C₄	—	178	2	no	—
180	C₂×C₄	—	178	2	no	—
181	C₂×C₄	—	712	2	no	—
182	C₂×C₄	C₂×D₃₅₆	1	2	yes	—
183	C₂×C₄	C₂×D₃₅₆	1	2	yes	—
184	C₂×C₄	C₂×D₃₅₆	1	1	yes	characteristic
185	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
186	C₈	—	2	2	no	—
187	C₈	—	2	1	no	—
188	C₈	—	2	1	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	C₂²×D₃₅₆	1	1	yes	characteristic
195	C₄	—	2	2	no	—
196	C₄	—	2	2	no	—
197	C₄	—	2	1	no	—
198	C₄	—	356	2	no	—
199	C₄	—	356	2	no	—
200	C₄	—	356	2	no	—
201	C₄	#15668057	1	1	yes	characteristic
202	C₄	#15668060	1	1	yes	characteristic
203	C₂	—	4	1	no	—
204	C₂	—	712	2	no	—
205	C₂	#19152179	1	1	yes	characteristic
206	C₂	#19152410	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#19152219

C₂

yes

maximal, complemented

#19151889

C₂

yes

maximal, complemented

#19151884

C₂

yes

maximal, complemented

#19151856

C₂

yes

maximal, characteristic, complemented

#19151851

C₂

yes

maximal, characteristic

#19151827

C₂

yes

maximal, complemented

#19151821

C₂

yes

maximal, complemented

#19151814

C₂

yes

maximal, complemented

#424×C₈₉

C₂

yes

maximal, characteristic, complemented

C₂×Dic₃₅₆

C₂²

yes

characteristic

C₂×#13083563

C₂²

yes

—

C₂×D₇₁₂

C₂²

yes

characteristic

#15668012

C₂²

yes

—

#15668007

—

#15668000

C₂²

yes

—

#15667998

—

#15667994

—

#15667991

C₂²

yes

—

#15667990

—

#15667986

C₂²

yes

—

C₄×Dic₁₇₈

C₂²

yes

—

C₄×D₃₅₆

C₂²

yes

—

#15667944

C₂²

yes

characteristic

#15667944

C₂²

yes

—

#15667944

C₂²

yes

characteristic

#15667937

C₂²

yes

—

#15667937

C₂²

yes

—

#15667934

C₂²

yes

—

#15667934

C₂²

yes

—

C₂×M₁₆×C₈₉

C₂²

yes

characteristic

C₂²×C₈×C₈₉

C₂²

yes

characteristic

#117×C₈₉

C₂²

yes

characteristic

C₄⋊C₈×C₈₉

C₂²

yes

—

C₄⋊C₈×C₈₉

C₂²

yes

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

C₂³

yes

—

C₂×D₃₅₆

C₂³

yes

—

C₂×C₂²⋊D₈₉

—

C₂×C₄×D₈₉

—

Dic₃₅₆

—

#13083563

—

D₇₁₂

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

C₂³

yes

—

C₄⋊Dic₈₉

C₂³

yes

—

#13083553

—

#13083553

—

C₂²⋊Dic₈₉

—

—

—

—

—

C₄×Dic₈₉

—

C₂²×C₄×C₈₉

C₂³

yes

characteristic

M₁₆×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

C₂³

yes

characteristic

C₂×C₈×C₈₉

C₂³

yes

—

C₂×C₈×C₈₉

C₂³

yes

characteristic

C₄⋊C₄×C₈₉

D₄

yes

—

C₂²⋊C₄×C₈₉

D₄

yes

characteristic

C₂²⋊C₄×C₈₉

D₄

yes

characteristic

C₄²×C₈₉

C₂³

yes

—

C₂³×C₈₉

C₂×D₄

yes

characteristic

C₂²×D₈₉

—

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

—

D₃₅₆

—

D₃₅₆

—

C₄×D₈₉

—

Dic₁₇₈

—

Dic₁₇₈

—

Dic₁₇₈

—

C₈×C₈₉

—

C₈×C₈₉

—

C₈×C₈₉

—

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

—

109

#469

—

110

#465

—

111

#464

—

112

#463

—

113

#424

D₈₉

yes

characteristic, complemented

114

C₂×Q₁₆

—

115

C₂×SD₁₆

—

116

C₂×D₈

—

117

C₂×M₁₆

C₂×D₈₉

yes

characteristic

118

C₂²×C₈

C₂×D₈₉

yes

characteristic

119

C₄⋊Q₈

—

120

#126

—

178

—

121

#124

—

122

#124

—

178

—

123

#123

—

178

—

124

C₂²⋊Q₈

—

125

C₂²⋊Q₈

—

178

—

126

#121

—

127

C₄×Q₈

—

128

C₄×D₄

—

129

#117

C₂×D₈₉

yes

characteristic

130

#106

—

131

#106

—

132

#106

—

133

C₄⋊C₈

C₂×D₈₉

yes

—

134

C₄⋊C₈

C₂×D₈₉

yes

—

135

#103

—

136

#103

—

137

#102

—

138

#102

—

139

C₂×Q₈

—

140

C₂×Q₈

—

178

—

141

C₂×D₄

—

142

C₂×D₄

—

178

—

143

C₂²×C₄

—

178

—

144

C₂²×C₄

C₂²×D₈₉

yes

characteristic

145

Q₁₆

—

356

—

146

SD₁₆

—

356

—

147

D₈

—

356

—

148

M₁₆

—

149

C₂×C₈

—

150

C₂×C₈

C₂²×D₈₉

yes

characteristic

151

C₂×C₈

C₂²×D₈₉

yes

—

152

C₂×C₈

C₂²×D₈₉

yes

characteristic

153

C₄⋊C₄

—

154

C₄⋊C₄

—

155

C₄⋊C₄

—

178

—

156

C₄⋊C₄

—

178

—

157

C₄⋊C₄

—

178

—

158

C₄⋊C₄

D₃₅₆

yes

—

159

C₂²⋊C₄

—

178

—

160

C₂²⋊C₄

—

178

—

161

C₂²⋊C₄

—

178

—

162

C₂²⋊C₄

—

178

—

163

C₂²⋊C₄

—

178

—

164

C₂²⋊C₄

D₃₅₆

yes

characteristic

165

C₂²⋊C₄

D₃₅₆

yes

characteristic

166

C₄²

—

178

—

167

C₄²

C₂²×D₈₉

yes

—

168

C₂³

—

178

—

169

C₂³

C₂×D₃₅₆

yes

characteristic

170

Q₈

—

178

—

171

Q₈

—

178

—

172

Q₈

—

712

—

173

D₄

—

178

—

174

D₄

—

178

—

175

D₄

—

712

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

178

—

179

C₂×C₄

—

178

—

180

C₂×C₄

—

178

—

181

C₂×C₄

—

712

—

182

C₂×C₄

C₂×D₃₅₆

yes

—

183

C₂×C₄

C₂×D₃₅₆

yes

—

184

C₂×C₄

C₂×D₃₅₆

yes

characteristic

185

C₂×C₄

C₂³×D₈₉

yes

characteristic

186

C₈

—

187

C₈

—

188

C₈

—

189

C₂²

—

190

C₂²

—

191

C₂²

—

356

—

192

C₂²

—

356

—

193

C₂²

—

356

—

194

C₂²

C₂²×D₃₅₆

yes

characteristic

195

C₄

—

196

C₄

—

197

C₄

—

198

C₄

—

356

—

199

C₄

—

356

—

200

C₄

—

356

—

201

C₄

#15668057

yes

characteristic

202

C₄

#15668060

yes

characteristic

203

C₂

—

204

C₂

—

712

—

205

C₂

#19152179

yes

characteristic

206

C₂

#19152410

yes

—

	subgroup	G/H
center	C₂²	C₂²×D₃₅₆
commutator, cosocle	C₂×C₄×C₈₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₉
Fitting	#424×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₄
prime radical	#19151851	C₂
hypercenter	#424	D₈₉
hypocenter	C₈₉	#3263
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ #117 ◁ #424

subgroup

G/H

center

C₂²

C₂²×D₃₅₆

commutator, cosocle

C₂×C₄×C₈₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₈₉

Fitting

#424×C₈₉

C₂

socle

C₂²×C₈₉

C₂²×D₄

C₂

D₈₉

C₈₉

G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ #117 ◁ #424

N	H ≅ G/N	type
1. #19152219	C₂	semidirect
2. #19151889	C₂	semidirect
3. #19151884	C₂	semidirect
4. #19151856	C₂	semidirect
6. #19151827	C₂	semidirect
7. #19151821	C₂	semidirect
8. #19151814	C₂	semidirect
9. #424×C₈₉	C₂	semidirect
104. C₈₉	#3263	semidirect
113. #424	D₈₉	semidirect

H ≅ G/N

type

1. #19152219

C₂

semidirect

2. #19151889

C₂

semidirect

3. #19151884

C₂

semidirect

4. #19151856

C₂

semidirect

6. #19151827

C₂

semidirect

7. #19151821

C₂

semidirect

8. #19151814

C₂

semidirect

9. #424×C₈₉

C₂

semidirect

104. C₈₉

#3263

semidirect

113. #424

D₈₉

semidirect

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

subgroup	index
#19152219	2
#19151889	2
#19151884	2
#19151856	2
#19151851	2
#19151827	2
#19151821	2
#19151814	2
#424×C₈₉	2
#3263	89

subgroup

index

#424×C₈₉

Subgroups — #26031723

Conjugacy classes of subgroups

Complements

Maximal subgroups