Group #26369655 — subgroups

Subgroups — #26369655

Order 12032.

646 conjugacy classes of subgroups; only the 211 normal subgroups are shown.

normal subgroups only

This group has 646 conjugacy classes of subgroups — too many to draw. Only its 211 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	C₂²×Dic₃₇₆	C₂	1	1	yes	maximal, characteristic
2	#20252255	C₂	1	2	yes	maximal
3	#20245258	C₂	1	4	yes	maximal, complemented
4	#20245246	C₂	1	4	yes	maximal, complemented
5	#20245107	C₂	1	2	yes	maximal
6	#20244984	C₂	1	1	yes	maximal, characteristic
7	C₄₇×#2158	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₁₈₈	C₂²	1	2	yes	—
9	#15689600	—	2	2	no	—
10	C₂×Dic₃₇₆	C₄	1	8	yes	complemented
11	C₂×Dic₃₇₆	C₂²	1	2	yes	—
12	C₂×Dic₃₇₆	C₂²	1	2	yes	—
13	C₂×#13093964	C₂²	1	1	yes	characteristic
14	#15689229	—	2	4	no	—
15	#15689217	C₂²	1	8	yes	—
16	#15689131	—	2	2	no	—
17	#15689078	C₂²	1	1	yes	characteristic
18	#15688926	C₂²	1	8	yes	—
19	#15688898	C₂²	1	2	yes	—
20	C₁₆×Dic₄₇	C₂²	1	2	yes	—
21	C₂²×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
22	C₄₇×#427	C₂²	1	2	yes	—
23	C₄₇×#367	C₂²	1	2	yes	—
24	C₄₇×#366	C₂²	1	2	yes	—
25	C₂²×Dic₉₄	—	2	2	no	—
26	C₂×Dic₁₈₈	—	2	2	no	—
27	C₂×Dic₁₈₈	—	2	4	no	—
28	C₂×Dic₁₈₈	C₂×C₄	1	8	yes	—
29	C₂×Dic₁₈₈	C₂³	1	4	yes	—
30	C₂×#10625329	D₄	1	1	yes	characteristic
31	C₄×Dic₉₄	—	4	4	no	—
32	#13093989	—	2	8	no	—
33	#13093983	D₄	1	1	yes	characteristic
34	#13093980	—	4	4	no	—
35	Dic₃₇₆	C₂×C₄	1	8	yes	—
36	Dic₃₇₆	C₂×C₄	1	8	yes	—
37	#13093964	—	2	2	no	—
38	C₂×C₄₇⋊C₁₆	C₂³	1	2	yes	—
39	#13093947	—	2	8	no	—
40	#13093934	D₄	1	2	yes	—
41	C₈×Dic₄₇	C₂³	1	2	yes	—
42	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
43	#117×C₄₇	—	2	2	no	—
44	C₂×C₁₆×C₄₇	C₂×C₄	1	4	yes	—
45	C₂×C₁₆×C₄₇	C₂³	1	1	yes	characteristic
46	C₂×C₁₆×C₄₇	C₂³	1	1	yes	characteristic
47	#107×C₄₇	C₂³	1	4	yes	—
48	#106×C₄₇	—	2	2	no	—
49	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
50	C₂²×Dic₄₇	—	4	2	no	—
51	C₂×Dic₉₄	—	2	8	no	—
52	C₂×Dic₉₄	—	2	4	no	—
53	C₂×Dic₉₄	—	4	2	no	—
54	C₂×Dic₉₄	—	4	4	no	—
55	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
56	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
57	C₂×C₈×C₄₇	#41	1	2	yes	—
58	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
59	C₄⋊C₄×C₄₇	—	2	4	no	—
60	C₂²⋊C₄×C₄₇	—	4	2	no	—
61	C₄²×C₄₇	—	4	2	no	—
62	C₂²⋊Dic₄₇	D₈	1	2	yes	—
63	C₄⋊Dic₄₇	D₈	1	2	yes	—
64	Dic₄₇⋊C₄	—	4	8	no	—
65	C₄×Dic₄₇	—	4	8	no	—
66	C₄×Dic₄₇	C₂×D₄	1	2	yes	—
67	#10625329	—	2	2	no	—
68	C₂×C₄₇⋊C₈	C₂×D₄	1	2	yes	—
69	Dic₁₈₈	—	2	8	no	—
70	Dic₁₈₈	—	2	4	no	—
71	Dic₁₈₈	C₂²×C₄	1	8	yes	—
72	C₁₆×C₄₇	C₂²×C₄	1	2	yes	—
73	C₁₆×C₄₇	C₂²×C₄	1	2	yes	—
74	C₄₇⋊C₁₆	—	2	2	no	—
75	C₂³×C₄₇	C₂×D₈	1	1	yes	characteristic
76	C₂×C₄×C₄₇	—	4	4	no	—
77	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
78	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
79	C₂×C₄×C₄₇	#135	1	2	yes	—
80	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
81	C₂×Dic₄₇	—	4	4	no	—
82	C₂×Dic₄₇	—	4	8	no	—
83	C₂×Dic₄₇	C₂×D₈	1	4	yes	—
84	Dic₉₄	—	2	8	no	—
85	Dic₉₄	—	4	4	no	—
86	Dic₉₄	—	4	8	no	—
87	C₈×C₄₇	C₄×D₄	1	2	yes	—
88	C₈×C₄₇	C₂³×C₄	1	1	yes	characteristic
89	C₈×C₄₇	C₂×#41	1	1	yes	characteristic
90	C₄₇⋊C₈	—	2	2	no	—
91	#33458	—	47	1	no	maximal, Sylow
92	C₂²×C₄₇	—	2	2	no	—
93	C₂²×C₄₇	C₄×D₈	1	2	yes	—
94	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
95	C₄×C₄₇	—	8	4	no	—
96	C₄×C₄₇	C₄×D₈	1	2	yes	—
97	C₄×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
98	C₄×C₄₇	C₂×#135	1	1	yes	characteristic
99	Dic₄₇	—	2	4	no	—
100	Dic₄₇	—	4	8	no	—
101	C₂²×Q₃₂	—	47	1	no	—
102	#2950	—	47	2	no	—
103	#2176	—	47	4	no	—
104	C₄×Q₃₂	—	47	4	no	—
105	#2158	D₄₇	1	1	yes	characteristic
106	#2141	—	47	2	no	—
107	#2108	—	47	1	no	—
108	C₂×C₄₇	—	2	2	no	—
109	C₂×C₄₇	C₂×C₄×D₈	1	1	yes	characteristic
110	C₂×C₄₇	#3417	1	2	yes	—
111	C₂²×Q₁₆	—	47	2	no	—
112	#519	—	94	2	no	—
113	C₂×Q₃₂	—	47	8	no	—
114	C₂×Q₃₂	—	47	2	no	—
115	C₂×Q₃₂	—	47	2	no	—
116	C₂×M₃₂	—	47	1	no	—
117	C₂²×C₁₆	C₂×D₄₇	1	1	yes	characteristic
118	#441	—	94	4	no	—
119	C₄×Q₁₆	—	47	8	no	—
120	#427	C₂×D₄₇	1	2	yes	—
121	#419	—	94	2	no	—
122	#405	—	47	1	no	—
123	#367	C₂×D₄₇	1	2	yes	—
124	#366	C₂×D₄₇	1	2	yes	—
125	#358	—	47	8	no	—
126	#346	—	47	2	no	—
127	C₄×C₁₆	—	47	2	no	—
128	C₄₇	#33458	1	1	yes	characteristic, complemented, Sylow
129	C₂²×Q₈	—	94	2	no	—
130	C₂×Q₁₆	—	47	8	no	—
131	C₂×Q₁₆	—	47	4	no	—
132	C₂×Q₁₆	—	94	2	no	—
133	C₂×Q₁₆	—	94	4	no	—
134	C₂×M₁₆	—	47	1	no	—
135	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
136	C₄×Q₈	—	94	8	no	—
137	C₄×Q₈	—	188	4	no	—
138	#117	—	2	2	no	—
139	#117	—	47	1	no	—
140	#117	—	188	4	no	—
141	Q₃₂	—	47	8	no	—
142	Q₃₂	—	47	8	no	—
143	M₃₂	—	94	2	no	—
144	C₂×C₁₆	—	47	2	no	—
145	C₂×C₁₆	C₄×D₄₇	1	4	yes	—
146	C₂×C₁₆	C₂²×D₄₇	1	1	yes	characteristic
147	C₂×C₁₆	C₂²×D₄₇	1	1	yes	characteristic
148	#107	C₂²×D₄₇	1	4	yes	—
149	#106	—	2	2	no	—
150	#103	—	94	8	no	—
151	#97	—	47	2	no	—
152	C₄×C₈	—	47	2	no	—
153	C₂×Q₈	—	94	8	no	—
154	C₂×Q₈	—	94	4	no	—
155	C₂×Q₈	—	188	2	no	—
156	C₂×Q₈	—	188	4	no	—
157	C₂²×C₄	—	188	2	no	—
158	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
159	Q₁₆	—	47	8	no	—
160	Q₁₆	—	94	8	no	—
161	Q₁₆	—	94	4	no	—
162	M₁₆	—	94	2	no	—
163	C₂×C₈	—	47	2	no	—
164	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
165	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
166	C₂×C₈	#10625354	1	2	yes	—
167	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
168	C₄⋊C₄	—	2	4	no	—
169	C₄⋊C₄	—	47	2	no	—
170	C₄⋊C₄	—	188	8	no	—
171	C₂²⋊C₄	—	4	2	no	—
172	C₂²⋊C₄	—	47	2	no	—
173	C₄²	—	4	2	no	—
174	C₄²	—	47	2	no	—
175	C₄²	—	188	8	no	—
176	C₁₆	—	94	2	no	—
177	C₁₆	C₂×C₄×D₄₇	1	2	yes	—
178	C₁₆	C₂×C₄×D₄₇	1	2	yes	—
179	C₂³	D₈×D₄₇	1	1	yes	characteristic
180	Q₈	—	94	8	no	—
181	Q₈	—	188	4	no	—
182	Q₈	—	188	8	no	—
183	C₂×C₄	—	4	4	no	—
184	C₂×C₄	—	47	4	no	—
185	C₂×C₄	—	188	4	no	—
186	C₂×C₄	—	188	8	no	—
187	C₂×C₄	#13093986	1	2	yes	—
188	C₂×C₄	D₈×D₄₇	1	1	yes	characteristic
189	C₂×C₄	#13094047	1	2	yes	—
190	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
191	C₈	—	94	2	no	—
192	C₈	#13093986	1	2	yes	—
193	C₈	C₂²×C₄×D₄₇	1	1	yes	characteristic
194	C₈	C₂×#10625354	1	1	yes	characteristic
195	C₂²	—	2	2	no	—
196	C₂²	#15689205	1	2	yes	—
197	C₂²	C₂×D₈×D₄₇	1	1	yes	characteristic
198	C₄	—	8	4	no	—
199	C₄	—	94	4	no	—
200	C₄	—	188	8	no	—
201	C₄	#15689205	1	2	yes	—
202	C₄	C₂×#13093986	1	1	yes	characteristic
203	C₄	C₂×#13094047	1	1	yes	characteristic
204	C₂	—	2	2	no	—
205	C₂	C₂×#15689205	1	1	yes	characteristic
206	C₂	#20256538	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₃₇₆

C₂

yes

maximal, characteristic

#20252255

C₂

yes

maximal

#20245258

C₂

yes

maximal, complemented

#20245246

C₂

yes

maximal, complemented

#20245107

C₂

yes

maximal

#20244984

C₂

yes

maximal, characteristic

C₄₇×#2158

C₂

yes

maximal, characteristic

C₂²×Dic₁₈₈

C₂²

yes

—

#15689600

—

C₂×Dic₃₇₆

C₄

yes

complemented

C₂×Dic₃₇₆

C₂²

yes

—

C₂×Dic₃₇₆

C₂²

yes

—

C₂×#13093964

C₂²

yes

characteristic

#15689229

—

#15689217

C₂²

yes

—

#15689131

—

#15689078

C₂²

yes

characteristic

#15688926

C₂²

yes

—

#15688898

C₂²

yes

—

C₁₆×Dic₄₇

C₂²

yes

—

C₂²×C₁₆×C₄₇

C₂²

yes

characteristic

C₄₇×#427

C₂²

yes

—

C₄₇×#367

C₂²

yes

—

C₄₇×#366

C₂²

yes

—

C₂²×Dic₉₄

—

C₂×Dic₁₈₈

—

C₂×Dic₁₈₈

—

C₂×Dic₁₈₈

C₂×C₄

yes

—

C₂×Dic₁₈₈

C₂³

yes

—

C₂×#10625329

D₄

yes

characteristic

C₄×Dic₉₄

—

#13093989

—

#13093983

D₄

yes

characteristic

#13093980

—

Dic₃₇₆

C₂×C₄

yes

—

Dic₃₇₆

C₂×C₄

yes

—

#13093964

—

C₂×C₄₇⋊C₁₆

C₂³

yes

—

#13093947

—

#13093934

D₄

yes

—

C₈×Dic₄₇

C₂³

yes

—

C₂²×C₈×C₄₇

C₂³

yes

characteristic

#117×C₄₇

—

C₂×C₁₆×C₄₇

C₂×C₄

yes

—

C₂×C₁₆×C₄₇

C₂³

yes

characteristic

C₂×C₁₆×C₄₇

C₂³

yes

characteristic

#107×C₄₇

C₂³

yes

—

#106×C₄₇

—

C₂²×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂²×Dic₄₇

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₂×C₈×C₄₇

C₂²×C₄

yes

—

C₂×C₈×C₄₇

C₂×D₄

yes

characteristic

C₂×C₈×C₄₇

#41

yes

—

C₂×C₈×C₄₇

C₂⁴

yes

characteristic

C₄⋊C₄×C₄₇

—

C₂²⋊C₄×C₄₇

—

C₄²×C₄₇

—

C₂²⋊Dic₄₇

D₈

yes

—

C₄⋊Dic₄₇

D₈

yes

—

Dic₄₇⋊C₄

—

C₄×Dic₄₇

—

C₄×Dic₄₇

C₂×D₄

yes

—

#10625329

—

C₂×C₄₇⋊C₈

C₂×D₄

yes

—

Dic₁₈₈

—

Dic₁₈₈

—

Dic₁₈₈

C₂²×C₄

yes

—

C₁₆×C₄₇

C₂²×C₄

yes

—

C₁₆×C₄₇

C₂²×C₄

yes

—

C₄₇⋊C₁₆

—

C₂³×C₄₇

C₂×D₈

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₄×D₄

yes

—

C₂×C₄×C₄₇

C₂×D₈

yes

characteristic

C₂×C₄×C₄₇

#135

yes

—

C₂×C₄×C₄₇

C₂²×D₄

yes

characteristic

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

C₂×D₈

yes

—

Dic₉₄

—

Dic₉₄

—

Dic₉₄

—

C₈×C₄₇

C₄×D₄

yes

—

C₈×C₄₇

C₂³×C₄

yes

characteristic

C₈×C₄₇

C₂×#41

yes

characteristic

C₄₇⋊C₈

—

#33458

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

C₄×D₈

yes

—

C₂²×C₄₇

C₂²×D₈

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

C₄×D₈

yes

—

C₄×C₄₇

C₂×C₄×D₄

yes

characteristic

C₄×C₄₇

C₂×#135

yes

characteristic

Dic₄₇

—

100

Dic₄₇

—

101

C₂²×Q₃₂

—

102

#2950

—

103

#2176

—

104

C₄×Q₃₂

—

105

#2158

D₄₇

yes

characteristic

106

#2141

—

107

#2108

—

108

C₂×C₄₇

—

109

C₂×C₄₇

C₂×C₄×D₈

yes

characteristic

110

C₂×C₄₇

#3417

yes

—

111

C₂²×Q₁₆

—

112

#519

—

113

C₂×Q₃₂

—

114

C₂×Q₃₂

—

115

C₂×Q₃₂

—

116

C₂×M₃₂

—

117

C₂²×C₁₆

C₂×D₄₇

yes

characteristic

118

#441

—

119

C₄×Q₁₆

—

120

#427

C₂×D₄₇

yes

—

121

#419

—

122

#405

—

123

#367

C₂×D₄₇

yes

—

124

#366

C₂×D₄₇

yes

—

125

#358

—

126

#346

—

127

C₄×C₁₆

—

128

C₄₇

#33458

yes

characteristic, complemented, Sylow

129

C₂²×Q₈

—

130

C₂×Q₁₆

—

131

C₂×Q₁₆

—

132

C₂×Q₁₆

—

133

C₂×Q₁₆

—

134

C₂×M₁₆

—

135

C₂²×C₈

C₂²×D₄₇

yes

characteristic

136

C₄×Q₈

—

137

C₄×Q₈

—

188

—

138

#117

—

139

#117

—

140

#117

—

188

—

141

Q₃₂

—

142

Q₃₂

—

143

M₃₂

—

144

C₂×C₁₆

—

145

C₂×C₁₆

C₄×D₄₇

yes

—

146

C₂×C₁₆

C₂²×D₄₇

yes

characteristic

147

C₂×C₁₆

C₂²×D₄₇

yes

characteristic

148

#107

C₂²×D₄₇

yes

—

149

#106

—

150

#103

—

151

#97

—

152

C₄×C₈

—

153

C₂×Q₈

—

154

C₂×Q₈

—

155

C₂×Q₈

—

188

—

156

C₂×Q₈

—

188

—

157

C₂²×C₄

—

188

—

158

C₂²×C₄

D₄×D₄₇

yes

characteristic

159

Q₁₆

—

160

Q₁₆

—

161

Q₁₆

—

162

M₁₆

—

163

C₂×C₈

—

164

C₂×C₈

C₂×C₄×D₄₇

yes

—

165

C₂×C₈

D₄×D₄₇

yes

characteristic

166

C₂×C₈

#10625354

yes

—

167

C₂×C₈

C₂³×D₄₇

yes

characteristic

168

C₄⋊C₄

—

169

C₄⋊C₄

—

170

C₄⋊C₄

—

188

—

171

C₂²⋊C₄

—

172

C₂²⋊C₄

—

173

C₄²

—

174

C₄²

—

175

C₄²

—

188

—

176

C₁₆

—

177

C₁₆

C₂×C₄×D₄₇

yes

—

178

C₁₆

C₂×C₄×D₄₇

yes

—

179

C₂³

D₈×D₄₇

yes

characteristic

180

Q₈

—

181

Q₈

—

188

—

182

Q₈

—

188

—

183

C₂×C₄

—

184

C₂×C₄

—

185

C₂×C₄

—

188

—

186

C₂×C₄

—

188

—

187

C₂×C₄

#13093986

yes

—

188

C₂×C₄

D₈×D₄₇

yes

characteristic

189

C₂×C₄

#13094047

yes

—

190

C₂×C₄

C₂×D₄×D₄₇

yes

characteristic

191

C₈

—

192

C₈

#13093986

yes

—

193

C₈

C₂²×C₄×D₄₇

yes

characteristic

194

C₈

C₂×#10625354

yes

characteristic

195

C₂²

—

196

C₂²

#15689205

yes

—

197

C₂²

C₂×D₈×D₄₇

yes

characteristic

198

C₄

—

199

C₄

—

200

C₄

—

188

—

201

C₄

#15689205

yes

—

202

C₄

C₂×#13093986

yes

characteristic

203

C₄

C₂×#13094047

yes

characteristic

204

C₂

—

205

C₂

C₂×#15689205

yes

characteristic

206

C₂

#20256538

yes

—

	subgroup	G/H
center	C₂²	C₂×D₈×D₄₇
commutator	C₈×C₄₇	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₄₇
Fitting	C₄₇×#2158	C₂
socle	C₂²×C₄₇	C₂²×D₈
prime radical	C₂³×C₄₇	C₂×D₈
cosocle	C₂×C₈×C₄₇	C₂⁴
hypercenter	#2158	D₄₇
hypocenter	C₄₇	#33458
LCS	G ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2158

subgroup

G/H

center

C₂²

C₂×D₈×D₄₇

commutator

C₈×C₄₇

C₂³×C₄

Frattini

C₂×C₈

C₂³×D₄₇

Fitting

C₄₇×#2158

C₂

socle

C₂²×C₄₇

C₂²×D₈

prime radical

C₂³×C₄₇

C₂×D₈

cosocle

C₂×C₈×C₄₇

C₂⁴

D₄₇

C₄₇

G ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2158

N	H ≅ G/N	type
3. #20245258	C₂	semidirect
4. #20245246	C₂	semidirect
10. C₂×Dic₃₇₆	C₄	semidirect
128. C₄₇	#33458	semidirect

H ≅ G/N

type

3. #20245258

C₂

semidirect

4. #20245246

C₂

semidirect

10. C₂×Dic₃₇₆

C₄

semidirect

128. C₄₇

#33458

semidirect

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

subgroup	index
C₂²×Dic₃₇₆	2
#20252255	2
#20245258	2
#20245246	2
#20245107	2
#20244984	2
C₄₇×#2158	2
#33458	47

subgroup

index

C₂²×Dic₃₇₆

C₄₇×#2158

Subgroups — #26369655

Conjugacy classes of subgroups

Complements

Maximal subgroups