Group #26116305 — subgroups

Subgroups — #26116305

Order 12032.

434 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 434 conjugacy classes of subgroups — too many to draw. Only its 151 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	C₄×#13094016	C₂	1	1	yes	maximal, characteristic
2	#20243505	C₂	1	1	yes	maximal, characteristic
3	#20241228	C₂	1	4	yes	maximal
4	C₄₇×#1744	C₂	1	1	yes	maximal, characteristic
5	C₂×#13094016	C₄	1	1	yes	characteristic
6	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
7	#15689021	C₄	1	2	yes	—
8	#15689007	—	2	2	no	—
9	#15688989	C₄	1	1	yes	characteristic
10	C₄×Dic₄₇⋊C₄	—	2	2	no	—
11	C₄×C₄⋊Dic₄₇	C₂²	1	2	yes	—
12	#15688878	—	2	2	no	—
13	C₂×C₄×C₈×C₄₇	C₂²	1	2	yes	—
14	C₄³×C₄₇	C₂²	1	1	yes	characteristic
15	C₄₇×#336	—	2	2	no	—
16	C₂²×C₄₇⋊C₈	—	2	2	no	—
17	#13094016	—	2	4	no	—
18	C₂×C₄⋊Dic₄₇	C₂×C₄	1	2	yes	—
19	C₂×C₄⋊Dic₄₇	C₂×C₄	1	2	yes	—
20	C₂×Dic₄₇⋊C₄	—	2	2	no	—
21	C₂×Dic₄₇⋊C₄	—	2	2	no	—
22	C₂×C₄×Dic₄₇	—	2	2	no	—
23	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
24	C₄₇⋊#95	—	2	2	no	—
25	C₄₇⋊#95	—	2	2	no	—
26	C₄₇⋊#95	—	2	2	no	—
27	C₂²×C₈×C₄₇	—	2	2	no	—
28	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄²×C₄₇	D₄	1	2	yes	—
31	C₂×C₄²×C₄₇	D₄	1	2	yes	—
32	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
33	C₄×C₈×C₄₇	—	2	2	no	—
34	C₄×C₈×C₄₇	Q₈	1	4	yes	—
35	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
36	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
37	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
38	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
39	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
40	C₂²×Dic₄₇	—	2	2	no	—
41	C₂²×Dic₄₇	—	2	2	no	—
42	C₂×C₈×C₄₇	—	2	4	no	—
43	C₂×C₈×C₄₇	—	4	4	no	—
44	C₄²×C₄₇	—	2	2	no	—
45	C₄²×C₄₇	—	2	2	no	—
46	C₄²×C₄₇	—	2	4	no	—
47	C₄²×C₄₇	—	4	2	no	—
48	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
49	C₄²×C₄₇	#41	1	2	yes	—
50	C₄⋊Dic₄₇	—	2	2	no	—
51	C₄⋊Dic₄₇	—	2	4	no	—
52	C₄⋊Dic₄₇	M₁₆	1	4	yes	—
53	Dic₄₇⋊C₄	—	2	8	no	—
54	Dic₄₇⋊C₄	—	4	4	no	—
55	C₄×Dic₄₇	—	4	4	no	—
56	C₂×C₄₇⋊C₈	—	4	2	no	—
57	C₂×C₄₇⋊C₈	—	4	4	no	—
58	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
59	C₂×C₄×C₄₇	—	2	2	no	—
60	C₂×C₄×C₄₇	—	2	2	no	—
61	C₂×C₄×C₄₇	—	2	2	no	—
62	C₂×C₄×C₄₇	—	2	2	no	—
63	C₂×C₄×C₄₇	—	2	2	no	—
64	C₂×C₄×C₄₇	—	2	2	no	—
65	C₂×C₄×C₄₇	—	2	2	no	—
66	C₂×C₄×C₄₇	C₄×Q₈	1	2	yes	—
67	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
68	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
69	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
70	C₂×C₄×C₄₇	C₂×M₁₆	1	2	yes	—
71	C₂×C₄×C₄₇	#131	1	2	yes	—
72	C₂×C₄×C₄₇	#135	1	4	yes	—
73	C₂×Dic₄₇	—	2	4	no	—
74	C₂×Dic₄₇	—	4	2	no	—
75	C₂×Dic₄₇	—	4	2	no	—
76	C₂×Dic₄₇	—	4	2	no	—
77	C₂×Dic₄₇	—	4	2	no	—
78	C₂×Dic₄₇	—	4	2	no	—
79	C₈×C₄₇	—	4	4	no	—
80	C₄₇⋊C₈	—	8	2	no	—
81	#9875	—	47	1	no	maximal, Sylow
82	C₂²×C₄₇	#391	1	1	yes	characteristic
83	C₂²×C₄₇	#407	1	1	yes	characteristic
84	C₂²×C₄₇	#408	1	1	yes	characteristic
85	C₂²×C₄₇	#416	1	2	yes	—
86	C₂²×C₄₇	#421	1	2	yes	—
87	C₄×C₄₇	—	2	2	no	—
88	C₄×C₄₇	—	4	2	no	—
89	C₄×C₄₇	—	4	2	no	—
90	C₄×C₄₇	#446	1	4	yes	—
91	C₄×C₄₇	#488	1	2	yes	—
92	C₄×C₄₇	#499	1	2	yes	—
93	Dic₄₇	—	4	4	no	—
94	Dic₄₇	—	8	2	no	—
95	C₄×#125	—	47	1	no	—
96	#1831	—	47	1	no	—
97	#1744	D₄₇	1	1	yes	characteristic
98	#1279	—	47	4	no	—
99	C₂×C₄₇	#1482	1	2	yes	—
100	C₂×C₄₇	#1488	1	2	yes	—
101	C₂×C₄₇	#1985	1	1	yes	characteristic
102	C₂×C₄₇	#1991	1	1	yes	characteristic
103	C₂×C₄₇	#1995	1	1	yes	characteristic
104	C₂×#125	—	47	1	no	—
105	C₂×C₄⋊C₈	—	47	2	no	—
106	C₂×C₄×C₈	C₂×D₄₇	1	2	yes	—
107	#389	—	47	2	no	—
108	#387	—	94	2	no	—
109	#382	—	47	1	no	—
110	C₄×C₄⋊C₄	—	47	2	no	—
111	C₄×C₄⋊C₄	—	94	2	no	—
112	C₄³	C₂×D₄₇	1	1	yes	characteristic
113	#336	—	2	2	no	—
114	#336	—	94	2	no	—
115	C₄₇	#9875	1	1	yes	characteristic, complemented, Sylow
116	C₂²×C₈	—	2	2	no	—
117	C₂²×C₈	—	94	2	no	—
118	#125	—	94	4	no	—
119	C₂×C₄⋊C₄	—	47	2	no	—
120	C₂×C₄⋊C₄	—	47	2	no	—
121	C₂×C₄⋊C₄	—	94	2	no	—
122	C₂×C₄⋊C₄	—	94	2	no	—
123	C₂×C₄²	—	94	2	no	—
124	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
125	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
126	C₂×C₄²	D₁₈₈	1	2	yes	—
127	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
128	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
129	C₄⋊C₈	—	94	4	no	—
130	C₄×C₈	—	2	2	no	—
131	C₄×C₈	Dic₉₄	1	4	yes	—
132	#95	—	94	2	no	—
133	#95	—	94	2	no	—
134	#95	—	94	2	no	—
135	C₂²×C₄	—	94	2	no	—
136	C₂²×C₄	—	94	2	no	—
137	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
138	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
139	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
140	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
141	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
142	C₂×C₈	—	2	4	no	—
143	C₂×C₈	—	4	4	no	—
144	C₂×C₈	—	188	2	no	—
145	C₂×C₈	—	188	4	no	—
146	C₄⋊C₄	—	47	4	no	—
147	C₄⋊C₄	—	94	8	no	—
148	C₄⋊C₄	—	94	2	no	—
149	C₄⋊C₄	—	94	4	no	—
150	C₄⋊C₄	—	188	4	no	—
151	C₄²	—	2	2	no	—
152	C₄²	—	2	2	no	—
153	C₄²	—	2	4	no	—
154	C₄²	—	4	2	no	—
155	C₄²	—	188	4	no	—
156	C₄²	C₂×Dic₉₄	1	2	yes	—
157	C₄²	#10625350	1	2	yes	—
158	C₂³	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	94	4	no	—
167	C₂×C₄	—	188	2	no	—
168	C₂×C₄	—	188	2	no	—
169	C₂×C₄	—	188	2	no	—
170	C₂×C₄	—	188	2	no	—
171	C₂×C₄	—	188	2	no	—
172	C₂×C₄	C₄×Dic₉₄	1	2	yes	—
173	C₂×C₄	#13094003	1	2	yes	—
174	C₂×C₄	#13094012	1	1	yes	characteristic
175	C₂×C₄	#13094024	1	1	yes	characteristic
176	C₂×C₄	M₁₆×D₄₇	1	2	yes	—
177	C₂×C₄	#13094032	1	2	yes	—
178	C₂×C₄	#13094046	1	4	yes	—
179	C₈	—	4	4	no	—
180	C₈	—	376	2	no	—
181	C₂²	#15689029	1	1	yes	characteristic
182	C₂²	#15689086	1	1	yes	characteristic
183	C₂²	#15689091	1	1	yes	characteristic
184	C₂²	#15689120	1	2	yes	—
185	C₂²	#15689141	1	2	yes	—
186	C₄	—	2	2	no	—
187	C₄	—	4	2	no	—
188	C₄	—	4	2	no	—
189	C₄	—	376	2	no	—
190	C₄	—	188	4	no	—
191	C₄	#15689251	1	4	yes	—
192	C₄	#15689478	1	2	yes	—
193	C₄	#15689525	1	2	yes	—
194	C₂	#20241797	1	2	yes	—
195	C₂	#20241823	1	2	yes	—
196	C₂	#20244328	1	1	yes	characteristic
197	C₂	#20244353	1	1	yes	characteristic
198	C₂	#20244369	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#13094016

C₂

yes

maximal, characteristic

#20243505

C₂

yes

maximal, characteristic

#20241228

C₂

yes

maximal

C₄₇×#1744

C₂

yes

maximal, characteristic

C₂×#13094016

C₄

yes

characteristic

C₂×C₄⋊(C₄₇⋊C₈)

C₂²

yes

—

#15689021

C₄

yes

—

#15689007

—

#15688989

C₄

yes

characteristic

C₄×Dic₄₇⋊C₄

—

C₄×C₄⋊Dic₄₇

C₂²

yes

—

#15688878

—

C₂×C₄×C₈×C₄₇

C₂²

yes

—

C₄³×C₄₇

C₂²

yes

characteristic

C₄₇×#336

—

C₂²×C₄₇⋊C₈

—

#13094016

—

C₂×C₄⋊Dic₄₇

C₂×C₄

yes

—

C₂×C₄⋊Dic₄₇

C₂×C₄

yes

—

C₂×Dic₄₇⋊C₄

—

C₂×Dic₄₇⋊C₄

—

C₂×C₄×Dic₄₇

—

C₄⋊(C₄₇⋊C₈)

—

C₄₇⋊#95

—

C₄₇⋊#95

—

C₄₇⋊#95

—

C₂²×C₈×C₄₇

—

C₂×C₄²×C₄₇

C₂×C₄

yes

characteristic

C₂×C₄²×C₄₇

C₂×C₄

yes

characteristic

C₂×C₄²×C₄₇

D₄

yes

—

C₂×C₄²×C₄₇

D₄

yes

—

C₂×C₄²×C₄₇

C₂³

yes

characteristic

C₄×C₈×C₄₇

—

C₄×C₈×C₄₇

Q₈

yes

—

C₂²×C₄×C₄₇

C₂²⋊C₄

yes

—

C₂²×C₄×C₄₇

C₂²⋊C₄

yes

—

C₂²×C₄×C₄₇

C₂²×C₄

yes

characteristic

C₂²×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂²×Dic₄₇

—

C₂²×Dic₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

C₂×Q₈

yes

—

C₄²×C₄₇

#41

yes

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

M₁₆

yes

—

Dic₄₇⋊C₄

—

Dic₄₇⋊C₄

—

C₄×Dic₄₇

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₄×Q₈

yes

—

C₂×C₄×C₄₇

C₂²⋊Q₈

yes

—

C₂×C₄×C₄₇

#124

yes

characteristic

C₂×C₄×C₄₇

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₄₇

C₂×M₁₆

yes

—

C₂×C₄×C₄₇

#131

yes

—

C₂×C₄×C₄₇

#135

yes

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₈×C₄₇

—

C₄₇⋊C₈

—

—

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

—

C₂²×C₄₇

#421

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#446

yes

—

C₄×C₄₇

#488

yes

—

C₄×C₄₇

#499

yes

—

Dic₄₇

—

Dic₄₇

—

C₄×#125

—

#1831

—

#1744

D₄₇

yes

characteristic

#1279

—

C₂×C₄₇

#1482

yes

—

100

C₂×C₄₇

#1488

yes

—

101

C₂×C₄₇

#1985

yes

characteristic

102

C₂×C₄₇

#1991

yes

characteristic

103

C₂×C₄₇

#1995

yes

characteristic

104

C₂×#125

—

105

C₂×C₄⋊C₈

—

106

C₂×C₄×C₈

C₂×D₄₇

yes

—

107

#389

—

108

#387

—

109

#382

—

110

C₄×C₄⋊C₄

—

111

C₄×C₄⋊C₄

—

112

C₄³

C₂×D₄₇

yes

characteristic

113

#336

—

114

#336

—

115

C₄₇

#9875

yes

characteristic, complemented, Sylow

116

C₂²×C₈

—

117

C₂²×C₈

—

118

#125

—

119

C₂×C₄⋊C₄

—

120

C₂×C₄⋊C₄

—

121

C₂×C₄⋊C₄

—

122

C₂×C₄⋊C₄

—

123

C₂×C₄²

—

124

C₂×C₄²

C₄×D₄₇

yes

characteristic

125

C₂×C₄²

C₄×D₄₇

yes

characteristic

126

C₂×C₄²

D₁₈₈

yes

—

127

C₂×C₄²

C₂²⋊D₄₇

yes

—

128

C₂×C₄²

C₂²×D₄₇

yes

characteristic

129

C₄⋊C₈

—

130

C₄×C₈

—

131

C₄×C₈

Dic₉₄

yes

—

132

#95

—

133

#95

—

134

#95

—

135

C₂²×C₄

—

136

C₂²×C₄

—

137

C₂²×C₄

(C₂×D₄₇)⋊C₄

yes

—

138

C₂²×C₄

(C₂×D₄₇)⋊C₄

yes

—

139

C₂²×C₄

C₂×C₄×D₄₇

yes

characteristic

140

C₂²×C₄

C₂×D₁₈₈

yes

characteristic

141

C₂²×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

142

C₂×C₈

—

143

C₂×C₈

—

144

C₂×C₈

—

188

—

145

C₂×C₈

—

188

—

146

C₄⋊C₄

—

147

C₄⋊C₄

—

148

C₄⋊C₄

—

149

C₄⋊C₄

—

150

C₄⋊C₄

—

188

—

151

C₄²

—

152

C₄²

—

153

C₄²

—

154

C₄²

—

155

C₄²

—

188

—

156

C₄²

C₂×Dic₉₄

yes

—

157

C₄²

#10625350

yes

—

158

C₂³

C₂×(C₂×D₄₇)⋊C₄

yes

characteristic

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

188

—

168

C₂×C₄

—

188

—

169

C₂×C₄

—

188

—

170

C₂×C₄

—

188

—

171

C₂×C₄

—

188

—

172

C₂×C₄

C₄×Dic₉₄

yes

—

173

C₂×C₄

#13094003

yes

—

174

C₂×C₄

#13094012

yes

characteristic

175

C₂×C₄

#13094024

yes

characteristic

176

C₂×C₄

M₁₆×D₄₇

yes

—

177

C₂×C₄

#13094032

yes

—

178

C₂×C₄

#13094046

yes

—

179

C₈

—

180

C₈

—

376

—

181

C₂²

#15689029

yes

characteristic

182

C₂²

#15689086

yes

characteristic

183

C₂²

#15689091

yes

characteristic

184

C₂²

#15689120

yes

—

185

C₂²

#15689141

yes

—

186

C₄

—

187

C₄

—

188

C₄

—

189

C₄

—

376

—

190

C₄

—

188

—

191

C₄

#15689251

yes

—

192

C₄

#15689478

yes

—

193

C₄

#15689525

yes

—

194

C₂

#20241797

yes

—

195

C₂

#20241823

yes

—

196

C₂

#20244328

yes

characteristic

197

C₂

#20244353

yes

characteristic

198

C₂

#20244369

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₁₈₈
commutator	C₂²×C₄×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1744	C₂
socle, prime radical	C₂³×C₄₇	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#1744	D₄₇
hypocenter	C₄₇	#9875
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1744

subgroup

G/H

center

C₂²×C₄

C₂×D₁₈₈

commutator

C₂²×C₄×C₄₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₄₇

Fitting

C₄₇×#1744

C₂

socle, prime radical

C₂³×C₄₇

C₂×C₂²⋊C₄

cosocle

C₂×C₄²×C₄₇

C₂³

D₄₇

C₄₇

G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1744

N	H ≅ G/N	type
115. C₄₇	#9875	semidirect

H ≅ G/N

type

115. C₄₇

#9875

semidirect

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

subgroup	index
C₄×#13094016	2
#20243505	2
#20241228	2
C₄₇×#1744	2
#9875	47

subgroup

index

C₄₇×#1744

Subgroups — #26116305

Conjugacy classes of subgroups

Complements

Maximal subgroups