Group #26109234 — subgroups

Subgroups — #26109234

Order 12032.

362 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 113 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	C₂×Q₈×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic
2	#20242149	C₂	1	2	yes	maximal
3	#20241732	C₂	1	1	yes	maximal, characteristic
4	#20241257	C₂	1	2	yes	maximal
5	C₄₇×#1467	C₂	1	1	yes	maximal, characteristic
6	Q₈×C₄₇⋊C₈	—	2	2	no	—
7	Q₈×C₄₇⋊C₈	—	2	2	no	—
8	#15689180	C₂²	1	2	yes	—
9	C₂×C₄⋊(C₄₇⋊C₈)	C₄	1	2	yes	—
10	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₄₇⋊C₈	C₄	1	2	yes	—
12	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
13	#15688875	—	2	2	no	—
14	#15688873	—	2	2	no	—
15	#15688849	—	2	2	no	—
16	C₂×C₄×Q₈×C₄₇	C₂²	1	1	yes	characteristic
17	C₄₇×#432	C₂²	1	2	yes	—
18	C₄₇×#324	—	2	2	no	—
19	C₂²×C₄₇⋊C₈	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
21	C₂²×C₄₇⋊C₈	C₂×C₄	1	1	yes	characteristic
22	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
23	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
24	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
25	C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
26	#13093935	—	2	2	no	—
27	C₄×C₄₇⋊C₈	—	2	2	no	—
28	C₄×C₄₇⋊C₈	—	2	2	no	—
29	C₄×C₄₇⋊C₈	D₄	1	2	yes	—
30	C₄×C₄₇⋊C₈	D₄	1	2	yes	—
31	C₂²×Q₈×C₄₇	Q₈	1	1	yes	characteristic
32	C₄×Q₈×C₄₇	—	2	2	no	—
33	C₄×Q₈×C₄₇	—	2	2	no	—
34	C₂×C₄⋊C₄×C₄₇	D₄	1	2	yes	—
35	C₂×C₄⋊C₄×C₄₇	Q₈	1	1	yes	characteristic
36	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
37	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
38	C₄⋊C₈×C₄₇	—	2	2	no	—
39	C₂²⋊C₈×C₄₇	—	2	2	no	—
40	#97×C₄₇	—	2	2	no	—
41	C₂×Q₈×C₄₇	—	2	4	no	—
42	C₂×Q₈×C₄₇	—	2	2	no	—
43	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
44	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
45	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
46	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
47	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
48	C₂×C₈×C₄₇	—	4	2	no	—
49	C₄⋊C₄×C₄₇	—	2	4	no	—
50	C₄⋊C₄×C₄₇	—	2	2	no	—
51	C₄²×C₄₇	—	2	2	no	—
52	C₄²×C₄₇	—	2	2	no	—
53	C₄²×C₄₇	C₂×D₄	1	2	yes	—
54	C₄²×C₄₇	#41	1	2	yes	—
55	C₂×C₄₇⋊C₈	—	2	2	no	—
56	C₂×C₄₇⋊C₈	—	2	2	no	—
57	C₂×C₄₇⋊C₈	—	4	1	no	—
58	C₂×C₄₇⋊C₈	—	4	2	no	—
59	C₂×C₄₇⋊C₈	—	4	2	no	—
60	C₂×C₄₇⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
61	C₂×C₄₇⋊C₈	C₂²⋊C₄	1	2	yes	—
62	C₂×C₄₇⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
63	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
64	Q₈×C₄₇	—	2	8	no	—
65	C₂×C₄×C₄₇	—	2	2	no	—
66	C₂×C₄×C₄₇	—	2	2	no	—
67	C₂×C₄×C₄₇	—	2	2	no	—
68	C₂×C₄×C₄₇	—	2	1	no	—
69	C₂×C₄×C₄₇	—	2	1	no	—
70	C₂×C₄×C₄₇	—	2	2	no	—
71	C₂×C₄×C₄₇	—	4	1	no	—
72	C₂×C₄×C₄₇	—	4	2	no	—
73	C₂×C₄×C₄₇	#108	1	4	yes	—
74	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
76	C₂×C₄×C₄₇	#121	1	2	yes	—
77	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
78	C₈×C₄₇	—	8	2	no	—
79	C₄₇⋊C₈	—	2	1	no	—
80	C₄₇⋊C₈	—	2	2	no	—
81	C₄₇⋊C₈	—	4	2	no	—
82	C₄₇⋊C₈	—	8	2	no	—
83	C₄₇⋊C₈	#101	1	2	yes	—
84	#8623	—	47	1	no	maximal, Sylow
85	C₂²×C₄₇	—	2	1	no	—
86	C₂²×C₄₇	—	2	1	no	—
87	C₂²×C₄₇	#380	1	1	yes	characteristic
88	C₂²×C₄₇	C₂×#108	1	2	yes	—
89	C₄×C₄₇	—	2	1	no	—
90	C₄×C₄₇	—	2	4	no	—
91	C₄×C₄₇	—	2	2	no	—
92	C₄×C₄₇	—	2	2	no	—
93	C₄×C₄₇	—	4	2	no	—
94	C₄×C₄₇	C₂×#101	1	1	yes	characteristic
95	C₄×C₄₇	#413	1	1	yes	characteristic
96	C₂×C₈×Q₈	—	47	1	no	—
97	#1568	—	47	2	no	—
98	#1467	—	47	1	no	—
99	#1467	D₄₇	1	1	yes	characteristic
100	#1290	—	47	2	no	—
101	C₂×C₄₇	—	2	2	no	—
102	C₂×C₄₇	#1801	1	1	yes	characteristic
103	C₂×C₄₇	#1810	1	1	yes	characteristic
104	C₂×C₄₇	#1812	1	1	yes	characteristic
105	C₂×C₄×Q₈	C₂×D₄₇	1	1	yes	characteristic
106	C₈×Q₈	—	94	2	no	—
107	C₈×Q₈	—	94	2	no	—
108	#432	—	47	2	no	—
109	#432	C₂×D₄₇	1	2	yes	—
110	C₂×C₄⋊C₈	—	47	2	no	—
111	C₂×C₄⋊C₈	—	47	1	no	—
112	C₂×C₄×C₈	—	47	1	no	—
113	C₂×C₄×C₈	—	47	2	no	—
114	#335	—	94	2	no	—
115	#334	—	94	2	no	—
116	#324	—	2	2	no	—
117	#324	—	94	2	no	—
118	C₄₇	#8623	1	1	yes	characteristic, complemented, Sylow
119	C₂²×Q₈	Dic₉₄	1	1	yes	characteristic
120	C₂²×C₈	—	47	1	no	—
121	C₂²×C₈	—	47	1	no	—
122	C₂²×C₈	—	47	2	no	—
123	C₄×Q₈	—	2	2	no	—
124	C₄×Q₈	—	2	2	no	—
125	C₂×C₄⋊C₄	Dic₉₄	1	1	yes	characteristic
126	C₂×C₄⋊C₄	C₂²⋊D₄₇	1	2	yes	—
127	C₂×C₄²	C₄×D₄₇	1	2	yes	—
128	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
129	C₄⋊C₈	—	2	2	no	—
130	C₄⋊C₈	—	94	4	no	—
131	C₄⋊C₈	—	94	2	no	—
132	C₄⋊C₈	—	94	2	no	—
133	C₂²⋊C₈	—	2	2	no	—
134	C₂²⋊C₈	—	94	2	no	—
135	#97	—	2	2	no	—
136	#97	—	94	2	no	—
137	C₄×C₈	—	47	2	no	—
138	C₄×C₈	—	47	2	no	—
139	C₄×C₈	—	94	2	no	—
140	C₄×C₈	—	94	2	no	—
141	C₂×Q₈	—	2	4	no	—
142	C₂×Q₈	—	2	2	no	—
143	C₂²×C₄	Dic₄₇⋊C₄	1	2	yes	—
144	C₂²×C₄	Dic₄₇⋊C₄	1	2	yes	—
145	C₂²×C₄	C₂×Dic₉₄	1	1	yes	characteristic
146	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
147	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
148	C₂×C₈	—	4	2	no	—
149	C₂×C₈	—	47	1	no	—
150	C₂×C₈	—	47	1	no	—
151	C₂×C₈	—	47	2	no	—
152	C₂×C₈	—	94	2	no	—
153	C₂×C₈	—	94	2	no	—
154	C₂×C₈	—	188	1	no	—
155	C₂×C₈	—	188	2	no	—
156	C₂×C₈	—	188	2	no	—
157	C₄⋊C₄	—	2	4	no	—
158	C₄⋊C₄	—	2	2	no	—
159	C₄²	—	2	2	no	—
160	C₄²	—	2	2	no	—
161	C₄²	D₄×D₄₇	1	2	yes	—
162	C₄²	#10625352	1	2	yes	—
163	C₂³	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
164	Q₈	—	2	8	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	2	1	no	—
169	C₂×C₄	—	2	1	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	4	1	no	—
172	C₂×C₄	—	4	2	no	—
173	C₂×C₄	#13093959	1	4	yes	—
174	C₂×C₄	C₂²⋊C₄×D₄₇	1	1	yes	characteristic
175	C₂×C₄	#13093980	1	1	yes	characteristic
176	C₂×C₄	#13093996	1	2	yes	—
177	C₂×C₄	#13094001	1	2	yes	—
178	C₈	—	8	2	no	—
179	C₈	—	47	2	no	—
180	C₈	—	94	1	no	—
181	C₈	—	94	2	no	—
182	C₈	—	376	2	no	—
183	C₈	—	188	2	no	—
184	C₂²	—	2	1	no	—
185	C₂²	—	2	1	no	—
186	C₂²	#15688981	1	1	yes	characteristic
187	C₂²	C₂×#13093959	1	2	yes	—
188	C₄	—	2	1	no	—
189	C₄	—	2	4	no	—
190	C₄	—	2	2	no	—
191	C₄	—	2	2	no	—
192	C₄	—	4	2	no	—
193	C₄	#101×D₄₇	1	1	yes	characteristic
194	C₄	#15689106	1	1	yes	characteristic
195	C₂	—	2	2	no	—
196	C₂	#20243382	1	1	yes	characteristic
197	C₂	#20243424	1	1	yes	characteristic
198	C₂	#20243432	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Q₈×C₄₇⋊C₈

C₂

yes

maximal, characteristic

#20242149

C₂

yes

maximal

#20241732

C₂

yes

maximal, characteristic

#20241257

C₂

yes

maximal

C₄₇×#1467

C₂

yes

maximal, characteristic

Q₈×C₄₇⋊C₈

—

Q₈×C₄₇⋊C₈

—

#15689180

C₂²

yes

—

C₂×C₄⋊(C₄₇⋊C₈)

C₄

yes

—

C₂×C₄⋊(C₄₇⋊C₈)

C₂²

yes

characteristic

C₂×C₄×C₄₇⋊C₈

C₄

yes

—

C₂×C₄×C₄₇⋊C₈

C₂²

yes

—

—

—

C₂×C₄×Q₈×C₄₇

C₂²

yes

characteristic

C₄₇×#432

C₂²

yes

—

C₄₇×#324

—

C₂²×C₄₇⋊C₈

C₂×C₄

yes

characteristic

C₂²×C₄₇⋊C₈

C₂×C₄

yes

—

C₂²×C₄₇⋊C₈

C₂×C₄

yes

characteristic

C₄⋊(C₄₇⋊C₈)

—

C₄⋊(C₄₇⋊C₈)

—

C₄⋊(C₄₇⋊C₈)

—

C₂²⋊(C₄₇⋊C₈)

—

#13093935

—

C₄×C₄₇⋊C₈

—

C₄×C₄₇⋊C₈

—

C₄×C₄₇⋊C₈

D₄

yes

—

C₄×C₄₇⋊C₈

D₄

yes

—

C₂²×Q₈×C₄₇

Q₈

yes

characteristic

C₄×Q₈×C₄₇

—

C₄×Q₈×C₄₇

—

C₂×C₄⋊C₄×C₄₇

D₄

yes

—

C₂×C₄⋊C₄×C₄₇

Q₈

yes

characteristic

C₂×C₄²×C₄₇

C₂×C₄

yes

—

C₂×C₄²×C₄₇

C₂³

yes

characteristic

C₄⋊C₈×C₄₇

—

C₂²⋊C₈×C₄₇

—

#97×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂²×C₄×C₄₇

C₄⋊C₄

yes

—

C₂²×C₄×C₄₇

C₄⋊C₄

yes

—

C₂²×C₄×C₄₇

C₂²×C₄

yes

characteristic

C₂²×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₄₇

C₂×Q₈

yes

characteristic

C₂×C₈×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

C₂×D₄

yes

—

C₄²×C₄₇

#41

yes

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

C₂²⋊C₄

yes

characteristic

C₂×C₄₇⋊C₈

C₂²⋊C₄

yes

—

C₂×C₄₇⋊C₈

C₂²⋊C₄

yes

characteristic

C₂³×C₄₇

C₂×C₄⋊C₄

yes

characteristic

Q₈×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

#108

yes

—

C₂×C₄×C₄₇

C₂×C₂²⋊C₄

yes

C₂×C₄×C₄₇

yes

C₂×C₄×C₄₇

yes

—

C₂×C₄×C₄₇

C₂²⋊Q₈

yes

—

C₈×C₄₇

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₄₇⋊C₈

#101

yes

—

#8623

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

#380

yes

characteristic

C₂²×C₄₇

C₂×#108

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

yes

C₄×C₄₇

yes

C₂×C₈×Q₈

—

#1568

—

#1467

—

#1467

D₄₇

yes

characteristic

100

#1290

—

101

C₂×C₄₇

—

102

C₂×C₄₇

#1801

yes

characteristic

103

C₂×C₄₇

#1810

yes

characteristic

104

C₂×C₄₇

#1812

yes

characteristic

105

C₂×C₄×Q₈

C₂×D₄₇

yes

characteristic

106

C₈×Q₈

—

107

C₈×Q₈

—

108

#432

—

109

#432

C₂×D₄₇

yes

—

110

C₂×C₄⋊C₈

—

111

C₂×C₄⋊C₈

—

112

C₂×C₄×C₈

—

113

C₂×C₄×C₈

—

114

#335

—

115

#334

—

116

#324

—

117

#324

—

118

C₄₇

#8623

yes

characteristic, complemented, Sylow

119

C₂²×Q₈

Dic₉₄

yes

characteristic

120

C₂²×C₈

—

121

C₂²×C₈

—

122

C₂²×C₈

—

123

C₄×Q₈

—

124

C₄×Q₈

—

125

C₂×C₄⋊C₄

Dic₉₄

yes

characteristic

126

C₂×C₄⋊C₄

C₂²⋊D₄₇

yes

—

127

C₂×C₄²

C₄×D₄₇

yes

—

128

C₂×C₄²

C₂²×D₄₇

yes

characteristic

129

C₄⋊C₈

—

130

C₄⋊C₈

—

131

C₄⋊C₈

—

132

C₄⋊C₈

—

133

C₂²⋊C₈

—

134

C₂²⋊C₈

—

135

#97

—

136

#97

—

137

C₄×C₈

—

138

C₄×C₈

—

139

C₄×C₈

—

140

C₄×C₈

—

141

C₂×Q₈

—

142

C₂×Q₈

—

143

C₂²×C₄

Dic₄₇⋊C₄

yes

—

144

C₂²×C₄

Dic₄₇⋊C₄

yes

—

145

C₂²×C₄

C₂×Dic₉₄

yes

characteristic

146

C₂²×C₄

C₂×C₄×D₄₇

yes

characteristic

147

C₂²×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

148

C₂×C₈

—

149

C₂×C₈

—

150

C₂×C₈

—

151

C₂×C₈

—

152

C₂×C₈

—

153

C₂×C₈

—

154

C₂×C₈

—

188

—

155

C₂×C₈

—

188

—

156

C₂×C₈

—

188

—

157

C₄⋊C₄

—

158

C₄⋊C₄

—

159

C₄²

—

160

C₄²

—

161

C₄²

D₄×D₄₇

yes

—

162

C₄²

#10625352

yes

—

163

C₂³

C₂×Dic₄₇⋊C₄

yes

characteristic

164

Q₈

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

#13093959

yes

—

174

C₂×C₄

C₂²⋊C₄×D₄₇

yes

characteristic

175

C₂×C₄

#13093980

yes

characteristic

176

C₂×C₄

#13093996

yes

—

177

C₂×C₄

#13094001

yes

—

178

C₈

—

179

C₈

—

180

C₈

—

181

C₈

—

182

C₈

—

376

—

183

C₈

—

188

—

184

C₂²

—

185

C₂²

—

186

C₂²

#15688981

yes

characteristic

187

C₂²

C₂×#13093959

yes

—

188

C₄

—

189

C₄

—

190

C₄

—

191

C₄

—

192

C₄

—

193

C₄

#101×D₄₇

yes

characteristic

194

C₄

#15689106

yes

characteristic

195

C₂

—

196

C₂

#20243382

yes

characteristic

197

C₂

#20243424

yes

characteristic

198

C₂

#20243432

yes

characteristic

	subgroup	G/H
center	C₂²	#15688981
commutator	C₂²×C₄×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1467	C₂
socle	C₂²×C₄₇	#380
prime radical	C₂³×C₄₇	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#1467	D₄₇
hypocenter	C₄₇	#8623
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1467

subgroup

G/H

center

C₂²

#15688981

commutator

C₂²×C₄×C₄₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₄₇

Fitting

C₄₇×#1467

C₂

socle

C₂²×C₄₇

#380

prime radical

C₂³×C₄₇

C₂×C₄⋊C₄

cosocle

C₂×C₄²×C₄₇

C₂³

D₄₇

C₄₇

G ▷ C₂²×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1467

N	H ≅ G/N	type
118. C₄₇	#8623	semidirect

H ≅ G/N

type

118. C₄₇

#8623

semidirect

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

subgroup	index
C₂×Q₈×C₄₇⋊C₈	2
#20242149	2
#20241732	2
#20241257	2
C₄₇×#1467	2
#8623	47

subgroup

index

C₂×Q₈×C₄₇⋊C₈

C₄₇×#1467

Subgroups — #26109234

Conjugacy classes of subgroups

Complements

Maximal subgroups