Group #26121027 — subgroups

Subgroups — #26121027

Order 12032.

350 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 123 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	#20243516	C₂	1	1	yes	maximal, characteristic
2	#20243393	C₂	1	2	yes	maximal
3	#20241272	C₂	1	2	yes	maximal
4	#20241223	C₂	1	1	yes	maximal, characteristic
5	C₄₇×#1984	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
7	C₂×#13093948	—	2	2	no	—
8	#15688993	C₂²	1	2	yes	—
9	#15688886	—	2	2	no	—
10	#15688885	—	2	2	no	—
11	C₂×C₄×Q₈×C₄₇	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
14	C₄₇×#336	—	2	2	no	—
15	C₂²×C₄₇⋊C₈	—	2	2	no	—
16	C₂×C₄⋊Dic₄₇	—	2	2	no	—
17	C₂×C₄⋊Dic₄₇	—	2	2	no	—
18	C₂×C₄⋊Dic₄₇	—	2	2	no	—
19	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
20	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
21	#13093948	—	4	4	no	—
22	C₂²×Q₈×C₄₇	D₄	1	1	yes	characteristic
23	C₂²×C₈×C₄₇	—	2	1	no	—
24	C₂²×C₈×C₄₇	D₄	1	2	yes	—
25	C₄×Q₈×C₄₇	—	2	2	no	—
26	C₄×Q₈×C₄₇	—	2	2	no	—
27	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
28	C₂×C₄⋊C₄×C₄₇	D₄	1	1	yes	characteristic
29	C₂×C₄²×C₄₇	—	2	1	no	—
30	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
31	C₄⋊C₈×C₄₇	—	2	2	no	—
32	C₄×C₈×C₄₇	—	2	2	no	—
33	C₂×Q₈×C₄₇	—	2	1	no	—
34	C₂×Q₈×C₄₇	—	2	2	no	—
35	C₂×Q₈×C₄₇	—	2	2	no	—
36	C₂×Q₈×C₄₇	Q₁₆	1	2	yes	—
37	C₂²×C₄×C₄₇	—	2	1	no	—
38	C₂²×C₄×C₄₇	—	2	1	no	—
39	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
40	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
41	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
42	C₂²×Dic₄₇	—	4	2	no	—
43	C₂×C₈×C₄₇	—	2	2	no	—
44	C₂×C₈×C₄₇	—	2	2	no	—
45	C₂×C₈×C₄₇	—	2	2	no	—
46	C₂×C₈×C₄₇	—	4	1	no	—
47	C₂×C₈×C₄₇	—	4	2	no	—
48	C₄⋊C₄×C₄₇	—	2	2	no	—
49	C₄⋊C₄×C₄₇	—	4	2	no	—
50	C₄²×C₄₇	—	4	1	no	—
51	C₄²×C₄₇	—	4	1	no	—
52	C₄²×C₄₇	#41	1	2	yes	—
53	C₄²×C₄₇	#41	1	2	yes	—
54	C₄⋊Dic₄₇	—	4	2	no	—
55	C₄⋊Dic₄₇	—	4	2	no	—
56	C₄⋊Dic₄₇	—	4	2	no	—
57	C₄⋊Dic₄₇	—	4	2	no	—
58	C₄⋊Dic₄₇	—	8	2	no	—
59	C₂×C₄₇⋊C₈	—	4	2	no	—
60	C₂×C₄₇⋊C₈	—	4	2	no	—
61	C₂×C₄₇⋊C₈	—	4	2	no	—
62	C₂³×C₄₇	#121	1	1	yes	characteristic
63	Q₈×C₄₇	—	2	4	no	—
64	Q₈×C₄₇	—	4	2	no	—
65	C₂×C₄×C₄₇	—	2	2	no	—
66	C₂×C₄×C₄₇	—	2	2	no	—
67	C₂×C₄×C₄₇	—	4	1	no	—
68	C₂×C₄×C₄₇	—	4	1	no	—
69	C₂×C₄×C₄₇	—	4	1	no	—
70	C₂×C₄×C₄₇	—	4	2	no	—
71	C₂×C₄×C₄₇	#123	1	1	yes	characteristic
72	C₂×C₄×C₄₇	#123	1	2	yes	—
73	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
74	C₂×C₄×C₄₇	#126	1	2	yes	—
75	C₂×C₄×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
76	C₂×C₄×C₄₇	#135	1	2	yes	—
77	C₂×C₄×C₄₇	#135	1	1	yes	characteristic
78	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
79	C₂×C₄×C₄₇	#137	1	2	yes	—
80	C₂×Dic₄₇	—	8	2	no	—
81	C₂×Dic₄₇	—	8	2	no	—
82	C₂×Dic₄₇	—	8	2	no	—
83	C₈×C₄₇	—	4	2	no	—
84	C₈×C₄₇	—	8	1	no	—
85	C₄₇⋊C₈	—	8	2	no	—
86	#10624	—	47	1	no	maximal, Sylow
87	C₂²×C₄₇	#397	1	1	yes	characteristic
88	C₂²×C₄₇	#464	1	2	yes	—
89	C₂²×C₄₇	#467	1	1	yes	characteristic
90	C₂²×C₄₇	#469	1	1	yes	characteristic
91	C₂²×C₄₇	#471	1	2	yes	—
92	C₄×C₄₇	—	2	2	no	—
93	C₄×C₄₇	—	4	2	no	—
94	C₄×C₄₇	—	8	1	no	—
95	C₄×C₄₇	#482	1	2	yes	—
96	C₄×C₄₇	#484	1	2	yes	—
97	C₄×C₄₇	#485	1	2	yes	—
98	C₄×C₄₇	#488	1	1	yes	characteristic
99	C₄×C₄₇	#491	1	1	yes	characteristic
100	Dic₄₇	—	16	2	no	—
101	#1984	D₄₇	1	1	yes	characteristic
102	#1834	—	47	1	no	—
103	#1803	—	47	2	no	—
104	#1296	—	47	2	no	—
105	#1277	—	47	1	no	—
106	C₂×C₄₇	#1684	1	2	yes	—
107	C₂×C₄₇	#1702	1	2	yes	—
108	C₂×C₄₇	#2074	1	1	yes	characteristic
109	C₂×C₄₇	#2076	1	1	yes	characteristic
110	C₂×C₄₇	#2079	1	1	yes	characteristic
111	C₂×C₄×Q₈	C₂×D₄₇	1	1	yes	characteristic
112	C₂×C₄⋊C₈	—	47	2	no	—
113	C₂×C₄⋊C₈	C₂×D₄₇	1	1	yes	characteristic
114	C₂×#103	—	94	2	no	—
115	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
116	#384	—	47	2	no	—
117	#340	—	94	2	no	—
118	#340	—	94	2	no	—
119	#336	—	2	2	no	—
120	C₄₇	#10624	1	1	yes	characteristic, complemented, Sylow
121	C₂²×Q₈	D₁₈₈	1	1	yes	characteristic
122	C₂²×C₈	—	2	1	no	—
123	C₂²×C₈	—	94	2	no	—
124	C₂²×C₈	C₂²⋊D₄₇	1	2	yes	—
125	C₄×Q₈	—	2	2	no	—
126	C₄×Q₈	—	2	2	no	—
127	C₂×C₄⋊C₄	—	2	1	no	—
128	C₂×C₄⋊C₄	—	94	2	no	—
129	C₂×C₄⋊C₄	—	94	2	no	—
130	C₂×C₄⋊C₄	—	94	2	no	—
131	C₂×C₄⋊C₄	D₁₈₈	1	1	yes	characteristic
132	C₂×C₄²	—	2	1	no	—
133	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
134	C₄⋊C₈	—	2	2	no	—
135	C₄⋊C₈	—	94	2	no	—
136	C₄⋊C₈	—	94	2	no	—
137	#103	—	188	4	no	—
138	C₄×C₈	—	2	2	no	—
139	C₂×Q₈	—	2	1	no	—
140	C₂×Q₈	—	2	2	no	—
141	C₂×Q₈	—	2	2	no	—
142	C₂×Q₈	Dic₁₈₈	1	2	yes	—
143	C₂²×C₄	—	2	1	no	—
144	C₂²×C₄	—	2	1	no	—
145	C₂²×C₄	—	188	2	no	—
146	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
147	C₂²×C₄	#10625350	1	1	yes	characteristic
148	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
149	C₂×C₈	—	2	2	no	—
150	C₂×C₈	—	2	2	no	—
151	C₂×C₈	—	2	2	no	—
152	C₂×C₈	—	4	1	no	—
153	C₂×C₈	—	4	2	no	—
154	C₂×C₈	—	188	2	no	—
155	C₂×C₈	—	188	2	no	—
156	C₂×C₈	—	188	2	no	—
157	C₄⋊C₄	—	2	2	no	—
158	C₄⋊C₄	—	4	2	no	—
159	C₄⋊C₄	—	376	2	no	—
160	C₄⋊C₄	—	188	2	no	—
161	C₄⋊C₄	—	188	2	no	—
162	C₄⋊C₄	—	188	2	no	—
163	C₄⋊C₄	—	188	2	no	—
164	C₄²	—	4	1	no	—
165	C₄²	—	4	1	no	—
166	C₄²	#10625352	1	2	yes	—
167	C₄²	#10625352	1	2	yes	—
168	C₂³	#13093998	1	1	yes	characteristic
169	Q₈	—	2	4	no	—
170	Q₈	—	4	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	4	1	no	—
174	C₂×C₄	—	4	1	no	—
175	C₂×C₄	—	4	1	no	—
176	C₂×C₄	—	4	2	no	—
177	C₂×C₄	—	376	2	no	—
178	C₂×C₄	—	376	2	no	—
179	C₂×C₄	—	376	2	no	—
180	C₂×C₄	#13094005	1	1	yes	characteristic
181	C₂×C₄	#13094008	1	2	yes	—
182	C₂×C₄	#13094011	1	1	yes	characteristic
183	C₂×C₄	#13094017	1	2	yes	—
184	C₂×C₄	C₂×Dic₁₈₈	1	1	yes	characteristic
185	C₂×C₄	#13094044	1	2	yes	—
186	C₂×C₄	#13094046	1	1	yes	characteristic
187	C₂×C₄	#13094052	1	2	yes	—
188	C₂×C₄	#13094055	1	2	yes	—
189	C₈	—	4	2	no	—
190	C₈	—	8	1	no	—
191	C₈	—	376	2	no	—
192	C₂²	#15689048	1	1	yes	characteristic
193	C₂²	#15689345	1	2	yes	—
194	C₂²	#15689364	1	1	yes	characteristic
195	C₂²	#15689376	1	1	yes	characteristic
196	C₂²	#15689386	1	2	yes	—
197	C₄	—	2	2	no	—
198	C₄	—	4	2	no	—
199	C₄	—	8	1	no	—
200	C₄	—	752	2	no	—
201	C₄	#15689448	1	2	yes	—
202	C₄	#15689459	1	2	yes	—
203	C₄	#15689461	1	2	yes	—
204	C₄	#15689477	1	1	yes	characteristic
205	C₄	#15689491	1	1	yes	characteristic
206	C₂	#20242846	1	2	yes	—
207	C₂	#20242956	1	2	yes	—
208	C₂	#20244806	1	1	yes	characteristic
209	C₂	#20244816	1	1	yes	characteristic
210	C₂	#20244833	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20243516

C₂

yes

maximal, characteristic

#20243393

C₂

yes

maximal

#20241272

C₂

yes

maximal

#20241223

C₂

yes

maximal, characteristic

C₄₇×#1984

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₄₇⋊C₈)

C₂²

yes

—

C₂×#13093948

—

#15688993

C₂²

yes

—

#15688886

—

#15688885

—

C₂×C₄×Q₈×C₄₇

C₂²

yes

characteristic

C₂×C₄⋊C₈×C₄₇

C₂²

yes

characteristic

C₂×C₄×C₈×C₄₇

C₂²

yes

characteristic

C₄₇×#336

—

C₂²×C₄₇⋊C₈

—

C₂×C₄⋊Dic₄₇

—

C₂×C₄⋊Dic₄₇

—

C₂×C₄⋊Dic₄₇

—

C₄⋊(C₄₇⋊C₈)

—

C₄⋊(C₄₇⋊C₈)

—

#13093948

—

C₂²×Q₈×C₄₇

D₄

yes

characteristic

C₂²×C₈×C₄₇

—

C₂²×C₈×C₄₇

D₄

yes

—

C₄×Q₈×C₄₇

—

C₄×Q₈×C₄₇

—

C₂×C₄⋊C₄×C₄₇

—

C₂×C₄⋊C₄×C₄₇

D₄

yes

characteristic

C₂×C₄²×C₄₇

—

C₂×C₄²×C₄₇

C₂³

yes

characteristic

C₄⋊C₈×C₄₇

—

C₄×C₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

Q₁₆

yes

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₄₇

#41

yes

characteristic

C₂²×Dic₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

#41

yes

—

C₄²×C₄₇

#41

yes

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

#121

yes

characteristic

Q₈×C₄₇

—

Q₈×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

#123

yes

characteristic

C₂×C₄×C₄₇

#123

yes

—

C₂×C₄×C₄₇

#124

yes

characteristic

C₂×C₄×C₄₇

#126

yes

—

C₂×C₄×C₄₇

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₄₇

#135

yes

—

C₂×C₄×C₄₇

#135

yes

characteristic

C₂×C₄×C₄₇

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₄₇

#137

yes

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₈×C₄₇

—

C₈×C₄₇

—

C₄₇⋊C₈

—

—

C₂²×C₄₇

yes

C₂²×C₄₇

yes

—

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#482

yes

—

C₄×C₄₇

#484

yes

—

C₄×C₄₇

#485

yes

—

C₄×C₄₇

yes

C₄×C₄₇

yes

100

Dic₄₇

—

101

#1984

D₄₇

yes

characteristic

102

#1834

—

103

#1803

—

104

#1296

—

105

#1277

—

106

C₂×C₄₇

#1684

yes

—

107

C₂×C₄₇

#1702

yes

—

108

C₂×C₄₇

#2074

yes

characteristic

109

C₂×C₄₇

#2076

yes

characteristic

110

C₂×C₄₇

#2079

yes

characteristic

111

C₂×C₄×Q₈

C₂×D₄₇

yes

characteristic

112

C₂×C₄⋊C₈

—

113

C₂×C₄⋊C₈

C₂×D₄₇

yes

characteristic

114

C₂×#103

—

115

C₂×C₄×C₈

C₂×D₄₇

yes

characteristic

116

#384

—

117

#340

—

118

#340

—

119

#336

—

120

C₄₇

#10624

yes

characteristic, complemented, Sylow

121

C₂²×Q₈

D₁₈₈

yes

characteristic

122

C₂²×C₈

—

123

C₂²×C₈

—

124

C₂²×C₈

C₂²⋊D₄₇

yes

—

125

C₄×Q₈

—

126

C₄×Q₈

—

127

C₂×C₄⋊C₄

—

128

C₂×C₄⋊C₄

—

129

C₂×C₄⋊C₄

—

130

C₂×C₄⋊C₄

—

131

C₂×C₄⋊C₄

D₁₈₈

yes

characteristic

132

C₂×C₄²

—

133

C₂×C₄²

C₂²×D₄₇

yes

characteristic

134

C₄⋊C₈

—

135

C₄⋊C₈

—

136

C₄⋊C₈

—

137

#103

—

188

—

138

C₄×C₈

—

139

C₂×Q₈

—

140

C₂×Q₈

—

141

C₂×Q₈

—

142

C₂×Q₈

Dic₁₈₈

yes

—

143

C₂²×C₄

—

144

C₂²×C₄

—

145

C₂²×C₄

—

188

—

146

C₂²×C₄

C₂×D₁₈₈

yes

characteristic

147

C₂²×C₄

#10625350

yes

characteristic

148

C₂²×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

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

C₄⋊C₄

—

376

—

160

C₄⋊C₄

—

188

—

161

C₄⋊C₄

—

188

—

162

C₄⋊C₄

—

188

—

163

C₄⋊C₄

—

188

—

164

C₄²

—

165

C₄²

—

166

C₄²

#10625352

yes

—

167

C₄²

#10625352

yes

—

168

C₂³

#13093998

yes

characteristic

169

Q₈

—

170

Q₈

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

376

—

178

C₂×C₄

—

376

—

179

C₂×C₄

—

376

—

180

C₂×C₄

#13094005

yes

characteristic

181

C₂×C₄

#13094008

yes

—

182

C₂×C₄

#13094011

yes

characteristic

183

C₂×C₄

#13094017

yes

—

184

C₂×C₄

C₂×Dic₁₈₈

yes

characteristic

185

C₂×C₄

#13094044

yes

—

186

C₂×C₄

#13094046

yes

characteristic

187

C₂×C₄

#13094052

yes

—

188

C₂×C₄

#13094055

yes

—

189

C₈

—

190

C₈

—

191

C₈

—

376

—

192

C₂²

#15689048

yes

characteristic

193

C₂²

#15689345

yes

—

194

C₂²

#15689364

yes

characteristic

195

C₂²

#15689376

yes

characteristic

196

C₂²

#15689386

yes

—

197

C₄

—

198

C₄

—

199

C₄

—

200

C₄

—

752

—

201

C₄

#15689448

yes

—

202

C₄

#15689459

yes

—

203

C₄

#15689461

yes

—

204

C₄

#15689477

yes

characteristic

205

C₄

#15689491

yes

characteristic

206

C₂

#20242846

yes

—

207

C₂

#20242956

yes

—

208

C₂

#20244806

yes

characteristic

209

C₂

#20244816

yes

characteristic

210

C₂

#20244833

yes

characteristic

	subgroup	G/H
center	C₂³	#13093998
commutator, cosocle	C₂×C₄²×C₄₇	C₂³
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1984	C₂
socle, prime radical	C₂³×C₄₇	#121
hypercenter	#1984	D₄₇
hypocenter	C₄₇	#10624
LCS	G ▷ C₂×C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1984

subgroup

G/H

center

C₂³

#13093998

commutator, cosocle

C₂×C₄²×C₄₇

C₂³

Frattini

C₂×C₄²

C₂²×D₄₇

Fitting

C₄₇×#1984

C₂

C₂³×C₄₇

D₄₇

C₄₇

G ▷ C₂×C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1984

N	H ≅ G/N	type
120. C₄₇	#10624	semidirect

H ≅ G/N

type

120. C₄₇

#10624

semidirect

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

subgroup	index
#20243516	2
#20243393	2
#20241272	2
#20241223	2
C₄₇×#1984	2
#10624	47

subgroup

index

C₄₇×#1984

Subgroups — #26121027

Conjugacy classes of subgroups

Complements

Maximal subgroups