Group #26121304 — subgroups

Subgroups — #26121304

Order 12032.

296 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 93 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	#20244369	C₂	1	1	yes	maximal, characteristic
2	#20243508	C₂	1	1	yes	maximal, characteristic
3	#20242163	C₂	1	1	yes	maximal, characteristic
4	#20242159	C₂	1	1	yes	maximal, characteristic
5	#20241792	C₂	1	1	yes	maximal, characteristic
6	#20241792	C₂	1	1	yes	maximal, characteristic
7	C₄₇×#1274	C₂	1	1	yes	maximal, characteristic
8	#15689680	C₂²	1	1	yes	characteristic
9	#15689147	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
12	#15688883	—	2	2	no	—
13	#15688878	—	2	2	no	—
14	#15688876	—	2	2	no	—
15	#15688874	—	2	2	no	—
16	#15688855	—	2	2	no	—
17	#15688855	—	2	2	no	—
18	C₄₇×#431	C₂²	1	1	yes	characteristic
19	C₄₇×#431	C₂²	1	1	yes	characteristic
20	C₄₇×#376	C₂²	1	1	yes	characteristic
21	C₂×#10625329	—	2	1	no	—
22	C₂²×C₄₇⋊C₈	—	2	1	no	—
23	C₂²×C₄₇⋊C₈	—	2	1	no	—
24	#13094024	Q₈	1	1	yes	characteristic
25	#13094024	Q₈	1	1	yes	characteristic
26	#13094016	—	2	1	no	—
27	#13094003	—	2	1	no	—
28	#13094003	—	2	1	no	—
29	C₄×Dic₉₄	—	2	1	no	—
30	C₄×Dic₉₄	—	2	1	no	—
31	#13093983	—	2	1	no	—
32	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
33	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
34	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
35	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
36	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
37	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
38	C₂×C₄²×C₄₇	D₄	1	2	yes	—
39	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
40	C₄⋊C₈×C₄₇	—	2	1	no	—
41	C₄⋊C₈×C₄₇	—	2	1	no	—
42	C₂²⋊C₈×C₄₇	—	2	1	no	—
43	C₂²⋊C₈×C₄₇	—	2	1	no	—
44	C₄×C₈×C₄₇	D₄	1	2	yes	—
45	C₄×C₈×C₄₇	D₄	1	2	yes	—
46	#95×C₄₇	—	2	1	no	—
47	#95×C₄₇	—	2	1	no	—
48	C₂²×C₄×C₄₇	—	2	2	no	—
49	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
51	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
52	C₂×Dic₉₄	—	2	1	no	—
53	C₂×Dic₉₄	—	2	1	no	—
54	C₂×C₈×C₄₇	—	2	2	no	—
55	C₂×C₈×C₄₇	—	2	2	no	—
56	C₄²×C₄₇	—	2	2	no	—
57	C₄²×C₄₇	—	2	2	no	—
58	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
59	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
60	C₄²×C₄₇	C₂×Q₈	1	1	yes	characteristic
61	C₄²×C₄₇	#41	1	1	yes	characteristic
62	C₂²⋊Dic₄₇	—	4	1	no	—
63	C₄⋊Dic₄₇	—	2	1	no	—
64	C₄⋊Dic₄₇	—	2	1	no	—
65	C₄⋊Dic₄₇	—	2	1	no	—
66	C₄⋊Dic₄₇	—	2	1	no	—
67	Dic₄₇⋊C₄	—	4	1	no	—
68	Dic₄₇⋊C₄	—	4	1	no	—
69	C₄×Dic₄₇	—	4	1	no	—
70	#10625329	—	4	2	no	—
71	C₂×C₄₇⋊C₈	—	4	1	no	—
72	C₂×C₄₇⋊C₈	—	4	1	no	—
73	C₂×C₄₇⋊C₈	—	4	1	no	—
74	C₂×C₄₇⋊C₈	—	4	2	no	—
75	C₂×C₄₇⋊C₈	—	4	2	no	—
76	C₂³×C₄₇	#123	1	1	yes	characteristic
77	C₂×C₄×C₄₇	—	4	1	no	—
78	C₂×C₄×C₄₇	—	4	1	no	—
79	C₂×C₄×C₄₇	—	4	2	no	—
80	C₂×C₄×C₄₇	—	4	2	no	—
81	C₂×C₄×C₄₇	—	4	2	no	—
82	C₂×C₄×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
83	C₂×C₄×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
84	C₂×C₄×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
85	C₂×C₄×C₄₇	#123	1	1	yes	characteristic
86	C₂×C₄×C₄₇	#123	1	1	yes	characteristic
87	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
88	C₂×C₄×C₄₇	#135	1	2	yes	—
89	C₂×C₄×C₄₇	#137	1	2	yes	—
90	C₂×Dic₄₇	—	4	1	no	—
91	C₂×Dic₄₇	—	4	1	no	—
92	Dic₉₄	—	4	2	no	—
93	Dic₉₄	—	4	2	no	—
94	C₈×C₄₇	—	4	2	no	—
95	C₈×C₄₇	—	4	2	no	—
96	C₄₇⋊C₈	—	8	1	no	—
97	C₄₇⋊C₈	—	8	1	no	—
98	C₄₇⋊C₈	—	8	1	no	—
99	#10669	—	47	1	no	maximal, Sylow
100	C₂²×C₄₇	—	2	1	no	—
101	C₂²×C₄₇	—	2	1	no	—
102	C₂²×C₄₇	#399	1	1	yes	characteristic
103	C₂²×C₄₇	#485	1	2	yes	—
104	C₄×C₄₇	—	2	1	no	—
105	C₄×C₄₇	—	2	1	no	—
106	C₄×C₄₇	—	2	1	no	—
107	C₄×C₄₇	—	2	1	no	—
108	C₄×C₄₇	—	8	2	no	—
109	C₄×C₄₇	#479	1	2	yes	—
110	C₄×C₄₇	#495	1	1	yes	characteristic
111	C₄×C₄₇	#497	1	1	yes	characteristic
112	Dic₄₇	—	8	1	no	—
113	Dic₄₇	—	8	1	no	—
114	#1995	—	47	1	no	—
115	#1832	—	47	1	no	—
116	#1571	—	47	1	no	—
117	#1570	—	47	1	no	—
118	#1481	—	47	1	no	—
119	#1481	—	47	1	no	—
120	#1274	D₄₇	1	1	yes	characteristic
121	C₂×C₄₇	—	2	2	no	—
122	C₂×C₄₇	#2097	1	1	yes	characteristic
123	C₂×C₄₇	#2098	1	1	yes	characteristic
124	C₂×C₄₇	#2098	1	1	yes	characteristic
125	#533	—	47	1	no	—
126	#431	C₂×D₄₇	1	1	yes	characteristic
127	#431	C₂×D₄₇	1	1	yes	characteristic
128	#423	—	47	1	no	—
129	C₂×C₄⋊C₈	—	47	1	no	—
130	C₂×C₄⋊C₈	—	47	1	no	—
131	#376	C₂×D₄₇	1	1	yes	characteristic
132	#339	—	94	2	no	—
133	#336	—	94	2	no	—
134	#335	—	94	2	no	—
135	#334	—	94	2	no	—
136	Q₈⋊C₈	—	94	2	no	—
137	Q₈⋊C₈	—	94	2	no	—
138	C₄₇	#10669	1	1	yes	characteristic, complemented, Sylow
139	C₂×M₁₆	—	94	1	no	—
140	C₂²×C₈	—	94	1	no	—
141	C₂²×C₈	—	94	1	no	—
142	C₄⋊Q₈	—	47	1	no	—
143	C₄⋊Q₈	—	47	1	no	—
144	#125	—	94	1	no	—
145	C₂²⋊Q₈	—	94	1	no	—
146	C₂²⋊Q₈	—	94	1	no	—
147	C₄×Q₈	—	94	1	no	—
148	C₄×Q₈	—	94	1	no	—
149	#117	—	94	1	no	—
150	C₂×C₄²	D₁₈₈	1	2	yes	—
151	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
152	C₄⋊C₈	—	2	1	no	—
153	C₄⋊C₈	—	2	1	no	—
154	C₄⋊C₈	—	94	1	no	—
155	C₄⋊C₈	—	94	1	no	—
156	C₄⋊C₈	—	94	1	no	—
157	C₄⋊C₈	—	94	1	no	—
158	C₄⋊C₈	—	94	1	no	—
159	C₄⋊C₈	—	94	1	no	—
160	C₂²⋊C₈	—	2	1	no	—
161	C₂²⋊C₈	—	2	1	no	—
162	C₄×C₈	D₁₈₈	1	2	yes	—
163	C₄×C₈	D₁₈₈	1	2	yes	—
164	#95	—	2	1	no	—
165	#95	—	2	1	no	—
166	C₂×Q₈	—	94	1	no	—
167	C₂×Q₈	—	94	1	no	—
168	C₂²×C₄	—	2	2	no	—
169	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
170	C₂²×C₄	#10625352	1	1	yes	characteristic
171	C₂²×C₄	#10625352	1	1	yes	characteristic
172	M₁₆	—	188	2	no	—
173	C₂×C₈	—	2	2	no	—
174	C₂×C₈	—	2	2	no	—
175	C₂×C₈	—	188	1	no	—
176	C₂×C₈	—	188	1	no	—
177	C₂×C₈	—	188	1	no	—
178	C₂×C₈	—	188	2	no	—
179	C₂×C₈	—	188	2	no	—
180	C₄⋊C₄	—	94	1	no	—
181	C₄⋊C₄	—	94	1	no	—
182	C₄⋊C₄	—	94	1	no	—
183	C₄⋊C₄	—	94	1	no	—
184	C₄⋊C₄	—	188	1	no	—
185	C₄⋊C₄	—	188	1	no	—
186	C₂²⋊C₄	—	188	1	no	—
187	C₄²	—	2	2	no	—
188	C₄²	—	2	2	no	—
189	C₄²	—	188	1	no	—
190	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
191	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
192	C₄²	#10625352	1	1	yes	characteristic
193	C₄²	Q₈×D₄₇	1	1	yes	characteristic
194	C₂³	#13094008	1	1	yes	characteristic
195	Q₈	—	188	2	no	—
196	Q₈	—	188	2	no	—
197	C₂×C₄	—	4	1	no	—
198	C₂×C₄	—	4	1	no	—
199	C₂×C₄	—	4	2	no	—
200	C₂×C₄	—	4	2	no	—
201	C₂×C₄	—	4	2	no	—
202	C₂×C₄	—	188	1	no	—
203	C₂×C₄	—	188	1	no	—
204	C₂×C₄	#13094004	1	1	yes	characteristic
205	C₂×C₄	#13094004	1	1	yes	characteristic
206	C₂×C₄	#13094004	1	1	yes	characteristic
207	C₂×C₄	#13094008	1	1	yes	characteristic
208	C₂×C₄	#13094008	1	1	yes	characteristic
209	C₂×C₄	#13094021	1	1	yes	characteristic
210	C₂×C₄	#13094046	1	2	yes	—
211	C₂×C₄	#13094057	1	2	yes	—
212	C₈	—	4	2	no	—
213	C₈	—	4	2	no	—
214	C₈	—	376	1	no	—
215	C₈	—	376	1	no	—
216	C₈	—	376	1	no	—
217	C₂²	—	2	1	no	—
218	C₂²	—	2	1	no	—
219	C₂²	#15689060	1	1	yes	characteristic
220	C₂²	#15689465	1	2	yes	—
221	C₄	—	2	1	no	—
222	C₄	—	2	1	no	—
223	C₄	—	2	1	no	—
224	C₄	—	2	1	no	—
225	C₄	—	8	2	no	—
226	C₄	—	376	1	no	—
227	C₄	—	376	1	no	—
228	C₄	#15689431	1	2	yes	—
229	C₄	#15689507	1	1	yes	characteristic
230	C₄	#15689517	1	1	yes	characteristic
231	C₂	—	2	2	no	—
232	C₂	#20244935	1	1	yes	characteristic
233	C₂	#20244942	1	1	yes	characteristic
234	C₂	#20244942	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20244369

C₂

yes

maximal, characteristic

#20243508

C₂

yes

maximal, characteristic

#20242163

C₂

yes

maximal, characteristic

#20242159

C₂

yes

maximal, characteristic

#20241792

C₂

yes

maximal, characteristic

#20241792

C₂

yes

maximal, characteristic

C₄₇×#1274

C₂

yes

maximal, characteristic

#15689680

C₂²

yes

characteristic

#15689147

C₂²

yes

characteristic

C₂×C₄⋊(C₄₇⋊C₈)

C₂²

yes

characteristic

C₂×C₄⋊(C₄₇⋊C₈)

C₂²

yes

—

—

—

—

—

—

C₄₇×#431

C₂²

yes

characteristic

C₄₇×#431

C₂²

yes

characteristic

C₄₇×#376

C₂²

yes

characteristic

C₂×#10625329

—

C₂²×C₄₇⋊C₈

—

C₂²×C₄₇⋊C₈

—

#13094024

Q₈

yes

characteristic

#13094024

Q₈

yes

—

—

—

C₄×Dic₉₄

—

C₄×Dic₉₄

—

#13093983

—

C₄⋊(C₄₇⋊C₈)

—

C₄⋊(C₄₇⋊C₈)

—

C₄⋊(C₄₇⋊C₈)

—

C₄⋊(C₄₇⋊C₈)

—

C₄⋊(C₄₇⋊C₈)

—

C₄⋊(C₄₇⋊C₈)

—

C₂×C₄²×C₄₇

D₄

yes

—

C₂×C₄²×C₄₇

C₂³

yes

characteristic

C₄⋊C₈×C₄₇

—

C₄⋊C₈×C₄₇

—

C₂²⋊C₈×C₄₇

—

C₂²⋊C₈×C₄₇

—

C₄×C₈×C₄₇

D₄

yes

—

C₄×C₈×C₄₇

D₄

yes

—

#95×C₄₇

—

#95×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂×D₄

yes

C₂²×C₄×C₄₇

yes

C₂²×C₄×C₄₇

yes

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

C₂×D₄

yes

characteristic

C₄²×C₄₇

C₂×D₄

yes

characteristic

C₄²×C₄₇

C₂×Q₈

yes

characteristic

C₄²×C₄₇

#41

yes

characteristic

C₂²⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

Dic₄₇⋊C₄

—

Dic₄₇⋊C₄

—

C₄×Dic₄₇

—

#10625329

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

#123

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₄₇

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₄₇

C₂²⋊Q₈

yes

C₂×C₄×C₄₇

yes

C₂×C₄×C₄₇

yes

C₂×C₄×C₄₇

yes

C₂×C₄×C₄₇

yes

—

C₂×C₄×C₄₇

#137

yes

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

Dic₉₄

—

Dic₉₄

—

C₈×C₄₇

—

C₈×C₄₇

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

#10669

—

maximal, Sylow

100

C₂²×C₄₇

—

101

C₂²×C₄₇

—

102

C₂²×C₄₇

#399

yes

characteristic

103

C₂²×C₄₇

#485

yes

—

104

C₄×C₄₇

—

105

C₄×C₄₇

—

106

C₄×C₄₇

—

107

C₄×C₄₇

—

108

C₄×C₄₇

—

109

C₄×C₄₇

#479

yes

—

110

C₄×C₄₇

#495

yes

characteristic

111

C₄×C₄₇

#497

yes

characteristic

112

Dic₄₇

—

113

Dic₄₇

—

114

#1995

—

115

#1832

—

116

#1571

—

117

#1570

—

118

#1481

—

119

#1481

—

120

#1274

D₄₇

yes

characteristic

121

C₂×C₄₇

—

122

C₂×C₄₇

#2097

yes

characteristic

123

C₂×C₄₇

#2098

yes

characteristic

124

C₂×C₄₇

#2098

yes

characteristic

125

#533

—

126

#431

C₂×D₄₇

yes

characteristic

127

#431

C₂×D₄₇

yes

characteristic

128

#423

—

129

C₂×C₄⋊C₈

—

130

C₂×C₄⋊C₈

—

131

#376

C₂×D₄₇

yes

characteristic

132

#339

—

133

#336

—

134

#335

—

135

#334

—

136

Q₈⋊C₈

—

137

Q₈⋊C₈

—

138

C₄₇

#10669

yes

characteristic, complemented, Sylow

139

C₂×M₁₆

—

140

C₂²×C₈

—

141

C₂²×C₈

—

142

C₄⋊Q₈

—

143

C₄⋊Q₈

—

144

#125

—

145

C₂²⋊Q₈

—

146

C₂²⋊Q₈

—

147

C₄×Q₈

—

148

C₄×Q₈

—

149

#117

—

150

C₂×C₄²

D₁₈₈

yes

—

151

C₂×C₄²

C₂²×D₄₇

yes

characteristic

152

C₄⋊C₈

—

153

C₄⋊C₈

—

154

C₄⋊C₈

—

155

C₄⋊C₈

—

156

C₄⋊C₈

—

157

C₄⋊C₈

—

158

C₄⋊C₈

—

159

C₄⋊C₈

—

160

C₂²⋊C₈

—

161

C₂²⋊C₈

—

162

C₄×C₈

D₁₈₈

yes

—

163

C₄×C₈

D₁₈₈

yes

—

164

#95

—

165

#95

—

166

C₂×Q₈

—

167

C₂×Q₈

—

168

C₂²×C₄

—

169

C₂²×C₄

C₂×D₁₈₈

yes

characteristic

170

C₂²×C₄

#10625352

yes

characteristic

171

C₂²×C₄

#10625352

yes

characteristic

172

M₁₆

—

188

—

173

C₂×C₈

—

174

C₂×C₈

—

175

C₂×C₈

—

188

—

176

C₂×C₈

—

188

—

177

C₂×C₈

—

188

—

178

C₂×C₈

—

188

—

179

C₂×C₈

—

188

—

180

C₄⋊C₄

—

181

C₄⋊C₄

—

182

C₄⋊C₄

—

183

C₄⋊C₄

—

184

C₄⋊C₄

—

188

—

185

C₄⋊C₄

—

188

—

186

C₂²⋊C₄

—

188

—

187

C₄²

—

188

C₄²

—

189

C₄²

—

188

—

190

C₄²

C₂×D₁₈₈

yes

characteristic

191

C₄²

C₂×D₁₈₈

yes

characteristic

192

C₄²

#10625352

yes

characteristic

193

C₄²

Q₈×D₄₇

yes

characteristic

194

C₂³

#13094008

yes

characteristic

195

Q₈

—

188

—

196

Q₈

—

188

—

197

C₂×C₄

—

198

C₂×C₄

—

199

C₂×C₄

—

200

C₂×C₄

—

201

C₂×C₄

—

202

C₂×C₄

—

188

—

203

C₂×C₄

—

188

—

204

C₂×C₄

#13094004

yes

characteristic

205

C₂×C₄

#13094004

yes

characteristic

206

C₂×C₄

#13094004

yes

characteristic

207

C₂×C₄

#13094008

yes

characteristic

208

C₂×C₄

#13094008

yes

characteristic

209

C₂×C₄

#13094021

yes

characteristic

210

C₂×C₄

#13094046

yes

—

211

C₂×C₄

#13094057

yes

—

212

C₈

—

213

C₈

—

214

C₈

—

376

—

215

C₈

—

376

—

216

C₈

—

376

—

217

C₂²

—

218

C₂²

—

219

C₂²

#15689060

yes

characteristic

220

C₂²

#15689465

yes

—

221

C₄

—

222

C₄

—

223

C₄

—

224

C₄

—

225

C₄

—

226

C₄

—

376

—

227

C₄

—

376

—

228

C₄

#15689431

yes

—

229

C₄

#15689507

yes

characteristic

230

C₄

#15689517

yes

characteristic

231

C₂

—

232

C₂

#20244935

yes

characteristic

233

C₂

#20244942

yes

characteristic

234

C₂

#20244942

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#13094021
commutator, cosocle	C₂×C₄²×C₄₇	C₂³
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1274	C₂
socle	C₂²×C₄₇	#399
prime radical	C₂³×C₄₇	#123
hypercenter	#1274	D₄₇
hypocenter	C₄₇	#10669
LCS	G ▷ C₂×C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1274

subgroup

G/H

center

C₂×C₄

#13094021

commutator, cosocle

C₂×C₄²×C₄₇

C₂³

Frattini

C₂×C₄²

C₂²×D₄₇

Fitting

C₄₇×#1274

C₂

C₂²×C₄₇

C₂³×C₄₇

D₄₇

C₄₇

G ▷ C₂×C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1274

N	H ≅ G/N	type
138. C₄₇	#10669	semidirect

H ≅ G/N

type

138. C₄₇

#10669

semidirect

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

subgroup	index
#20244369	2
#20243508	2
#20242163	2
#20242159	2
#20241792	2
C₄₇×#1274	2
#10669	47

subgroup

index

C₄₇×#1274

Subgroups — #26121304

Conjugacy classes of subgroups

Complements

Maximal subgroups