Group #26110785 — subgroups

Subgroups — #26110785

Order 12032.

412 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 103 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	#20252455	C₂	1	1	yes	maximal, characteristic
2	#20242179	C₂	1	2	yes	maximal, complemented
3	#20242050	C₂	1	1	yes	maximal, characteristic
4	#20241221	C₂	1	2	yes	maximal, complemented
5	C₄₇×#1962	C₂	1	1	yes	maximal, characteristic
6	#15689201	—	2	2	no	—
7	D₄×C₄₇⋊C₈	—	2	2	no	—
8	#15689176	C₂²	1	2	yes	—
9	#15689176	C₂²	1	2	yes	—
10	#15689090	—	2	2	no	—
11	#15688875	—	2	2	no	—
12	#15688873	—	2	2	no	—
13	#15688872	C₄	1	1	yes	characteristic, complemented
14	#15688872	C₄	1	1	yes	characteristic, complemented
15	#15688867	C₄	1	2	yes	complemented
16	C₄₇×#532	C₂²	1	1	yes	characteristic
17	C₄₇×#423	C₂²	1	1	yes	characteristic
18	C₂×#104×C₄₇	—	2	2	no	—
19	C₄³×C₄₇	C₂²	1	1	yes	characteristic
20	C₂×#10625329	—	2	2	no	—
21	C₂²×C₄₇⋊C₈	—	2	2	no	—
22	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	2	yes	—
23	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	2	yes	—
24	C₄⋊(C₄₇⋊C₈)	D₄	1	2	yes	—
25	C₄⋊(C₄₇⋊C₈)	D₄	1	2	yes	—
26	C₂²⋊(C₄₇⋊C₈)	—	2	4	no	—
27	C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
28	C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
29	C₄×C₄₇⋊C₈	—	2	2	no	—
30	C₄×C₄₇⋊C₈	—	2	2	no	—
31	C₂×#41×C₄₇	D₄	1	2	yes	—
32	C₂×M₁₆×C₄₇	—	2	1	no	—
33	C₄⋊Q₈×C₄₇	C₂×C₄	1	1	yes	characteristic
34	#127×C₄₇	C₂×C₄	1	1	yes	characteristic
35	#124×C₄₇	Q₈	1	2	yes	—
36	#121×C₄₇	—	2	2	no	—
37	C₄×D₄×C₄₇	—	2	2	no	—
38	C₂×C₄²×C₄₇	—	2	1	no	—
39	C₂×C₄²×C₄₇	—	2	2	no	—
40	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
41	C₄⋊C₈×C₄₇	—	2	2	no	—
42	#104×C₄₇	—	4	4	no	—
43	#41×C₄₇	—	4	2	no	—
44	#41×C₄₇	—	4	2	no	—
45	C₂×Q₈×C₄₇	C₄⋊C₄	1	2	yes	—
46	C₂×D₄×C₄₇	—	2	2	no	—
47	C₂×D₄×C₄₇	—	2	2	no	—
48	C₂×D₄×C₄₇	C₄⋊C₄	1	2	yes	—
49	C₂²×C₄×C₄₇	—	2	2	no	—
50	C₂²×C₄×C₄₇	—	4	1	no	—
51	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
52	C₂²×C₄×C₄₇	#41	1	2	yes	—
53	M₁₆×C₄₇	—	4	2	no	—
54	C₂×C₈×C₄₇	—	4	1	no	—
55	C₄⋊C₄×C₄₇	—	2	2	no	—
56	C₂²⋊C₄×C₄₇	—	2	4	no	—
57	C₄²×C₄₇	—	2	2	no	—
58	C₄²×C₄₇	—	2	4	no	—
59	C₄²×C₄₇	—	4	2	no	—
60	C₄²×C₄₇	—	4	1	no	—
61	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
62	C₄²×C₄₇	C₂×D₄	1	2	yes	—
63	C₄²×C₄₇	C₂×Q₈	1	1	yes	characteristic
64	#10625329	—	8	2	no	—
65	C₂×C₄₇⋊C₈	—	2	2	no	—
66	C₂×C₄₇⋊C₈	—	2	2	no	—
67	C₂×C₄₇⋊C₈	—	2	2	no	—
68	C₂×C₄₇⋊C₈	—	2	2	no	—
69	C₂×C₄₇⋊C₈	—	8	2	no	—
70	C₂³×C₄₇	—	2	2	no	—
71	C₂³×C₄₇	#123	1	1	yes	characteristic
72	Q₈×C₄₇	—	4	2	no	—
73	D₄×C₄₇	—	4	2	no	—
74	D₄×C₄₇	—	4	2	no	—
75	D₄×C₄₇	—	4	2	no	—
76	D₄×C₄₇	—	8	2	no	—
77	C₂×C₄×C₄₇	—	2	2	no	—
78	C₂×C₄×C₄₇	—	2	2	no	—
79	C₂×C₄×C₄₇	—	2	1	no	—
80	C₂×C₄×C₄₇	—	2	1	no	—
81	C₂×C₄×C₄₇	—	2	2	no	—
82	C₂×C₄×C₄₇	—	4	2	no	—
83	C₂×C₄×C₄₇	—	4	4	no	—
84	C₂×C₄×C₄₇	—	4	2	no	—
85	C₂×C₄×C₄₇	—	4	2	no	—
86	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
87	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
88	C₂×C₄×C₄₇	#120	1	1	yes	characteristic
89	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
90	C₈×C₄₇	—	8	1	no	—
91	C₄₇⋊C₈	—	4	2	no	—
92	C₄₇⋊C₈	—	4	2	no	—
93	C₄₇⋊C₈	—	4	2	no	—
94	C₄₇⋊C₈	—	4	2	no	—
95	#8935	—	47	1	no	maximal, Sylow
96	C₂²×C₄₇	—	2	1	no	—
97	C₂²×C₄₇	—	2	2	no	—
98	C₂²×C₄₇	—	4	2	no	—
99	C₂²×C₄₇	—	4	4	no	—
100	C₂²×C₄₇	#385	1	1	yes	characteristic
101	C₄×C₄₇	—	2	1	no	—
102	C₄×C₄₇	—	2	2	no	—
103	C₄×C₄₇	—	4	4	no	—
104	C₄×C₄₇	—	4	2	no	—
105	C₄×C₄₇	#443	1	2	yes	—
106	C₄×C₄₇	#444	1	2	yes	—
107	C₄×C₄₇	#453	1	1	yes	characteristic
108	C₄×C₄₇	#455	1	1	yes	characteristic
109	#2979	—	47	1	no	—
110	#1962	D₄₇	1	1	yes	characteristic
111	#1574	—	47	2	no	—
112	#1543	—	47	1	no	—
113	#1276	—	47	2	no	—
114	C₂×C₄₇	—	4	1	no	—
115	C₂×C₄₇	—	8	2	no	—
116	C₂×C₄₇	#1856	1	1	yes	characteristic
117	C₂×C₄₇	#1866	1	2	yes	—
118	#532	C₂×D₄₇	1	1	yes	characteristic
119	#436	—	94	2	no	—
120	C₈×D₄	—	94	2	no	—
121	#431	—	47	2	no	—
122	#431	—	47	2	no	—
123	#423	C₂×D₄₇	1	1	yes	characteristic
124	C₂×#104	—	2	2	no	—
125	#408	—	94	2	no	—
126	C₄³	C₂×D₄₇	1	1	yes	characteristic
127	#335	—	94	2	no	—
128	#334	—	94	2	no	—
129	#333	—	47	1	no	—
130	#333	—	47	1	no	—
131	#331	—	47	2	no	—
132	C₄₇	#8935	1	1	yes	characteristic, complemented, Sylow
133	C₂×#41	C₂²⋊D₄₇	1	2	yes	—
134	C₂×M₁₆	—	2	1	no	—
135	C₂×M₁₆	—	94	2	no	—
136	C₂²×C₈	—	94	2	no	—
137	C₄⋊Q₈	C₄×D₄₇	1	1	yes	characteristic
138	#127	C₄×D₄₇	1	1	yes	characteristic
139	#124	Dic₉₄	1	2	yes	—
140	#121	—	2	2	no	—
141	C₄×D₄	—	2	2	no	—
142	C₂×C₄²	—	2	2	no	—
143	C₂×C₄²	—	2	1	no	—
144	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
145	C₄⋊C₈	—	2	2	no	—
146	C₄⋊C₈	—	47	2	no	—
147	C₄⋊C₈	—	47	2	no	—
148	C₄⋊C₈	—	47	2	no	—
149	C₄⋊C₈	—	47	2	no	—
150	#104	—	4	4	no	—
151	C₂²⋊C₈	—	94	4	no	—
152	C₂²⋊C₈	—	94	2	no	—
153	C₂²⋊C₈	—	94	2	no	—
154	C₄×C₈	—	94	2	no	—
155	C₄×C₈	—	94	2	no	—
156	#41	—	4	2	no	—
157	#41	—	4	2	no	—
158	C₂×Q₈	Dic₄₇⋊C₄	1	2	yes	—
159	C₂×D₄	—	2	2	no	—
160	C₂×D₄	—	2	2	no	—
161	C₂×D₄	Dic₄₇⋊C₄	1	2	yes	—
162	C₂²×C₄	—	2	2	no	—
163	C₂²×C₄	—	4	1	no	—
164	C₂²×C₄	#10625352	1	2	yes	—
165	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
166	M₁₆	—	4	2	no	—
167	M₁₆	—	376	2	no	—
168	C₂×C₈	—	4	1	no	—
169	C₂×C₈	—	94	2	no	—
170	C₂×C₈	—	94	2	no	—
171	C₂×C₈	—	94	2	no	—
172	C₂×C₈	—	94	2	no	—
173	C₂×C₈	—	376	2	no	—
174	C₄⋊C₄	—	2	2	no	—
175	C₂²⋊C₄	—	2	4	no	—
176	C₄²	—	2	2	no	—
177	C₄²	—	2	4	no	—
178	C₄²	—	4	2	no	—
179	C₄²	—	4	1	no	—
180	C₄²	C₂×Dic₉₄	1	1	yes	characteristic
181	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
182	C₄²	D₄×D₄₇	1	2	yes	—
183	C₂³	—	2	2	no	—
184	C₂³	#13094005	1	1	yes	characteristic
185	Q₈	—	4	2	no	—
186	D₄	—	4	2	no	—
187	D₄	—	4	2	no	—
188	D₄	—	4	2	no	—
189	D₄	—	8	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	1	no	—
193	C₂×C₄	—	2	1	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	4	2	no	—
196	C₂×C₄	—	4	4	no	—
197	C₂×C₄	—	4	2	no	—
198	C₂×C₄	—	4	2	no	—
199	C₂×C₄	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
200	C₂×C₄	(C₂²⋊D₄₇)⋊C₄	1	2	yes	—
201	C₂×C₄	#13093992	1	1	yes	characteristic
202	C₂×C₄	#13094001	1	2	yes	—
203	C₈	—	8	1	no	—
204	C₈	—	188	2	no	—
205	C₈	—	188	2	no	—
206	C₈	—	188	2	no	—
207	C₈	—	188	2	no	—
208	C₂²	—	2	1	no	—
209	C₂²	—	2	2	no	—
210	C₂²	—	4	2	no	—
211	C₂²	—	4	4	no	—
212	C₂²	#15688995	1	1	yes	characteristic
213	C₄	—	2	1	no	—
214	C₄	—	2	2	no	—
215	C₄	—	4	4	no	—
216	C₄	—	4	2	no	—
217	C₄	#15689237	1	2	yes	—
218	C₄	#15689242	1	2	yes	—
219	C₄	#15689290	1	1	yes	characteristic
220	C₄	#15689300	1	1	yes	characteristic
221	C₂	—	4	1	no	—
222	C₂	—	8	2	no	—
223	C₂	#20243597	1	1	yes	characteristic
224	C₂	#20243661	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20252455

C₂

yes

maximal, characteristic

#20242179

C₂

yes

maximal, complemented

#20242050

C₂

yes

maximal, characteristic

#20241221

C₂

yes

maximal, complemented

C₄₇×#1962

C₂

yes

maximal, characteristic

#15689201

—

D₄×C₄₇⋊C₈

—

#15689176

C₂²

yes

—

#15689176

C₂²

yes

—

—

—

—

C₄

yes

characteristic, complemented

#15688872

C₄

yes

characteristic, complemented

#15688867

C₄

yes

complemented

C₄₇×#532

C₂²

yes

characteristic

C₄₇×#423

C₂²

yes

characteristic

C₂×#104×C₄₇

—

C₄³×C₄₇

C₂²

yes

characteristic

C₂×#10625329

—

C₂²×C₄₇⋊C₈

—

C₄⋊(C₄₇⋊C₈)

C₂×C₄

yes

—

C₄⋊(C₄₇⋊C₈)

C₂×C₄

yes

—

C₄⋊(C₄₇⋊C₈)

D₄

yes

—

C₄⋊(C₄₇⋊C₈)

D₄

yes

—

C₂²⋊(C₄₇⋊C₈)

—

C₂²⋊(C₄₇⋊C₈)

—

C₂²⋊(C₄₇⋊C₈)

—

C₄×C₄₇⋊C₈

—

C₄×C₄₇⋊C₈

—

C₂×#41×C₄₇

D₄

yes

—

C₂×M₁₆×C₄₇

—

C₄⋊Q₈×C₄₇

C₂×C₄

yes

characteristic

#127×C₄₇

C₂×C₄

yes

characteristic

#124×C₄₇

Q₈

yes

—

#121×C₄₇

—

C₄×D₄×C₄₇

—

C₂×C₄²×C₄₇

—

C₂×C₄²×C₄₇

—

C₂×C₄²×C₄₇

C₂³

yes

characteristic

C₄⋊C₈×C₄₇

—

#104×C₄₇

—

#41×C₄₇

—

#41×C₄₇

—

C₂×Q₈×C₄₇

C₄⋊C₄

yes

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

C₄⋊C₄

yes

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₄₇

#41

yes

—

M₁₆×C₄₇

—

C₂×C₈×C₄₇

—

C₄⋊C₄×C₄₇

—

C₂²⋊C₄×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

C₂²×C₄

yes

characteristic

C₄²×C₄₇

C₂×D₄

yes

—

C₄²×C₄₇

C₂×Q₈

yes

characteristic

#10625329

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

—

C₂³×C₄₇

#123

yes

characteristic

Q₈×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

D₄×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

—

C₂×C₄×C₄₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₄₇

C₄×D₄

yes

—

C₂×C₄×C₄₇

#120

yes

characteristic

C₂×C₄×C₄₇

C₂²⋊Q₈

yes

—

C₈×C₄₇

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

#8935

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

100

C₂²×C₄₇

#385

yes

characteristic

101

C₄×C₄₇

—

102

C₄×C₄₇

—

103

C₄×C₄₇

—

104

C₄×C₄₇

—

105

C₄×C₄₇

#443

yes

—

106

C₄×C₄₇

#444

yes

—

107

C₄×C₄₇

#453

yes

characteristic

108

C₄×C₄₇

#455

yes

characteristic

109

#2979

—

110

#1962

D₄₇

yes

characteristic

111

#1574

—

112

#1543

—

113

#1276

—

114

C₂×C₄₇

—

115

C₂×C₄₇

—

116

C₂×C₄₇

#1856

yes

characteristic

117

C₂×C₄₇

#1866

yes

—

118

#532

C₂×D₄₇

yes

characteristic

119

#436

—

120

C₈×D₄

—

121

#431

—

122

#431

—

123

#423

C₂×D₄₇

yes

characteristic

124

C₂×#104

—

125

#408

—

126

C₄³

C₂×D₄₇

yes

characteristic

127

#335

—

128

#334

—

129

#333

—

130

#333

—

131

#331

—

132

C₄₇

#8935

yes

characteristic, complemented, Sylow

133

C₂×#41

C₂²⋊D₄₇

yes

—

134

C₂×M₁₆

—

135

C₂×M₁₆

—

136

C₂²×C₈

—

137

C₄⋊Q₈

C₄×D₄₇

yes

characteristic

138

#127

C₄×D₄₇

yes

characteristic

139

#124

Dic₉₄

yes

—

140

#121

—

141

C₄×D₄

—

142

C₂×C₄²

—

143

C₂×C₄²

—

144

C₂×C₄²

C₂²×D₄₇

yes

characteristic

145

C₄⋊C₈

—

146

C₄⋊C₈

—

147

C₄⋊C₈

—

148

C₄⋊C₈

—

149

C₄⋊C₈

—

150

#104

—

151

C₂²⋊C₈

—

152

C₂²⋊C₈

—

153

C₂²⋊C₈

—

154

C₄×C₈

—

155

C₄×C₈

—

156

#41

—

157

#41

—

158

C₂×Q₈

Dic₄₇⋊C₄

yes

—

159

C₂×D₄

—

160

C₂×D₄

—

161

C₂×D₄

Dic₄₇⋊C₄

yes

—

162

C₂²×C₄

—

163

C₂²×C₄

—

164

C₂²×C₄

#10625352

yes

—

165

C₂²×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

166

M₁₆

—

167

M₁₆

—

376

—

168

C₂×C₈

—

169

C₂×C₈

—

170

C₂×C₈

—

171

C₂×C₈

—

172

C₂×C₈

—

173

C₂×C₈

—

376

—

174

C₄⋊C₄

—

175

C₂²⋊C₄

—

176

C₄²

—

177

C₄²

—

178

C₄²

—

179

C₄²

—

180

C₄²

C₂×Dic₉₄

yes

characteristic

181

C₄²

C₂×C₄×D₄₇

yes

characteristic

182

C₄²

D₄×D₄₇

yes

—

183

C₂³

—

184

C₂³

#13094005

yes

characteristic

185

Q₈

—

186

D₄

—

187

D₄

—

188

D₄

—

189

D₄

—

190

C₂×C₄

—

191

C₂×C₄

—

192

C₂×C₄

—

193

C₂×C₄

—

194

C₂×C₄

—

195

C₂×C₄

—

196

C₂×C₄

—

197

C₂×C₄

—

198

C₂×C₄

—

199

C₂×C₄

C₂×Dic₄₇⋊C₄

yes

characteristic

200

C₂×C₄

(C₂²⋊D₄₇)⋊C₄

yes

—

201

C₂×C₄

#13093992

yes

characteristic

202

C₂×C₄

#13094001

yes

—

203

C₈

—

204

C₈

—

188

—

205

C₈

—

188

—

206

C₈

—

188

—

207

C₈

—

188

—

208

C₂²

—

209

C₂²

—

210

C₂²

—

211

C₂²

—

212

C₂²

#15688995

yes

characteristic

213

C₄

—

214

C₄

—

215

C₄

—

216

C₄

—

217

C₄

#15689237

yes

—

218

C₄

#15689242

yes

—

219

C₄

#15689290

yes

characteristic

220

C₄

#15689300

yes

characteristic

221

C₂

—

222

C₂

—

223

C₂

#20243597

yes

characteristic

224

C₂

#20243661

yes

—

	subgroup	G/H
center	C₂×C₄	#13093992
commutator	C₄²×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1962	C₂
socle	C₂²×C₄₇	#385
prime radical	C₄₇×#532	C₂²
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#1962	D₄₇
hypocenter	C₄₇	#8935
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1962

subgroup

G/H

center

C₂×C₄

#13093992

commutator

C₄²×C₄₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₄₇

Fitting

C₄₇×#1962

C₂

socle

C₂²×C₄₇

#385

prime radical

C₄₇×#532

C₂²

cosocle

C₂×C₄²×C₄₇

C₂³

D₄₇

C₄₇

G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1962

N	H ≅ G/N	type
2. #20242179	C₂	semidirect
4. #20241221	C₂	semidirect
13. #15688872	C₄	semidirect
14. #15688872	C₄	semidirect
15. #15688867	C₄	semidirect
132. C₄₇	#8935	semidirect

H ≅ G/N

type

2. #20242179

C₂

semidirect

4. #20241221

C₂

semidirect

13. #15688872

C₄

semidirect

14. #15688872

C₄

semidirect

15. #15688867

C₄

semidirect

132. C₄₇

#8935

semidirect

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

subgroup	index
#20252455	2
#20242179	2
#20242050	2
#20241221	2
C₄₇×#1962	2
#8935	47

subgroup

index

C₄₇×#1962

Subgroups — #26110785

Conjugacy classes of subgroups

Complements

Maximal subgroups