Group #26135655 — subgroups

Subgroups — #26135655

Order 12032.

540 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 540 conjugacy classes of subgroups — too many to draw. Only its 101 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×#15688936	C₂	1	2	yes	maximal, complemented
2	#20245104	C₂	1	2	yes	maximal, complemented
3	#20244136	C₂	1	1	yes	maximal, characteristic, complemented
4	#20241470	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₄₇×#493	C₂	1	1	yes	maximal, characteristic
6	C₂×#13093964	C₂²	1	2	yes	—
7	C₂×#13093960	C₂²	1	1	yes	characteristic
8	#15689161	C₂²	1	1	yes	characteristic
9	#15689149	—	2	1	no	—
10	#15688936	—	2	4	no	—
11	#15688924	—	2	4	no	—
12	#15688883	—	2	1	no	—
13	C₂²×D₈×C₄₇	—	2	1	no	—
14	C₂²×D₈×C₄₇	C₂²	1	2	yes	—
15	C₂×#127×C₄₇	—	2	1	no	—
16	C₄₇×#493	—	2	4	no	—
17	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
18	C₂×#10625329	—	2	1	no	—
19	#13093983	—	2	1	no	—
20	#13093964	—	2	2	no	—
21	#13093964	—	2	2	no	—
22	C₂×C₄₇⋊C₁₆	—	2	2	no	—
23	#13093960	—	2	2	no	—
24	#13093958	D₄	1	2	yes	complemented
25	#13093956	—	2	1	no	—
26	Dic₄₇⋊C₈	—	4	2	no	—
27	C₂²×D₄×C₄₇	—	2	2	no	—
28	C₂²×D₄×C₄₇	—	4	2	no	—
29	C₂×D₈×C₄₇	—	2	4	no	—
30	C₂×D₈×C₄₇	—	2	2	no	—
31	C₂×D₈×C₄₇	—	2	2	no	—
32	C₂×D₈×C₄₇	—	2	2	no	—
33	C₂×D₈×C₄₇	—	2	2	no	—
34	C₂×D₈×C₄₇	—	2	2	no	—
35	C₂×D₈×C₄₇	—	4	2	no	—
36	C₂×D₈×C₄₇	—	4	2	no	—
37	C₂×D₈×C₄₇	D₄	1	4	yes	—
38	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
39	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
40	#127×C₄₇	—	2	4	no	—
41	#127×C₄₇	—	4	2	no	—
42	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
43	C₄×C₈×C₄₇	—	2	2	no	—
44	C₂⁴×C₄₇	—	4	2	no	—
45	C₂×D₄×C₄₇	—	2	4	no	—
46	C₂×D₄×C₄₇	—	4	8	no	—
47	C₂×D₄×C₄₇	—	4	2	no	—
48	C₂×D₄×C₄₇	—	4	2	no	—
49	C₂×D₄×C₄₇	—	4	2	no	—
50	C₂×D₄×C₄₇	—	4	2	no	—
51	C₂×D₄×C₄₇	—	4	2	no	—
52	C₂×D₄×C₄₇	—	8	8	no	—
53	C₂²×C₄×C₄₇	—	2	1	no	—
54	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
55	D₈×C₄₇	—	2	4	no	—
56	D₈×C₄₇	—	2	4	no	—
57	D₈×C₄₇	—	4	8	no	—
58	D₈×C₄₇	—	4	2	no	—
59	D₈×C₄₇	—	4	2	no	—
60	C₂×C₈×C₄₇	—	2	1	no	—
61	C₂×C₈×C₄₇	—	2	1	no	—
62	C₂×C₈×C₄₇	D₈	1	2	yes	—
63	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
64	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
65	C₂×C₈×C₄₇	#41	1	2	yes	—
66	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
67	C₄²×C₄₇	—	2	1	no	—
68	C₄²×C₄₇	SD₁₆	1	2	yes	—
69	C₂²⋊Dic₄₇	—	4	1	no	—
70	C₄⋊Dic₄₇	—	2	2	no	—
71	C₄×Dic₄₇	—	4	1	no	—
72	#10625329	—	4	1	no	—
73	#10625329	—	4	1	no	—
74	C₂×C₄₇⋊C₈	—	4	1	no	—
75	C₄₇⋊C₁₆	—	4	2	no	—
76	C₂³×C₄₇	—	4	4	no	—
77	C₂³×C₄₇	—	8	8	no	—
78	C₂³×C₄₇	—	8	2	no	—
79	C₂³×C₄₇	—	8	2	no	—
80	C₂³×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
81	D₄×C₄₇	—	4	4	no	—
82	D₄×C₄₇	—	4	4	no	—
83	D₄×C₄₇	—	8	16	no	—
84	D₄×C₄₇	—	8	2	no	—
85	D₄×C₄₇	—	8	2	no	—
86	C₂×C₄×C₄₇	—	2	4	no	—
87	C₂×C₄×C₄₇	—	2	2	no	—
88	C₂×C₄×C₄₇	#120	1	1	yes	characteristic
89	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
90	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
91	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
92	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
93	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
94	C₂×Dic₄₇	—	4	2	no	—
95	C₈×C₄₇	—	2	2	no	—
96	C₈×C₄₇	#121	1	2	yes	—
97	C₈×C₄₇	#123	1	2	yes	—
98	C₄₇⋊C₈	—	8	1	no	—
99	#13413	—	47	1	no	maximal, Sylow
100	C₂²×C₄₇	—	2	2	no	—
101	C₂²×C₄₇	—	8	8	no	—
102	C₂²×C₄₇	—	8	4	no	—
103	C₂²×C₄₇	—	16	4	no	—
104	C₂²×C₄₇	—	16	4	no	—
105	C₂²×C₄₇	#450	1	1	yes	characteristic
106	C₂²×C₄₇	#459	1	1	yes	characteristic
107	C₂²×C₄₇	#490	1	1	yes	characteristic
108	C₄×C₄₇	—	2	4	no	—
109	C₄×C₄₇	#394	1	1	yes	characteristic
110	C₄×C₄₇	#447	1	1	yes	characteristic
111	C₄×C₄₇	#460	1	1	yes	characteristic
112	C₄×C₄₇	#486	1	1	yes	characteristic
113	Dic₄₇	—	8	2	no	—
114	C₂×#493	D₄₇	1	1	yes	characteristic
115	C₂×#361	—	47	2	no	—
116	#2140	—	47	2	no	—
117	#1945	—	47	1	no	—
118	#1376	—	47	1	no	—
119	C₂×C₄₇	—	2	1	no	—
120	C₂×C₄₇	—	2	1	no	—
121	C₂×C₄₇	—	16	4	no	—
122	C₂×C₄₇	#2041	1	1	yes	characteristic
123	C₂×C₄₇	#2213	1	2	yes	—
124	C₂²×D₈	—	2	1	no	—
125	C₂²×D₈	C₂×D₄₇	1	2	yes	—
126	C₂×#127	—	2	1	no	—
127	C₂×M₃₂	—	47	2	no	—
128	#493	—	2	4	no	—
129	C₂×#108	—	47	1	no	—
130	#427	—	47	1	no	—
131	#424	—	94	1	no	—
132	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
133	#361	—	94	4	no	—
134	#357	—	94	4	no	—
135	#339	—	94	1	no	—
136	C₄₇	#13413	1	1	yes	characteristic, complemented, Sylow
137	C₂²×D₄	—	2	2	no	—
138	C₂²×D₄	—	4	2	no	—
139	C₂×D₈	—	2	4	no	—
140	C₂×D₈	—	2	2	no	—
141	C₂×D₈	—	2	2	no	—
142	C₂×D₈	—	2	2	no	—
143	C₂×D₈	—	2	2	no	—
144	C₂×D₈	—	2	2	no	—
145	C₂×D₈	—	4	2	no	—
146	C₂×D₈	—	4	2	no	—
147	C₂×D₈	C₂²⋊D₄₇	1	4	yes	—
148	C₂×M₁₆	—	94	1	no	—
149	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
150	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
151	#127	—	2	4	no	—
152	#127	—	4	2	no	—
153	#117	—	94	1	no	—
154	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
155	M₃₂	—	94	2	no	—
156	M₃₂	—	94	2	no	—
157	C₂×C₁₆	—	94	2	no	—
158	#108	—	94	2	no	—
159	#107	—	47	2	no	—
160	#106	—	94	1	no	—
161	C₄⋊C₈	—	188	2	no	—
162	C₄×C₈	—	2	2	no	—
163	C₂⁴	—	4	2	no	—
164	C₂×D₄	—	2	4	no	—
165	C₂×D₄	—	4	8	no	—
166	C₂×D₄	—	4	2	no	—
167	C₂×D₄	—	4	2	no	—
168	C₂×D₄	—	4	2	no	—
169	C₂×D₄	—	4	2	no	—
170	C₂×D₄	—	4	2	no	—
171	C₂×D₄	—	8	8	no	—
172	C₂²×C₄	—	2	1	no	—
173	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
174	D₈	—	2	4	no	—
175	D₈	—	2	4	no	—
176	D₈	—	4	8	no	—
177	D₈	—	4	2	no	—
178	D₈	—	4	2	no	—
179	M₁₆	—	188	1	no	—
180	M₁₆	—	188	1	no	—
181	C₂×C₈	—	2	1	no	—
182	C₂×C₈	—	2	1	no	—
183	C₂×C₈	—	188	1	no	—
184	C₂×C₈	#10625334	1	2	yes	—
185	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
186	C₂×C₈	#10625352	1	2	yes	—
187	C₂×C₈	#10625352	1	1	yes	characteristic
188	C₂×C₈	C₂×C₂²⋊D₄₇	1	2	yes	—
189	C₄⋊C₄	—	94	2	no	—
190	C₂²⋊C₄	—	188	1	no	—
191	C₄²	—	2	1	no	—
192	C₄²	—	188	1	no	—
193	C₄²	#10625335	1	2	yes	—
194	C₁₆	—	188	2	no	—
195	C₂³	—	4	4	no	—
196	C₂³	—	8	8	no	—
197	C₂³	—	8	2	no	—
198	C₂³	—	8	2	no	—
199	C₂³	#13094050	1	1	yes	characteristic
200	D₄	—	4	4	no	—
201	D₄	—	4	4	no	—
202	D₄	—	8	16	no	—
203	D₄	—	8	2	no	—
204	D₄	—	8	2	no	—
205	C₂×C₄	—	2	4	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	188	2	no	—
208	C₂×C₄	#13093993	1	1	yes	characteristic
209	C₂×C₄	#13093995	1	1	yes	characteristic
210	C₂×C₄	#13094011	1	1	yes	characteristic
211	C₂×C₄	C₂×#10625334	1	1	yes	characteristic
212	C₂×C₄	C₂×#10625335	1	1	yes	characteristic
213	C₂×C₄	#13094050	1	1	yes	characteristic
214	C₈	—	2	2	no	—
215	C₈	—	376	1	no	—
216	C₈	#13093996	1	2	yes	—
217	C₈	#13094005	1	2	yes	—
218	C₂²	—	2	2	no	—
219	C₂²	—	8	8	no	—
220	C₂²	—	8	4	no	—
221	C₂²	—	16	4	no	—
222	C₂²	—	16	4	no	—
223	C₂²	#15689274	1	1	yes	characteristic
224	C₂²	#15689315	1	1	yes	characteristic
225	C₂²	#15689487	1	1	yes	characteristic
226	C₄	—	2	4	no	—
227	C₄	—	376	2	no	—
228	C₄	#15689037	1	1	yes	characteristic
229	C₄	#15689257	1	1	yes	characteristic
230	C₄	#15689320	1	1	yes	characteristic
231	C₄	#15689469	1	1	yes	characteristic
232	C₂	—	2	1	no	—
233	C₂	—	2	1	no	—
234	C₂	—	16	4	no	—
235	C₂	#20244619	1	1	yes	characteristic
236	C₂	#20245461	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15688936

C₂

yes

maximal, complemented

#20245104

C₂

yes

maximal, complemented

#20244136

C₂

yes

maximal, characteristic, complemented

#20241470

C₂

yes

maximal, characteristic, complemented

C₂×C₄₇×#493

C₂

yes

maximal, characteristic

C₂×#13093964

C₂²

yes

—

C₂×#13093960

C₂²

yes

characteristic

#15689161

C₂²

yes

—

—

—

—

C₂²×D₈×C₄₇

—

C₂²×D₈×C₄₇

C₂²

yes

—

C₂×#127×C₄₇

—

C₄₇×#493

—

C₂×C₄×C₈×C₄₇

C₂²

yes

—

—

—

—

C₂×C₄₇⋊C₁₆

—

#13093960

—

#13093958

D₄

yes

complemented

#13093956

—

Dic₄₇⋊C₈

—

C₂²×D₄×C₄₇

—

C₂²×D₄×C₄₇

—

C₂×D₈×C₄₇

—

C₂×D₈×C₄₇

—

C₂×D₈×C₄₇

—

C₂×D₈×C₄₇

—

C₂×D₈×C₄₇

—

C₂×D₈×C₄₇

—

C₂×D₈×C₄₇

—

C₂×D₈×C₄₇

—

C₂×D₈×C₄₇

D₄

yes

—

C₂²×C₈×C₄₇

D₄

yes

characteristic

C₂²×C₈×C₄₇

C₂³

yes

characteristic

#127×C₄₇

—

#127×C₄₇

—

C₂×C₄²×C₄₇

D₄

yes

characteristic

C₄×C₈×C₄₇

—

C₂⁴×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂×D₄

yes

characteristic

D₈×C₄₇

—

D₈×C₄₇

—

D₈×C₄₇

—

D₈×C₄₇

—

D₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

D₈

yes

—

C₂×C₈×C₄₇

C₂×D₄

yes

characteristic

C₂×C₈×C₄₇

C₂×D₄

yes

—

C₂×C₈×C₄₇

#41

yes

—

C₂×C₈×C₄₇

#41

yes

characteristic

C₄²×C₄₇

—

C₄²×C₄₇

SD₁₆

yes

—

C₂²⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄×Dic₄₇

—

#10625329

—

#10625329

—

C₂×C₄₇⋊C₈

—

C₄₇⋊C₁₆

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

AGL(1,ℤ/8)

yes

characteristic

D₄×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

yes

C₂×C₄×C₄₇

yes

C₂×C₄×C₄₇

yes

C₂×C₄×C₄₇

C₂×D₈

yes

characteristic

C₂×C₄×C₄₇

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₄₇

AGL(1,ℤ/8)

yes

characteristic

C₂×Dic₄₇

—

C₈×C₄₇

—

C₈×C₄₇

#121

yes

—

C₈×C₄₇

#123

yes

—

C₄₇⋊C₈

—

#13413

—

maximal, Sylow

100

C₂²×C₄₇

—

101

C₂²×C₄₇

—

102

C₂²×C₄₇

—

103

C₂²×C₄₇

—

104

C₂²×C₄₇

—

105

C₂²×C₄₇

#450

yes

characteristic

106

C₂²×C₄₇

#459

yes

characteristic

107

C₂²×C₄₇

#490

yes

characteristic

108

C₄×C₄₇

—

109

C₄×C₄₇

#394

yes

characteristic

110

C₄×C₄₇

#447

yes

characteristic

111

C₄×C₄₇

#460

yes

characteristic

112

C₄×C₄₇

#486

yes

characteristic

113

Dic₄₇

—

114

C₂×#493

D₄₇

yes

characteristic

115

C₂×#361

—

116

#2140

—

117

#1945

—

118

#1376

—

119

C₂×C₄₇

—

120

C₂×C₄₇

—

121

C₂×C₄₇

—

122

C₂×C₄₇

#2041

yes

characteristic

123

C₂×C₄₇

#2213

yes

—

124

C₂²×D₈

—

125

C₂²×D₈

C₂×D₄₇

yes

—

126

C₂×#127

—

127

C₂×M₃₂

—

128

#493

—

129

C₂×#108

—

130

#427

—

131

#424

—

132

C₂×C₄×C₈

C₂×D₄₇

yes

characteristic

133

#361

—

134

#357

—

135

#339

—

136

C₄₇

#13413

yes

characteristic, complemented, Sylow

137

C₂²×D₄

—

138

C₂²×D₄

—

139

C₂×D₈

—

140

C₂×D₈

—

141

C₂×D₈

—

142

C₂×D₈

—

143

C₂×D₈

—

144

C₂×D₈

—

145

C₂×D₈

—

146

C₂×D₈

—

147

C₂×D₈

C₂²⋊D₄₇

yes

—

148

C₂×M₁₆

—

149

C₂²×C₈

C₂²⋊D₄₇

yes

characteristic

150

C₂²×C₈

C₂²×D₄₇

yes

characteristic

151

#127

—

152

#127

—

153

#117

—

154

C₂×C₄²

C₂²⋊D₄₇

yes

characteristic

155

M₃₂

—

156

M₃₂

—

157

C₂×C₁₆

—

158

#108

—

159

#107

—

160

#106

—

161

C₄⋊C₈

—

188

—

162

C₄×C₈

—

163

C₂⁴

—

164

C₂×D₄

—

165

C₂×D₄

—

166

C₂×D₄

—

167

C₂×D₄

—

168

C₂×D₄

—

169

C₂×D₄

—

170

C₂×D₄

—

171

C₂×D₄

—

172

C₂²×C₄

—

173

C₂²×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

174

D₈

—

175

D₈

—

176

D₈

—

177

D₈

—

178

D₈

—

179

M₁₆

—

188

—

180

M₁₆

—

188

—

181

C₂×C₈

—

182

C₂×C₈

—

183

C₂×C₈

—

188

—

184

C₂×C₈

#10625334

yes

—

185

C₂×C₈

D₄×D₄₇

yes

characteristic

186

C₂×C₈

#10625352

yes

—

187

C₂×C₈

#10625352

yes

characteristic

188

C₂×C₈

C₂×C₂²⋊D₄₇

yes

—

189

C₄⋊C₄

—

190

C₂²⋊C₄

—

188

—

191

C₄²

—

192

C₄²

—

188

—

193

C₄²

#10625335

yes

—

194

C₁₆

—

188

—

195

C₂³

—

196

C₂³

—

197

C₂³

—

198

C₂³

—

199

C₂³

#13094050

yes

characteristic

200

D₄

—

201

D₄

—

202

D₄

—

203

D₄

—

204

D₄

—

205

C₂×C₄

—

206

C₂×C₄

—

207

C₂×C₄

—

188

—

208

C₂×C₄

#13093993

yes

characteristic

209

C₂×C₄

#13093995

yes

characteristic

210

C₂×C₄

#13094011

yes

characteristic

211

C₂×C₄

C₂×#10625334

yes

characteristic

212

C₂×C₄

C₂×#10625335

yes

characteristic

213

C₂×C₄

#13094050

yes

characteristic

214

C₈

—

215

C₈

—

376

—

216

C₈

#13093996

yes

—

217

C₈

#13094005

yes

—

218

C₂²

—

219

C₂²

—

220

C₂²

—

221

C₂²

—

222

C₂²

—

223

C₂²

#15689274

yes

characteristic

224

C₂²

#15689315

yes

characteristic

225

C₂²

#15689487

yes

characteristic

226

C₄

—

227

C₄

—

376

—

228

C₄

#15689037

yes

characteristic

229

C₄

#15689257

yes

characteristic

230

C₄

#15689320

yes

characteristic

231

C₄

#15689469

yes

characteristic

232

C₂

—

233

C₂

—

234

C₂

—

235

C₂

#20244619

yes

characteristic

236

C₂

#20245461

yes

—

	subgroup	G/H
center	C₂²	#15689315
commutator, cosocle	C₂²×C₈×C₄₇	C₂³
Frattini	C₂²×C₈	C₂²×D₄₇
Fitting, prime radical	C₂×C₄₇×#493	C₂
socle	C₂²×C₄₇	#459
hypercenter	C₂×#493	D₄₇
hypocenter	C₄₇	#13413
LCS	G ▷ C₂²×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#493

subgroup

G/H

center

C₂²

#15689315

commutator, cosocle

C₂²×C₈×C₄₇

C₂³

Frattini

C₂²×C₈

C₂²×D₄₇

Fitting, prime radical

C₂×C₄₇×#493

C₂

C₂²×C₄₇

D₄₇

C₄₇

G ▷ C₂²×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#493

N	H ≅ G/N	type
1. C₂×#15688936	C₂	semidirect
2. #20245104	C₂	semidirect
3. #20244136	C₂	semidirect
4. #20241470	C₂	semidirect
24. #13093958	D₄	semidirect
136. C₄₇	#13413	semidirect

H ≅ G/N

type

1. C₂×#15688936

C₂

semidirect

2. #20245104

C₂

semidirect

3. #20244136

C₂

semidirect

4. #20241470

C₂

semidirect

24. #13093958

D₄

semidirect

136. C₄₇

#13413

semidirect

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

subgroup	index
C₂×#15688936	2
#20245104	2
#20244136	2
#20241470	2
C₂×C₄₇×#493	2
#13413	47

subgroup

index

C₂×C₄₇×#493

Subgroups — #26135655

Conjugacy classes of subgroups

Complements

Maximal subgroups