Group #26135667 — subgroups

Subgroups — #26135667

Order 12032.

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

normal subgroups only

This group has 436 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×#15688939	C₂	1	2	yes	maximal
2	#20245108	C₂	1	2	yes	maximal
3	#20244136	C₂	1	1	yes	maximal, characteristic
4	#20241470	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄₇×#494	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	#15688939	—	2	4	no	—
11	#15688927	—	2	4	no	—
12	#15688883	—	2	1	no	—
13	C₂²×Q₁₆×C₄₇	—	2	1	no	—
14	C₂²×Q₁₆×C₄₇	C₂²	1	2	yes	—
15	C₂×C₄⋊Q₈×C₄₇	—	2	1	no	—
16	C₄₇×#494	—	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	—
25	#13093956	—	2	1	no	—
26	Dic₄₇⋊C₈	—	4	2	no	—
27	C₂²×Q₈×C₄₇	—	2	2	no	—
28	C₂×Q₁₆×C₄₇	—	2	4	no	—
29	C₂×Q₁₆×C₄₇	—	2	2	no	—
30	C₂×Q₁₆×C₄₇	—	2	2	no	—
31	C₂×Q₁₆×C₄₇	—	2	2	no	—
32	C₂×Q₁₆×C₄₇	—	2	2	no	—
33	C₂×Q₁₆×C₄₇	—	2	2	no	—
34	C₂×Q₁₆×C₄₇	—	4	2	no	—
35	C₂×Q₁₆×C₄₇	—	4	2	no	—
36	C₂×Q₁₆×C₄₇	D₄	1	4	yes	—
37	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
38	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
39	C₄⋊Q₈×C₄₇	—	2	4	no	—
40	C₄⋊Q₈×C₄₇	—	4	2	no	—
41	C₂×C₄⋊C₄×C₄₇	—	4	2	no	—
42	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
43	C₄×C₈×C₄₇	—	2	2	no	—
44	C₂×Q₈×C₄₇	—	2	4	no	—
45	C₂×Q₈×C₄₇	—	4	2	no	—
46	C₂×Q₈×C₄₇	—	4	2	no	—
47	C₂×Q₈×C₄₇	—	4	2	no	—
48	C₂×Q₈×C₄₇	—	4	2	no	—
49	C₂×Q₈×C₄₇	—	4	2	no	—
50	C₂²×C₄×C₄₇	—	2	1	no	—
51	C₂²×C₄×C₄₇	—	4	2	no	—
52	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
53	Q₁₆×C₄₇	—	2	4	no	—
54	Q₁₆×C₄₇	—	2	4	no	—
55	Q₁₆×C₄₇	—	4	8	no	—
56	Q₁₆×C₄₇	—	4	2	no	—
57	Q₁₆×C₄₇	—	4	2	no	—
58	C₂×C₈×C₄₇	—	2	1	no	—
59	C₂×C₈×C₄₇	—	2	1	no	—
60	C₂×C₈×C₄₇	D₈	1	2	yes	—
61	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
62	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
63	C₂×C₈×C₄₇	#41	1	2	yes	—
64	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
65	C₄⋊C₄×C₄₇	—	4	8	no	—
66	C₄²×C₄₇	—	2	1	no	—
67	C₄²×C₄₇	SD₁₆	1	2	yes	—
68	C₂²⋊Dic₄₇	—	4	1	no	—
69	C₄⋊Dic₄₇	—	2	2	no	—
70	C₄×Dic₄₇	—	4	1	no	—
71	#10625329	—	4	1	no	—
72	#10625329	—	4	1	no	—
73	C₂×C₄₇⋊C₈	—	4	1	no	—
74	C₄₇⋊C₁₆	—	4	2	no	—
75	C₂³×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
76	Q₈×C₄₇	—	4	4	no	—
77	Q₈×C₄₇	—	4	4	no	—
78	Q₈×C₄₇	—	8	2	no	—
79	Q₈×C₄₇	—	8	2	no	—
80	C₂×C₄×C₄₇	—	2	4	no	—
81	C₂×C₄×C₄₇	—	2	2	no	—
82	C₂×C₄×C₄₇	—	4	4	no	—
83	C₂×C₄×C₄₇	—	8	2	no	—
84	C₂×C₄×C₄₇	—	8	2	no	—
85	C₂×C₄×C₄₇	#120	1	1	yes	characteristic
86	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
87	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
88	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
89	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
90	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
91	C₂×Dic₄₇	—	4	2	no	—
92	C₈×C₄₇	—	2	2	no	—
93	C₈×C₄₇	#121	1	2	yes	—
94	C₈×C₄₇	#123	1	2	yes	—
95	C₄₇⋊C₈	—	8	1	no	—
96	#13415	—	47	1	no	maximal, Sylow
97	C₂²×C₄₇	—	2	2	no	—
98	C₂²×C₄₇	#450	1	1	yes	characteristic
99	C₂²×C₄₇	#459	1	1	yes	characteristic
100	C₂²×C₄₇	#490	1	1	yes	characteristic
101	C₄×C₄₇	—	2	4	no	—
102	C₄×C₄₇	—	8	4	no	—
103	C₄×C₄₇	#394	1	1	yes	characteristic
104	C₄×C₄₇	#447	1	1	yes	characteristic
105	C₄×C₄₇	#460	1	1	yes	characteristic
106	C₄×C₄₇	#486	1	1	yes	characteristic
107	Dic₄₇	—	8	2	no	—
108	C₂×#494	D₄₇	1	1	yes	characteristic
109	C₂×#362	—	47	2	no	—
110	#2141	—	47	2	no	—
111	#1945	—	47	1	no	—
112	#1376	—	47	1	no	—
113	C₂×C₄₇	—	2	1	no	—
114	C₂×C₄₇	—	2	1	no	—
115	C₂×C₄₇	#2041	1	1	yes	characteristic
116	C₂×C₄₇	#2214	1	2	yes	—
117	C₂²×Q₁₆	—	2	1	no	—
118	C₂²×Q₁₆	C₂×D₄₇	1	2	yes	—
119	C₂×C₄⋊Q₈	—	2	1	no	—
120	C₂×M₃₂	—	47	2	no	—
121	#494	—	2	4	no	—
122	C₂×#108	—	47	1	no	—
123	#427	—	47	1	no	—
124	#424	—	94	1	no	—
125	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
126	#362	—	94	4	no	—
127	#358	—	94	4	no	—
128	#339	—	94	1	no	—
129	C₄₇	#13415	1	1	yes	characteristic, complemented, Sylow
130	C₂²×Q₈	—	2	2	no	—
131	C₂×Q₁₆	—	2	4	no	—
132	C₂×Q₁₆	—	2	2	no	—
133	C₂×Q₁₆	—	2	2	no	—
134	C₂×Q₁₆	—	2	2	no	—
135	C₂×Q₁₆	—	2	2	no	—
136	C₂×Q₁₆	—	2	2	no	—
137	C₂×Q₁₆	—	4	2	no	—
138	C₂×Q₁₆	—	4	2	no	—
139	C₂×Q₁₆	C₂²⋊D₄₇	1	4	yes	—
140	C₂×M₁₆	—	94	1	no	—
141	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
142	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
143	C₄⋊Q₈	—	2	4	no	—
144	C₄⋊Q₈	—	4	2	no	—
145	#117	—	94	1	no	—
146	C₂×C₄⋊C₄	—	4	2	no	—
147	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
148	M₃₂	—	94	2	no	—
149	M₃₂	—	94	2	no	—
150	C₂×C₁₆	—	94	2	no	—
151	#108	—	94	2	no	—
152	#107	—	47	2	no	—
153	#106	—	94	1	no	—
154	C₄⋊C₈	—	188	2	no	—
155	C₄×C₈	—	2	2	no	—
156	C₂×Q₈	—	2	4	no	—
157	C₂×Q₈	—	4	2	no	—
158	C₂×Q₈	—	4	2	no	—
159	C₂×Q₈	—	4	2	no	—
160	C₂×Q₈	—	4	2	no	—
161	C₂×Q₈	—	4	2	no	—
162	C₂²×C₄	—	2	1	no	—
163	C₂²×C₄	—	4	2	no	—
164	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
165	Q₁₆	—	2	4	no	—
166	Q₁₆	—	2	4	no	—
167	Q₁₆	—	4	8	no	—
168	Q₁₆	—	4	2	no	—
169	Q₁₆	—	4	2	no	—
170	M₁₆	—	188	1	no	—
171	M₁₆	—	188	1	no	—
172	C₂×C₈	—	2	1	no	—
173	C₂×C₈	—	2	1	no	—
174	C₂×C₈	—	188	1	no	—
175	C₂×C₈	#10625334	1	2	yes	—
176	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
177	C₂×C₈	#10625352	1	2	yes	—
178	C₂×C₈	#10625352	1	1	yes	characteristic
179	C₂×C₈	C₂×C₂²⋊D₄₇	1	2	yes	—
180	C₄⋊C₄	—	4	8	no	—
181	C₄⋊C₄	—	94	2	no	—
182	C₂²⋊C₄	—	188	1	no	—
183	C₄²	—	2	1	no	—
184	C₄²	—	188	1	no	—
185	C₄²	#10625335	1	2	yes	—
186	C₁₆	—	188	2	no	—
187	C₂³	#13094050	1	1	yes	characteristic
188	Q₈	—	4	4	no	—
189	Q₈	—	4	4	no	—
190	Q₈	—	8	2	no	—
191	Q₈	—	8	2	no	—
192	C₂×C₄	—	2	4	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	4	4	no	—
195	C₂×C₄	—	8	2	no	—
196	C₂×C₄	—	8	2	no	—
197	C₂×C₄	—	188	2	no	—
198	C₂×C₄	#13093993	1	1	yes	characteristic
199	C₂×C₄	#13093995	1	1	yes	characteristic
200	C₂×C₄	#13094011	1	1	yes	characteristic
201	C₂×C₄	C₂×#10625334	1	1	yes	characteristic
202	C₂×C₄	C₂×#10625335	1	1	yes	characteristic
203	C₂×C₄	#13094050	1	1	yes	characteristic
204	C₈	—	2	2	no	—
205	C₈	—	376	1	no	—
206	C₈	#13093996	1	2	yes	—
207	C₈	#13094005	1	2	yes	—
208	C₂²	—	2	2	no	—
209	C₂²	#15689274	1	1	yes	characteristic
210	C₂²	#15689315	1	1	yes	characteristic
211	C₂²	#15689487	1	1	yes	characteristic
212	C₄	—	2	4	no	—
213	C₄	—	8	4	no	—
214	C₄	—	376	2	no	—
215	C₄	#15689037	1	1	yes	characteristic
216	C₄	#15689257	1	1	yes	characteristic
217	C₄	#15689320	1	1	yes	characteristic
218	C₄	#15689469	1	1	yes	characteristic
219	C₂	—	2	1	no	—
220	C₂	—	2	1	no	—
221	C₂	#20244619	1	1	yes	characteristic
222	C₂	#20245466	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15688939

C₂

yes

maximal

#20245108

C₂

yes

maximal

#20244136

C₂

yes

maximal, characteristic

#20241470

C₂

yes

maximal, characteristic

C₂×C₄₇×#494

C₂

yes

maximal, characteristic

C₂×#13093964

C₂²

yes

—

C₂×#13093960

C₂²

yes

characteristic

#15689161

C₂²

yes

—

—

—

—

C₂²×Q₁₆×C₄₇

—

C₂²×Q₁₆×C₄₇

C₂²

yes

—

C₂×C₄⋊Q₈×C₄₇

—

C₄₇×#494

—

C₂×C₄×C₈×C₄₇

C₂²

yes

—

—

—

—

C₂×C₄₇⋊C₁₆

—

#13093960

—

#13093958

D₄

yes

—

#13093956

—

Dic₄₇⋊C₈

—

C₂²×Q₈×C₄₇

—

C₂×Q₁₆×C₄₇

—

C₂×Q₁₆×C₄₇

—

C₂×Q₁₆×C₄₇

—

C₂×Q₁₆×C₄₇

—

C₂×Q₁₆×C₄₇

—

C₂×Q₁₆×C₄₇

—

C₂×Q₁₆×C₄₇

—

C₂×Q₁₆×C₄₇

—

C₂×Q₁₆×C₄₇

D₄

yes

—

C₂²×C₈×C₄₇

D₄

yes

characteristic

C₂²×C₈×C₄₇

C₂³

yes

characteristic

C₄⋊Q₈×C₄₇

—

C₄⋊Q₈×C₄₇

—

C₂×C₄⋊C₄×C₄₇

—

C₂×C₄²×C₄₇

D₄

yes

characteristic

C₄×C₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂×D₄

yes

characteristic

Q₁₆×C₄₇

—

Q₁₆×C₄₇

—

Q₁₆×C₄₇

—

Q₁₆×C₄₇

—

Q₁₆×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

C₄⋊C₄×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

SD₁₆

yes

—

C₂²⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄×Dic₄₇

—

#10625329

—

#10625329

—

C₂×C₄₇⋊C₈

—

C₄₇⋊C₁₆

—

C₂³×C₄₇

AGL(1,ℤ/8)

yes

characteristic

Q₈×C₄₇

—

Q₈×C₄₇

—

Q₈×C₄₇

—

Q₈×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×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

—

#13415

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

yes

C₂²×C₄₇

yes

100

C₂²×C₄₇

#490

yes

characteristic

101

C₄×C₄₇

—

102

C₄×C₄₇

—

103

C₄×C₄₇

#394

yes

characteristic

104

C₄×C₄₇

#447

yes

characteristic

105

C₄×C₄₇

#460

yes

characteristic

106

C₄×C₄₇

#486

yes

characteristic

107

Dic₄₇

—

108

C₂×#494

D₄₇

yes

characteristic

109

C₂×#362

—

110

#2141

—

111

#1945

—

112

#1376

—

113

C₂×C₄₇

—

114

C₂×C₄₇

—

115

C₂×C₄₇

#2041

yes

characteristic

116

C₂×C₄₇

#2214

yes

—

117

C₂²×Q₁₆

—

118

C₂²×Q₁₆

C₂×D₄₇

yes

—

119

C₂×C₄⋊Q₈

—

120

C₂×M₃₂

—

121

#494

—

122

C₂×#108

—

123

#427

—

124

#424

—

125

C₂×C₄×C₈

C₂×D₄₇

yes

characteristic

126

#362

—

127

#358

—

128

#339

—

129

C₄₇

#13415

yes

characteristic, complemented, Sylow

130

C₂²×Q₈

—

131

C₂×Q₁₆

—

132

C₂×Q₁₆

—

133

C₂×Q₁₆

—

134

C₂×Q₁₆

—

135

C₂×Q₁₆

—

136

C₂×Q₁₆

—

137

C₂×Q₁₆

—

138

C₂×Q₁₆

—

139

C₂×Q₁₆

C₂²⋊D₄₇

yes

—

140

C₂×M₁₆

—

141

C₂²×C₈

C₂²⋊D₄₇

yes

characteristic

142

C₂²×C₈

C₂²×D₄₇

yes

characteristic

143

C₄⋊Q₈

—

144

C₄⋊Q₈

—

145

#117

—

146

C₂×C₄⋊C₄

—

147

C₂×C₄²

C₂²⋊D₄₇

yes

characteristic

148

M₃₂

—

149

M₃₂

—

150

C₂×C₁₆

—

151

#108

—

152

#107

—

153

#106

—

154

C₄⋊C₈

—

188

—

155

C₄×C₈

—

156

C₂×Q₈

—

157

C₂×Q₈

—

158

C₂×Q₈

—

159

C₂×Q₈

—

160

C₂×Q₈

—

161

C₂×Q₈

—

162

C₂²×C₄

—

163

C₂²×C₄

—

164

C₂²×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

165

Q₁₆

—

166

Q₁₆

—

167

Q₁₆

—

168

Q₁₆

—

169

Q₁₆

—

170

M₁₆

—

188

—

171

M₁₆

—

188

—

172

C₂×C₈

—

173

C₂×C₈

—

174

C₂×C₈

—

188

—

175

C₂×C₈

#10625334

yes

—

176

C₂×C₈

D₄×D₄₇

yes

characteristic

177

C₂×C₈

#10625352

yes

—

178

C₂×C₈

#10625352

yes

characteristic

179

C₂×C₈

C₂×C₂²⋊D₄₇

yes

—

180

C₄⋊C₄

—

181

C₄⋊C₄

—

182

C₂²⋊C₄

—

188

—

183

C₄²

—

184

C₄²

—

188

—

185

C₄²

#10625335

yes

—

186

C₁₆

—

188

—

187

C₂³

#13094050

yes

characteristic

188

Q₈

—

189

Q₈

—

190

Q₈

—

191

Q₈

—

192

C₂×C₄

—

193

C₂×C₄

—

194

C₂×C₄

—

195

C₂×C₄

—

196

C₂×C₄

—

197

C₂×C₄

—

188

—

198

C₂×C₄

#13093993

yes

characteristic

199

C₂×C₄

#13093995

yes

characteristic

200

C₂×C₄

#13094011

yes

characteristic

201

C₂×C₄

C₂×#10625334

yes

characteristic

202

C₂×C₄

C₂×#10625335

yes

characteristic

203

C₂×C₄

#13094050

yes

characteristic

204

C₈

—

205

C₈

—

376

—

206

C₈

#13093996

yes

—

207

C₈

#13094005

yes

—

208

C₂²

—

209

C₂²

#15689274

yes

characteristic

210

C₂²

#15689315

yes

characteristic

211

C₂²

#15689487

yes

characteristic

212

C₄

—

213

C₄

—

214

C₄

—

376

—

215

C₄

#15689037

yes

characteristic

216

C₄

#15689257

yes

characteristic

217

C₄

#15689320

yes

characteristic

218

C₄

#15689469

yes

characteristic

219

C₂

—

220

C₂

—

221

C₂

#20244619

yes

characteristic

222

C₂

#20245466

yes

—

	subgroup	G/H
center	C₂²	#15689315
commutator, cosocle	C₂²×C₈×C₄₇	C₂³
Frattini	C₂²×C₈	C₂²×D₄₇
Fitting	C₂×C₄₇×#494	C₂
socle	C₂²×C₄₇	#459
prime radical	C₂³×C₄₇	AGL(1,ℤ/8)
hypercenter	C₂×#494	D₄₇
hypocenter	C₄₇	#13415
LCS	G ▷ C₂²×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#494

subgroup

G/H

center

C₂²

#15689315

commutator, cosocle

C₂²×C₈×C₄₇

C₂³

Frattini

C₂²×C₈

C₂²×D₄₇

Fitting

C₂×C₄₇×#494

C₂

C₂²×C₄₇

C₂³×C₄₇

D₄₇

C₄₇

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

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#494

N	H ≅ G/N	type
129. C₄₇	#13415	semidirect

H ≅ G/N

type

129. C₄₇

#13415

semidirect

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

subgroup	index
C₂×#15688939	2
#20245108	2
#20244136	2
#20241470	2
C₂×C₄₇×#494	2
#13415	47

subgroup

index

C₂×C₄₇×#494

Subgroups — #26135667

Conjugacy classes of subgroups

Complements

Maximal subgroups