Group #26134445 — subgroups

Subgroups — #26134445

Order 12032.

442 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 442 conjugacy classes of subgroups — too many to draw. Only its 139 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	#20252101	C₂	1	1	yes	maximal, characteristic
2	#20245144	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#15688906	C₂	1	4	yes	maximal, complemented
4	C₂×C₄₇×#346	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#13094030	C₂²	1	2	yes	—
6	#15689673	C₄	1	1	yes	characteristic
7	C₂×#13093964	C₂²	1	2	yes	—
8	#15689202	—	2	2	no	—
9	C₈×D₁₈₈	—	2	2	no	—
10	#15689147	C₄	1	1	yes	characteristic
11	#15689089	C₄	1	2	yes	—
12	C₄⋊(C₄₇⋊C₁₆)	—	2	2	no	—
13	#15688906	—	2	8	no	—
14	C₂²×C₁₆×C₄₇	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
16	C₄₇×#346	—	2	2	no	—
17	C₂×#10625350	C₂×C₄	1	2	yes	—
18	#13094030	—	2	4	no	—
19	#13094030	—	2	4	no	—
20	C₂×#10625329	C₂×C₄	1	2	yes	—
21	C₂×#10625324	—	2	2	no	—
22	C₂×C₈×D₄₇	—	2	2	no	—
23	#13094024	C₈	1	1	yes	characteristic
24	#13094021	C₈	1	1	yes	characteristic
25	#13094012	C₈	1	2	yes	—
26	#13093998	—	2	2	no	—
27	C₄×D₁₈₈	—	2	2	no	—
28	#13093964	—	2	4	no	—
29	C₂×C₄₇⋊C₁₆	—	2	2	no	—
30	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
31	(C₂×D₄₇)⋊C₈	—	2	4	no	—
32	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
34	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
35	C₂×C₁₆×C₄₇	—	2	2	no	—
36	C₂×C₁₆×C₄₇	—	2	2	no	—
37	C₂×C₁₆×C₄₇	D₄	1	4	yes	complemented
38	C₄×C₈×C₄₇	D₄	1	2	yes	—
39	C₄×C₈×C₄₇	D₄	1	2	yes	—
40	C₂²×C₄×C₄₇	M₁₆	1	2	yes	—
41	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₂²⋊D₄₇	—	2	2	no	—
43	#10625350	—	2	4	no	—
44	#10625350	—	2	4	no	—
45	C₂×D₁₈₈	—	2	2	no	—
46	C₂×D₁₈₈	C₂×C₈	1	2	yes	—
47	C₂×C₄×D₄₇	—	2	2	no	—
48	C₂×Dic₉₄	C₂×C₈	1	2	yes	—
49	C₂×C₈×C₄₇	—	2	2	no	—
50	C₂×C₈×C₄₇	C₂²⋊C₄	1	2	yes	—
51	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
52	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
53	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
54	C₂×C₈×C₄₇	#41	1	2	yes	—
55	C₄²×C₄₇	C₂²⋊C₄	1	2	yes	—
56	C₄²×C₄₇	C₂×C₈	1	1	yes	characteristic
57	C₄²×C₄₇	C₂×C₈	1	1	yes	characteristic
58	(C₂×D₄₇)⋊C₄	—	2	4	no	—
59	C₄⋊Dic₄₇	—	2	2	no	—
60	#10625329	—	2	4	no	—
61	C₂×C₄₇⋊C₈	—	2	2	no	—
62	#10625324	—	4	4	no	—
63	C₈×D₄₇	—	4	2	no	—
64	C₈×D₄₇	—	4	2	no	—
65	C₁₆×C₄₇	—	2	4	no	—
66	C₄₇⋊C₁₆	—	4	2	no	—
67	C₂³×C₄₇	C₂×M₁₆	1	1	yes	characteristic
68	C₂²×D₄₇	—	2	2	no	—
69	C₂×C₄×C₄₇	—	4	2	no	—
70	C₂×C₄×C₄₇	C₂²⋊C₈	1	2	yes	—
71	C₂×C₄×C₄₇	C₂²⋊C₈	1	2	yes	—
72	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
74	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
75	C₂×C₄×C₄₇	#131	1	2	yes	—
76	C₂²⋊D₄₇	—	4	4	no	—
77	C₂×Dic₄₇	—	2	2	no	—
78	D₁₈₈	—	2	4	no	—
79	D₁₈₈	—	4	4	no	—
80	C₄×D₄₇	—	4	2	no	—
81	C₄×D₄₇	—	4	2	no	—
82	Dic₉₄	—	2	4	no	—
83	C₈×C₄₇	—	2	2	no	—
84	C₈×C₄₇	#121	1	2	yes	—
85	C₈×C₄₇	#124	1	1	yes	characteristic
86	C₈×C₄₇	#127	1	1	yes	characteristic
87	C₄₇⋊C₈	—	4	2	no	—
88	#13192	—	47	1	no	maximal, Sylow
89	C₂²×C₄₇	—	2	2	no	—
90	C₂²×C₄₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
91	C₂²×C₄₇	#436	1	2	yes	—
92	C₂×D₄₇	—	4	4	no	—
93	C₂×D₄₇	—	4	2	no	—
94	C₄×C₄₇	—	2	2	no	—
95	C₄×C₄₇	—	2	2	no	—
96	C₄×C₄₇	#390	1	1	yes	characteristic
97	C₄×C₄₇	#408	1	1	yes	characteristic
98	C₄×C₄₇	C₈×D₄	1	2	yes	—
99	Dic₄₇	—	4	2	no	—
100	#2931	—	47	1	no	—
101	#2150	—	47	1	no	—
102	C₂×#350	—	47	4	no	—
103	C₂×#346	D₄₇	1	1	yes	characteristic, complemented
104	C₂×C₄₇	—	2	2	no	—
105	C₂×C₄₇	#1954	1	1	yes	characteristic
106	C₂×C₄₇	#2117	1	2	yes	—
107	D₄₇	—	8	2	no	—
108	C₂×#131	—	47	2	no	—
109	#532	—	47	1	no	—
110	C₂×M₃₂	—	47	2	no	—
111	C₂²×C₁₆	C₂×D₄₇	1	2	yes	—
112	#436	—	94	2	no	—
113	C₈×D₄	—	94	2	no	—
114	#423	—	47	1	no	—
115	#408	—	47	2	no	—
116	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
117	C₄⋊C₁₆	—	94	2	no	—
118	#350	—	94	8	no	—
119	#346	—	2	2	no	—
120	C₄₇	#13192	1	1	yes	characteristic, complemented, Sylow
121	C₂×#41	—	47	2	no	—
122	#131	—	94	4	no	—
123	#131	—	94	4	no	—
124	C₂×M₁₆	—	47	2	no	—
125	C₂×M₁₆	—	94	2	no	—
126	C₂²×C₈	—	94	2	no	—
127	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
128	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
129	C₄⋊Q₈	—	47	1	no	—
130	#127	—	47	1	no	—
131	#124	—	47	2	no	—
132	#121	—	94	2	no	—
133	C₄×D₄	—	94	2	no	—
134	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
135	M₃₂	—	94	4	no	—
136	C₂×C₁₆	—	2	2	no	—
137	C₂×C₁₆	—	2	2	no	—
138	C₂×C₁₆	—	94	2	no	—
139	C₂×C₁₆	D₁₈₈	1	4	yes	complemented
140	C₄⋊C₈	—	94	2	no	—
141	C₂²⋊C₈	—	94	4	no	—
142	C₄×C₈	D₁₈₈	1	2	yes	—
143	C₄×C₈	C₂²⋊D₄₇	1	2	yes	—
144	#41	—	94	4	no	—
145	#41	—	94	4	no	—
146	C₂×Q₈	—	47	2	no	—
147	C₂×D₄	—	47	2	no	—
148	C₂×D₄	—	94	2	no	—
149	C₂×D₄	—	94	2	no	—
150	C₂²×C₄	—	94	2	no	—
151	C₂²×C₄	#10625324	1	2	yes	—
152	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
153	M₁₆	—	94	4	no	—
154	M₁₆	—	188	4	no	—
155	C₂×C₈	—	2	2	no	—
156	C₂×C₈	—	94	2	no	—
157	C₂×C₈	—	188	2	no	—
158	C₂×C₈	—	188	2	no	—
159	C₂×C₈	(C₂×D₄₇)⋊C₄	1	2	yes	—
160	C₂×C₈	C₂×D₁₈₈	1	2	yes	—
161	C₂×C₈	C₂×D₁₈₈	1	1	yes	characteristic
162	C₂×C₈	#10625350	1	2	yes	—
163	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
164	C₄⋊C₄	—	94	2	no	—
165	C₂²⋊C₄	—	94	4	no	—
166	C₄²	C₈×D₄₇	1	1	yes	characteristic
167	C₄²	C₈×D₄₇	1	1	yes	characteristic
168	C₄²	(C₂×D₄₇)⋊C₄	1	2	yes	—
169	C₁₆	—	2	4	no	—
170	C₁₆	—	188	2	no	—
171	C₂³	—	94	2	no	—
172	C₂³	C₂×#10625324	1	1	yes	characteristic
173	Q₈	—	94	4	no	—
174	D₄	—	94	4	no	—
175	D₄	—	188	4	no	—
176	D₄	—	188	4	no	—
177	C₂×C₄	—	4	2	no	—
178	C₂×C₄	—	94	2	no	—
179	C₂×C₄	—	188	2	no	—
180	C₂×C₄	—	188	2	no	—
181	C₂×C₄	(C₂×D₄₇)⋊C₈	1	2	yes	—
182	C₂×C₄	(C₂×D₄₇)⋊C₈	1	2	yes	—
183	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
184	C₂×C₄	C₄×D₁₈₈	1	2	yes	—
185	C₂×C₄	C₂×C₈×D₄₇	1	1	yes	characteristic
186	C₂×C₄	#13094030	1	2	yes	—
187	C₈	—	2	2	no	—
188	C₈	—	188	2	no	—
189	C₈	#13093998	1	2	yes	—
190	C₈	#13094012	1	1	yes	characteristic
191	C₈	#13094021	1	1	yes	characteristic
192	C₂²	—	2	2	no	—
193	C₂²	—	188	4	no	—
194	C₂²	—	188	2	no	—
195	C₂²	C₂×(C₂×D₄₇)⋊C₈	1	1	yes	characteristic
196	C₂²	#15689202	1	2	yes	—
197	C₄	—	2	2	no	—
198	C₄	—	2	2	no	—
199	C₄	—	188	2	no	—
200	C₄	#15689025	1	1	yes	characteristic
201	C₄	#15689089	1	1	yes	characteristic
202	C₄	C₈×D₁₈₈	1	2	yes	—
203	C₂	—	2	2	no	—
204	C₂	—	376	2	no	—
205	C₂	#20244182	1	1	yes	characteristic
206	C₂	#20245015	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20252101

C₂

yes

maximal, characteristic

#20245144

C₂

yes

maximal, characteristic, complemented

C₂×#15688906

C₂

yes

maximal, complemented

C₂×C₄₇×#346

C₂

yes

maximal, characteristic, complemented

C₂×#13094030

C₂²

yes

—

#15689673

C₄

yes

characteristic

C₂×#13093964

C₂²

yes

—

#15689202

—

C₈×D₁₈₈

—

#15689147

C₄

yes

characteristic

#15689089

C₄

yes

—

C₄⋊(C₄₇⋊C₁₆)

—

#15688906

—

C₂²×C₁₆×C₄₇

C₂²

yes

—

C₂×C₄×C₈×C₄₇

C₂²

yes

characteristic

C₄₇×#346

—

C₂×#10625350

C₂×C₄

yes

—

#13094030

—

#13094030

—

C₂×#10625329

C₂×C₄

yes

—

C₂×#10625324

—

C₂×C₈×D₄₇

—

#13094024

C₈

yes

characteristic

#13094021

C₈

yes

characteristic

#13094012

C₈

yes

—

#13093998

—

C₄×D₁₈₈

—

#13093964

—

C₂×C₄₇⋊C₁₆

—

C₄⋊(C₄₇⋊C₈)

—

(C₂×D₄₇)⋊C₈

—

C₂²×C₈×C₄₇

C₂×C₄

yes

characteristic

C₂²×C₈×C₄₇

C₂³

yes

characteristic

C₂×C₄²×C₄₇

C₂×C₄

yes

characteristic

C₂×C₁₆×C₄₇

—

C₂×C₁₆×C₄₇

—

C₂×C₁₆×C₄₇

D₄

yes

complemented

C₄×C₈×C₄₇

D₄

yes

—

C₄×C₈×C₄₇

D₄

yes

—

C₂²×C₄×C₄₇

M₁₆

yes

—

C₂²×C₄×C₄₇

C₂²×C₄

yes

characteristic

C₂×C₂²⋊D₄₇

—

#10625350

—

#10625350

—

C₂×D₁₈₈

—

C₂×D₁₈₈

C₂×C₈

yes

—

C₂×C₄×D₄₇

—

C₂×Dic₉₄

C₂×C₈

yes

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

C₂²⋊C₄

yes

—

C₂×C₈×C₄₇

C₂×D₄

yes

characteristic

C₂×C₈×C₄₇

C₂×D₄

yes

—

C₂×C₈×C₄₇

C₂×D₄

yes

characteristic

C₂×C₈×C₄₇

#41

yes

—

C₄²×C₄₇

C₂²⋊C₄

yes

—

C₄²×C₄₇

C₂×C₈

yes

characteristic

C₄²×C₄₇

C₂×C₈

yes

characteristic

(C₂×D₄₇)⋊C₄

—

C₄⋊Dic₄₇

—

#10625329

—

C₂×C₄₇⋊C₈

—

#10625324

—

C₈×D₄₇

—

C₈×D₄₇

—

C₁₆×C₄₇

—

C₄₇⋊C₁₆

—

C₂³×C₄₇

C₂×M₁₆

yes

characteristic

C₂²×D₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂²⋊C₈

yes

—

C₂×C₄×C₄₇

C₂²⋊C₈

yes

—

C₂×C₄×C₄₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₇

C₄×D₄

yes

—

C₂×C₄×C₄₇

C₂²×C₈

yes

characteristic

C₂×C₄×C₄₇

#131

yes

—

C₂²⋊D₄₇

—

C₂×Dic₄₇

—

D₁₈₈

—

D₁₈₈

—

C₄×D₄₇

—

C₄×D₄₇

—

Dic₉₄

—

C₈×C₄₇

—

C₈×C₄₇

#121

yes

—

C₈×C₄₇

yes

C₈×C₄₇

yes

C₄₇⋊C₈

—

#13192

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

C₂×C₂²⋊C₈

yes

characteristic

C₂²×C₄₇

#436

yes

—

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

C₈×D₄

yes

—

Dic₄₇

—

100

#2931

—

101

#2150

—

102

C₂×#350

—

103

C₂×#346

D₄₇

yes

characteristic, complemented

104

C₂×C₄₇

—

105

C₂×C₄₇

#1954

yes

characteristic

106

C₂×C₄₇

#2117

yes

—

107

D₄₇

—

108

C₂×#131

—

109

#532

—

110

C₂×M₃₂

—

111

C₂²×C₁₆

C₂×D₄₇

yes

—

112

#436

—

113

C₈×D₄

—

114

#423

—

115

#408

—

116

C₂×C₄×C₈

C₂×D₄₇

yes

characteristic

117

C₄⋊C₁₆

—

118

#350

—

119

#346

—

120

C₄₇

#13192

yes

characteristic, complemented, Sylow

121

C₂×#41

—

122

#131

—

123

#131

—

124

C₂×M₁₆

—

125

C₂×M₁₆

—

126

C₂²×C₈

—

127

C₂²×C₈

C₄×D₄₇

yes

characteristic

128

C₂²×C₈

C₂²×D₄₇

yes

characteristic

129

C₄⋊Q₈

—

130

#127

—

131

#124

—

132

#121

—

133

C₄×D₄

—

134

C₂×C₄²

C₄×D₄₇

yes

characteristic

135

M₃₂

—

136

C₂×C₁₆

—

137

C₂×C₁₆

—

138

C₂×C₁₆

—

139

C₂×C₁₆

D₁₈₈

yes

complemented

140

C₄⋊C₈

—

141

C₂²⋊C₈

—

142

C₄×C₈

D₁₈₈

yes

—

143

C₄×C₈

C₂²⋊D₄₇

yes

—

144

#41

—

145

#41

—

146

C₂×Q₈

—

147

C₂×D₄

—

148

C₂×D₄

—

149

C₂×D₄

—

150

C₂²×C₄

—

151

C₂²×C₄

#10625324

yes

—

152

C₂²×C₄

C₂×C₄×D₄₇

yes

characteristic

153

M₁₆

—

154

M₁₆

—

188

—

155

C₂×C₈

—

156

C₂×C₈

—

157

C₂×C₈

—

188

—

158

C₂×C₈

—

188

—

159

C₂×C₈

(C₂×D₄₇)⋊C₄

yes

—

160

C₂×C₈

C₂×D₁₈₈

yes

—

161

C₂×C₈

C₂×D₁₈₈

yes

characteristic

162

C₂×C₈

#10625350

yes

—

163

C₂×C₈

C₂×C₂²⋊D₄₇

yes

characteristic

164

C₄⋊C₄

—

165

C₂²⋊C₄

—

166

C₄²

C₈×D₄₇

yes

characteristic

167

C₄²

C₈×D₄₇

yes

characteristic

168

C₄²

(C₂×D₄₇)⋊C₄

yes

—

169

C₁₆

—

170

C₁₆

—

188

—

171

C₂³

—

172

C₂³

C₂×#10625324

yes

characteristic

173

Q₈

—

174

D₄

—

175

D₄

—

188

—

176

D₄

—

188

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

—

188

—

180

C₂×C₄

—

188

—

181

C₂×C₄

(C₂×D₄₇)⋊C₈

yes

—

182

C₂×C₄

(C₂×D₄₇)⋊C₈

yes

—

183

C₂×C₄

C₂×(C₂×D₄₇)⋊C₄

yes

characteristic

184

C₂×C₄

C₄×D₁₈₈

yes

—

185

C₂×C₄

C₂×C₈×D₄₇

yes

characteristic

186

C₂×C₄

#13094030

yes

—

187

C₈

—

188

C₈

—

188

—

189

C₈

#13093998

yes

—

190

C₈

#13094012

yes

characteristic

191

C₈

#13094021

yes

characteristic

192

C₂²

—

193

C₂²

—

188

—

194

C₂²

—

188

—

195

C₂²

C₂×(C₂×D₄₇)⋊C₈

yes

characteristic

196

C₂²

#15689202

yes

—

197

C₄

—

198

C₄

—

199

C₄

—

188

—

200

C₄

#15689025

yes

characteristic

201

C₄

#15689089

yes

characteristic

202

C₄

C₈×D₁₈₈

yes

—

203

C₂

—

204

C₂

—

376

—

205

C₂

#20244182

yes

characteristic

206

C₂

#20245015

yes

—

	subgroup	G/H
center	C₂×C₈	C₂×D₁₈₈
commutator	C₂×C₄×C₄₇	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₄₇
Fitting	C₂×C₄₇×#346	C₂
socle	C₂²×C₄₇	C₂×C₂²⋊C₈
prime radical	#15689673	C₄
cosocle	C₂²×C₈×C₄₇	C₂³
hypercenter	C₂×#346	D₄₇
hypocenter	C₄₇	#13192
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₂×#346

subgroup

G/H

center

C₂×C₈

C₂×D₁₈₈

commutator

C₂×C₄×C₄₇

C₂²×C₈

Frattini

C₂²×C₈

C₂²×D₄₇

Fitting

C₂×C₄₇×#346

C₂

socle

C₂²×C₄₇

C₂×C₂²⋊C₈

prime radical

#15689673

C₄

cosocle

C₂²×C₈×C₄₇

C₂³

D₄₇

C₄₇

G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₂×#346

N	H ≅ G/N	type
2. #20245144	C₂	semidirect
3. C₂×#15688906	C₂	semidirect
4. C₂×C₄₇×#346	C₂	semidirect
37. C₂×C₁₆×C₄₇	D₄	semidirect
103. C₂×#346	D₄₇	semidirect
120. C₄₇	#13192	semidirect
139. C₂×C₁₆	D₁₈₈	semidirect

H ≅ G/N

type

2. #20245144

C₂

semidirect

3. C₂×#15688906

C₂

semidirect

4. C₂×C₄₇×#346

C₂

semidirect

37. C₂×C₁₆×C₄₇

D₄

semidirect

103. C₂×#346

D₄₇

semidirect

120. C₄₇

#13192

semidirect

139. C₂×C₁₆

D₁₈₈

semidirect

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

subgroup	index
#20252101	2
#20245144	2
C₂×#15688906	2
C₂×C₄₇×#346	2
#13192	47

subgroup

index

C₂×C₄₇×#346

Subgroups — #26134445

Conjugacy classes of subgroups

Complements

Maximal subgroups