Group #26109926 — subgroups

Subgroups — #26109926

Order 12032.

310 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 123 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	#20243075	C₂	1	1	yes	maximal, characteristic
2	#20242134	C₂	1	2	yes	maximal
3	#20241232	C₂	1	2	yes	maximal
4	#20241217	C₂	1	1	yes	maximal, characteristic
5	C₄₇×#1274	C₂	1	1	yes	maximal, characteristic
6	#15689176	C₂²	1	2	yes	—
7	C₂×#13093935	C₄	1	1	yes	characteristic
8	C₂×#13093935	C₄	1	1	yes	characteristic
9	C₂×#13093935	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₄₇⋊C₈	C₄	1	1	yes	characteristic
11	C₂×C₄×C₄₇⋊C₈	C₄	1	1	yes	characteristic
12	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
13	#15688875	—	2	2	no	—
14	#15688873	—	2	2	no	—
15	C₄₇×#431	C₂²	1	2	yes	—
16	C₄₇×#376	C₂²	1	1	yes	characteristic
17	C₂²×C₄₇⋊C₈	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₄₇⋊C₈	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₄₇⋊C₈	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₄₇⋊C₈	C₂×C₄	1	1	yes	characteristic
21	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
22	C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
23	#13093935	—	2	2	no	—
24	#13093935	—	2	2	no	—
25	#13093935	C₈	1	4	yes	complemented
26	C₄×C₄₇⋊C₈	—	2	2	no	—
27	C₄×C₄₇⋊C₈	—	2	2	no	—
28	C₄×C₄₇⋊C₈	—	2	2	no	—
29	C₄×C₄₇⋊C₈	D₄	1	2	yes	—
30	C₄×C₄₇⋊C₈	Q₈	1	2	yes	—
31	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
34	C₄⋊C₈×C₄₇	—	2	2	no	—
35	C₂²⋊C₈×C₄₇	—	2	2	no	—
36	C₄×C₈×C₄₇	—	2	2	no	—
37	#95×C₄₇	—	2	2	no	—
38	C₂²×C₄×C₄₇	—	2	1	no	—
39	C₂²×C₄×C₄₇	—	2	1	no	—
40	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
41	C₂²×C₄×C₄₇	#41	1	2	yes	—
42	C₂×C₈×C₄₇	—	4	2	no	—
43	C₄²×C₄₇	—	2	1	no	—
44	C₄²×C₄₇	—	2	1	no	—
45	C₄²×C₄₇	C₂×C₈	1	2	yes	—
46	C₄²×C₄₇	M₁₆	1	2	yes	—
47	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
48	C₄²×C₄₇	C₂×Q₈	1	1	yes	characteristic
49	C₄²×C₄₇	#41	1	2	yes	—
50	C₂×C₄₇⋊C₈	—	2	1	no	—
51	C₂×C₄₇⋊C₈	—	2	1	no	—
52	C₂×C₄₇⋊C₈	—	2	1	no	—
53	C₂×C₄₇⋊C₈	—	2	1	no	—
54	C₂×C₄₇⋊C₈	—	2	1	no	—
55	C₂×C₄₇⋊C₈	—	2	1	no	—
56	C₂×C₄₇⋊C₈	—	4	2	no	—
57	C₂×C₄₇⋊C₈	C₄⋊C₄	1	2	yes	—
58	C₂×C₄₇⋊C₈	C₄⋊C₄	1	2	yes	—
59	C₂×C₄₇⋊C₈	C₂×C₈	1	2	yes	—
60	C₂×C₄₇⋊C₈	C₂×C₈	1	2	yes	—
61	C₂×C₄₇⋊C₈	M₁₆	1	2	yes	—
62	C₂×C₄₇⋊C₈	M₁₆	1	2	yes	—
63	C₂³×C₄₇	#117	1	1	yes	characteristic
64	C₂×C₄×C₄₇	—	2	2	no	—
65	C₂×C₄×C₄₇	—	2	2	no	—
66	C₂×C₄×C₄₇	—	4	2	no	—
67	C₂×C₄×C₄₇	—	4	2	no	—
68	C₂×C₄×C₄₇	—	4	2	no	—
69	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
70	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
71	C₂×C₄×C₄₇	#124	1	2	yes	—
72	C₂×C₄×C₄₇	#125	1	2	yes	—
73	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
74	C₂×C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
75	C₂×C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
76	C₂×C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
77	C₈×C₄₇	—	8	2	no	—
78	C₄₇⋊C₈	—	2	2	no	—
79	C₄₇⋊C₈	—	2	2	no	—
80	C₄₇⋊C₈	—	2	2	no	—
81	C₄₇⋊C₈	—	8	2	no	—
82	C₄₇⋊C₈	C₄⋊C₈	1	4	yes	—
83	#8781	—	47	1	no	maximal, Sylow
84	C₂²×C₄₇	—	2	2	no	—
85	C₂²×C₄₇	#382	1	1	yes	characteristic
86	C₂²×C₄₇	#431	1	1	yes	characteristic
87	C₂²×C₄₇	#432	1	1	yes	characteristic
88	C₄×C₄₇	—	2	2	no	—
89	C₄×C₄₇	—	2	2	no	—
90	C₄×C₄₇	—	4	2	no	—
91	C₄×C₄₇	—	4	2	no	—
92	C₄×C₄₇	C₂×C₄⋊C₈	1	1	yes	characteristic
93	C₄×C₄₇	#423	1	1	yes	characteristic
94	C₄×C₄₇	#431	1	1	yes	characteristic
95	C₄×C₄₇	#432	1	1	yes	characteristic
96	#1726	—	47	1	no	—
97	#1564	—	47	2	no	—
98	#1281	—	47	2	no	—
99	#1274	—	47	1	no	—
100	#1274	D₄₇	1	1	yes	characteristic
101	C₂×C₄₇	—	2	1	no	—
102	C₂×C₄₇	—	2	1	no	—
103	C₂×C₄₇	#1831	1	1	yes	characteristic
104	C₂×C₄₇	#1833	1	2	yes	—
105	#431	—	47	2	no	—
106	#431	C₂×D₄₇	1	2	yes	—
107	C₂×#97	—	47	1	no	—
108	C₂×#97	—	47	1	no	—
109	C₂×#97	—	47	1	no	—
110	C₂×C₄×C₈	—	47	1	no	—
111	C₂×C₄×C₈	—	47	1	no	—
112	C₂×C₄×C₈	—	47	1	no	—
113	#376	C₂×D₄₇	1	1	yes	characteristic
114	#335	—	94	2	no	—
115	#334	—	94	2	no	—
116	C₄₇	#8781	1	1	yes	characteristic, complemented, Sylow
117	C₂²×C₈	—	47	1	no	—
118	C₂²×C₈	—	47	1	no	—
119	C₂²×C₈	—	47	1	no	—
120	C₂²×C₈	—	47	1	no	—
121	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
122	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
123	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
124	C₄⋊C₈	—	2	2	no	—
125	C₄⋊C₈	—	94	2	no	—
126	C₂²⋊C₈	—	2	2	no	—
127	C₂²⋊C₈	—	94	2	no	—
128	#97	—	47	4	no	—
129	#97	—	94	2	no	—
130	#97	—	94	2	no	—
131	C₄×C₈	—	2	2	no	—
132	C₄×C₈	—	47	2	no	—
133	C₄×C₈	—	47	2	no	—
134	C₄×C₈	—	94	2	no	—
135	C₄×C₈	—	94	2	no	—
136	C₄×C₈	—	94	2	no	—
137	#95	—	2	2	no	—
138	C₂²×C₄	—	2	1	no	—
139	C₂²×C₄	—	2	1	no	—
140	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
141	C₂²×C₄	#10625352	1	2	yes	—
142	C₂×C₈	—	4	2	no	—
143	C₂×C₈	—	47	2	no	—
144	C₂×C₈	—	47	2	no	—
145	C₂×C₈	—	47	2	no	—
146	C₂×C₈	—	47	2	no	—
147	C₂×C₈	—	47	2	no	—
148	C₂×C₈	—	47	2	no	—
149	C₂×C₈	—	94	1	no	—
150	C₂×C₈	—	94	1	no	—
151	C₂×C₈	—	94	1	no	—
152	C₂×C₈	—	94	1	no	—
153	C₂×C₈	—	94	1	no	—
154	C₂×C₈	—	94	1	no	—
155	C₂×C₈	—	188	2	no	—
156	C₄²	—	2	1	no	—
157	C₄²	—	2	1	no	—
158	C₄²	C₈×D₄₇	1	2	yes	—
159	C₄²	#10625324	1	2	yes	—
160	C₄²	#10625350	1	2	yes	—
161	C₄²	D₄×D₄₇	1	1	yes	characteristic
162	C₄²	Q₈×D₄₇	1	1	yes	characteristic
163	C₂³	#13093980	1	1	yes	characteristic
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	4	2	no	—
167	C₂×C₄	—	4	2	no	—
168	C₂×C₄	—	4	2	no	—
169	C₂×C₄	C₄⋊C₄×D₄₇	1	1	yes	characteristic
170	C₂×C₄	#13093981	1	1	yes	characteristic
171	C₂×C₄	#13094010	1	2	yes	—
172	C₂×C₄	#13094014	1	2	yes	—
173	C₂×C₄	C₂×C₈×D₄₇	1	1	yes	characteristic
174	C₂×C₄	M₁₆×D₄₇	1	1	yes	characteristic
175	C₂×C₄	M₁₆×D₄₇	1	1	yes	characteristic
176	C₂×C₄	C₂×#10625324	1	1	yes	characteristic
177	C₈	—	8	2	no	—
178	C₈	—	47	4	no	—
179	C₈	—	94	2	no	—
180	C₈	—	94	2	no	—
181	C₈	—	94	2	no	—
182	C₈	—	376	2	no	—
183	C₂²	—	2	2	no	—
184	C₂²	#15688987	1	1	yes	characteristic
185	C₂²	#15689173	1	1	yes	characteristic
186	C₂²	#15689177	1	1	yes	characteristic
187	C₄	—	2	2	no	—
188	C₄	—	2	2	no	—
189	C₄	—	4	2	no	—
190	C₄	—	4	2	no	—
191	C₄	C₄⋊C₈×D₄₇	1	1	yes	characteristic
192	C₄	#15689145	1	1	yes	characteristic
193	C₄	#15689174	1	1	yes	characteristic
194	C₄	#15689178	1	1	yes	characteristic
195	C₂	—	2	1	no	—
196	C₂	—	2	1	no	—
197	C₂	#20243503	1	1	yes	characteristic
198	C₂	#20243510	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20243075

C₂

yes

maximal, characteristic

#20242134

C₂

yes

maximal

#20241232

C₂

yes

maximal

#20241217

C₂

yes

maximal, characteristic

C₄₇×#1274

C₂

yes

maximal, characteristic

#15689176

C₂²

yes

—

C₂×#13093935

C₄

yes

characteristic

C₂×#13093935

C₄

yes

characteristic

C₂×#13093935

C₂²

yes

characteristic

C₂×C₄×C₄₇⋊C₈

C₄

yes

characteristic

C₂×C₄×C₄₇⋊C₈

C₄

yes

characteristic

C₂×C₄×C₄₇⋊C₈

C₂²

yes

characteristic

#15688875

—

#15688873

—

C₄₇×#431

C₂²

yes

—

C₄₇×#376

C₂²

yes

characteristic

C₂²×C₄₇⋊C₈

C₂×C₄

yes

characteristic

C₂²×C₄₇⋊C₈

C₂×C₄

yes

characteristic

C₂²×C₄₇⋊C₈

C₂×C₄

yes

characteristic

C₂²×C₄₇⋊C₈

C₂×C₄

yes

characteristic

C₄⋊(C₄₇⋊C₈)

—

C₂²⋊(C₄₇⋊C₈)

—

#13093935

—

#13093935

—

#13093935

C₈

yes

complemented

C₄×C₄₇⋊C₈

—

C₄×C₄₇⋊C₈

—

C₄×C₄₇⋊C₈

—

C₄×C₄₇⋊C₈

D₄

yes

—

C₄×C₄₇⋊C₈

Q₈

yes

—

C₂×C₄²×C₄₇

C₂×C₄

yes

characteristic

C₂×C₄²×C₄₇

C₂×C₄

yes

characteristic

C₂×C₄²×C₄₇

C₂³

yes

characteristic

C₄⋊C₈×C₄₇

—

C₂²⋊C₈×C₄₇

—

C₄×C₈×C₄₇

—

#95×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂²×C₄

yes

characteristic

C₂²×C₄×C₄₇

#41

yes

—

C₂×C₈×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

C₂×C₈

yes

—

C₄²×C₄₇

M₁₆

yes

—

C₄²×C₄₇

C₂×D₄

yes

characteristic

C₄²×C₄₇

C₂×Q₈

yes

characteristic

C₄²×C₄₇

#41

yes

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

C₄⋊C₄

yes

—

C₂×C₄₇⋊C₈

C₄⋊C₄

yes

—

C₂×C₄₇⋊C₈

C₂×C₈

yes

—

C₂×C₄₇⋊C₈

C₂×C₈

yes

—

C₂×C₄₇⋊C₈

M₁₆

yes

—

C₂×C₄₇⋊C₈

M₁₆

yes

—

C₂³×C₄₇

#117

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×C₄⋊C₄

yes

C₂×C₄×C₄₇

yes

C₂×C₄×C₄₇

yes

—

C₂×C₄×C₄₇

#125

yes

—

C₂×C₄×C₄₇

C₂²×C₈

yes

characteristic

C₂×C₄×C₄₇

C₂×M₁₆

yes

characteristic

C₂×C₄×C₄₇

C₂×M₁₆

yes

characteristic

C₂×C₄×C₄₇

C₂×M₁₆

yes

characteristic

C₈×C₄₇

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₄₇⋊C₈

C₄⋊C₈

yes

—

#8781

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₂×C₄⋊C₈

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

—

—

—

—

100

#1274

D₄₇

yes

characteristic

101

C₂×C₄₇

—

102

C₂×C₄₇

—

103

C₂×C₄₇

#1831

yes

characteristic

104

C₂×C₄₇

#1833

yes

—

105

#431

—

106

#431

C₂×D₄₇

yes

—

107

C₂×#97

—

108

C₂×#97

—

109

C₂×#97

—

110

C₂×C₄×C₈

—

111

C₂×C₄×C₈

—

112

C₂×C₄×C₈

—

113

#376

C₂×D₄₇

yes

characteristic

114

#335

—

115

#334

—

116

C₄₇

#8781

yes

characteristic, complemented, Sylow

117

C₂²×C₈

—

118

C₂²×C₈

—

119

C₂²×C₈

—

120

C₂²×C₈

—

121

C₂×C₄²

C₄×D₄₇

yes

characteristic

122

C₂×C₄²

C₄×D₄₇

yes

characteristic

123

C₂×C₄²

C₂²×D₄₇

yes

characteristic

124

C₄⋊C₈

—

125

C₄⋊C₈

—

126

C₂²⋊C₈

—

127

C₂²⋊C₈

—

128

#97

—

129

#97

—

130

#97

—

131

C₄×C₈

—

132

C₄×C₈

—

133

C₄×C₈

—

134

C₄×C₈

—

135

C₄×C₈

—

136

C₄×C₈

—

137

#95

—

138

C₂²×C₄

—

139

C₂²×C₄

—

140

C₂²×C₄

C₂×C₄×D₄₇

yes

characteristic

141

C₂²×C₄

#10625352

yes

—

142

C₂×C₈

—

143

C₂×C₈

—

144

C₂×C₈

—

145

C₂×C₈

—

146

C₂×C₈

—

147

C₂×C₈

—

148

C₂×C₈

—

149

C₂×C₈

—

150

C₂×C₈

—

151

C₂×C₈

—

152

C₂×C₈

—

153

C₂×C₈

—

154

C₂×C₈

—

155

C₂×C₈

—

188

—

156

C₄²

—

157

C₄²

—

158

C₄²

C₈×D₄₇

yes

—

159

C₄²

#10625324

yes

—

160

C₄²

#10625350

yes

—

161

C₄²

D₄×D₄₇

yes

characteristic

162

C₄²

Q₈×D₄₇

yes

characteristic

163

C₂³

#13093980

yes

characteristic

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

C₄⋊C₄×D₄₇

yes

characteristic

170

C₂×C₄

#13093981

yes

characteristic

171

C₂×C₄

#13094010

yes

—

172

C₂×C₄

#13094014

yes

—

173

C₂×C₄

C₂×C₈×D₄₇

yes

characteristic

174

C₂×C₄

M₁₆×D₄₇

yes

characteristic

175

C₂×C₄

M₁₆×D₄₇

yes

characteristic

176

C₂×C₄

C₂×#10625324

yes

characteristic

177

C₈

—

178

C₈

—

179

C₈

—

180

C₈

—

181

C₈

—

182

C₈

—

376

—

183

C₂²

—

184

C₂²

#15688987

yes

characteristic

185

C₂²

#15689173

yes

characteristic

186

C₂²

#15689177

yes

characteristic

187

C₄

—

188

C₄

—

189

C₄

—

190

C₄

—

191

C₄

C₄⋊C₈×D₄₇

yes

characteristic

192

C₄

#15689145

yes

characteristic

193

C₄

#15689174

yes

characteristic

194

C₄

#15689178

yes

characteristic

195

C₂

—

196

C₂

—

197

C₂

#20243503

yes

characteristic

198

C₂

#20243510

yes

—

	subgroup	G/H
center	C₂×C₄	#13093981
commutator	C₂×C₄×C₄₇	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1274	C₂
socle	C₂²×C₄₇	#382
prime radical	C₂³×C₄₇	#117
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#1274	D₄₇
hypocenter	C₄₇	#8781
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ #376 ◁ #1274

subgroup

G/H

center

C₂×C₄

#13093981

commutator

C₂×C₄×C₄₇

C₂²×C₈

Frattini

C₂×C₄²

C₂²×D₄₇

Fitting

C₄₇×#1274

C₂

C₂²×C₄₇

C₂³×C₄₇

C₂×C₄²×C₄₇

C₂³

D₄₇

C₄₇

G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂×C₄ ◁ #376 ◁ #1274

N	H ≅ G/N	type
25. #13093935	C₈	semidirect
116. C₄₇	#8781	semidirect

H ≅ G/N

type

25. #13093935

C₈

semidirect

116. C₄₇

#8781

semidirect

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

subgroup	index
#20243075	2
#20242134	2
#20241232	2
#20241217	2
C₄₇×#1274	2
#8781	47

subgroup

index

C₄₇×#1274

Subgroups — #26109926

Conjugacy classes of subgroups

Complements

Maximal subgroups