Group #26109235 — subgroups

Subgroups — #26109235

Order 12032.

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

normal subgroups only

This group has 362 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×#15689247	C₂	1	1	yes	maximal, characteristic
2	#20242149	C₂	1	2	yes	maximal
3	#20241731	C₂	1	2	yes	maximal
4	#20241258	C₂	1	1	yes	maximal, characteristic
5	C₄₇×#1290	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×Dic₉₄	C₂²	1	1	yes	characteristic
7	#15689247	—	2	2	no	—
8	#15689247	—	2	2	no	—
9	#15689180	C₂²	1	2	yes	—
10	C₂×Dic₄₇⋊C₈	C₄	1	2	yes	—
11	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
12	C₂×C₈×Dic₄₇	C₄	1	2	yes	—
13	#15688875	—	2	2	no	—
14	#15688873	—	2	2	no	—
15	#15688848	—	2	4	no	—
16	C₄₇×#432	C₂²	1	2	yes	—
17	C₂×C₄⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
18	C₂²×Dic₉₄	Q₈	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×Dic₉₄	—	2	2	no	—
22	C₄×Dic₉₄	—	2	2	no	—
23	C₂×C₄⋊Dic₄₇	Q₈	1	1	yes	characteristic
24	C₂×Dic₄₇⋊C₄	D₄	1	2	yes	—
25	C₂×C₄×Dic₄₇	C₂×C₄	1	2	yes	—
26	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
27	Dic₄₇⋊C₈	—	2	4	no	—
28	C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
29	#13093935	—	2	2	no	—
30	C₄×C₄₇⋊C₈	D₄	1	2	yes	—
31	C₄×C₄₇⋊C₈	D₄	1	2	yes	—
32	C₈×Dic₄₇	—	2	2	no	—
33	C₈×Dic₄₇	—	2	2	no	—
34	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
35	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
36	C₄⋊C₈×C₄₇	—	2	2	no	—
37	C₄⋊C₈×C₄₇	—	2	2	no	—
38	C₂²⋊C₈×C₄₇	—	2	2	no	—
39	#97×C₄₇	—	2	2	no	—
40	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
41	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
42	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
43	C₂²×Dic₄₇	C₄⋊C₄	1	2	yes	—
44	C₂²×Dic₄₇	C₄⋊C₄	1	2	yes	—
45	C₂×Dic₉₄	—	2	2	no	—
46	C₂×Dic₉₄	—	2	4	no	—
47	C₂×C₈×C₄₇	—	2	2	no	—
48	C₂×C₈×C₄₇	—	4	2	no	—
49	C₂×C₈×C₄₇	—	4	2	no	—
50	C₄²×C₄₇	C₂×D₄	1	2	yes	—
51	C₄²×C₄₇	#41	1	2	yes	—
52	C₄⋊Dic₄₇	—	2	2	no	—
53	Dic₄₇⋊C₄	—	2	4	no	—
54	C₄×Dic₄₇	—	2	2	no	—
55	C₄×Dic₄₇	—	2	2	no	—
56	C₂×C₄₇⋊C₈	—	2	2	no	—
57	C₂×C₄₇⋊C₈	—	4	1	no	—
58	C₂×C₄₇⋊C₈	—	4	2	no	—
59	C₂×C₄₇⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
60	C₂×C₄₇⋊C₈	C₂²⋊C₄	1	2	yes	—
61	C₂×C₄₇⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
62	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
63	C₂×C₄×C₄₇	—	2	2	no	—
64	C₂×C₄×C₄₇	—	2	1	no	—
65	C₂×C₄×C₄₇	—	2	1	no	—
66	C₂×C₄×C₄₇	—	4	1	no	—
67	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
69	C₂×C₄×C₄₇	#121	1	2	yes	—
70	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
71	C₂×Dic₄₇	—	2	2	no	—
72	C₂×Dic₄₇	—	2	2	no	—
73	C₂×Dic₄₇	—	2	2	no	—
74	C₂×Dic₄₇	—	4	2	no	—
75	C₂×Dic₄₇	#108	1	4	yes	—
76	Dic₉₄	—	2	8	no	—
77	C₈×C₄₇	—	4	2	no	—
78	C₈×C₄₇	—	8	2	no	—
79	C₄₇⋊C₈	—	2	1	no	—
80	C₄₇⋊C₈	—	2	2	no	—
81	C₄₇⋊C₈	—	8	2	no	—
82	C₄₇⋊C₈	#101	1	2	yes	—
83	#8623	—	47	1	no	maximal, Sylow
84	C₂²×C₄₇	—	2	1	no	—
85	C₂²×C₄₇	—	2	1	no	—
86	C₂²×C₄₇	#380	1	1	yes	characteristic
87	C₂²×C₄₇	C₂×#108	1	2	yes	—
88	C₄×C₄₇	—	2	1	no	—
89	C₄×C₄₇	—	2	2	no	—
90	C₄×C₄₇	—	2	2	no	—
91	C₄×C₄₇	C₂×#101	1	1	yes	characteristic
92	C₄×C₄₇	#413	1	1	yes	characteristic
93	Dic₄₇	—	2	4	no	—
94	Dic₄₇	—	4	2	no	—
95	C₂×C₈×Q₈	—	47	1	no	—
96	#1568	—	47	2	no	—
97	#1467	—	47	2	no	—
98	#1290	—	47	1	no	—
99	#1290	D₄₇	1	1	yes	characteristic
100	C₂×C₄₇	—	2	2	no	—
101	C₂×C₄₇	#1801	1	1	yes	characteristic
102	C₂×C₄₇	#1810	1	1	yes	characteristic
103	C₂×C₄₇	#1812	1	1	yes	characteristic
104	C₂×C₄×Q₈	—	47	1	no	—
105	C₈×Q₈	—	94	2	no	—
106	C₈×Q₈	—	94	2	no	—
107	#432	—	47	2	no	—
108	#432	C₂×D₄₇	1	2	yes	—
109	C₂×C₄⋊C₈	—	47	2	no	—
110	C₂×C₄⋊C₈	C₂×D₄₇	1	1	yes	characteristic
111	C₂×C₄×C₈	—	47	1	no	—
112	C₂×C₄×C₈	—	47	2	no	—
113	#335	—	94	2	no	—
114	#334	—	94	2	no	—
115	#324	—	94	4	no	—
116	C₄₇	#8623	1	1	yes	characteristic, complemented, Sylow
117	C₂²×Q₈	—	47	1	no	—
118	C₂²×C₈	—	47	1	no	—
119	C₂²×C₈	—	47	1	no	—
120	C₂²×C₈	C₄×D₄₇	1	2	yes	—
121	C₄×Q₈	—	94	2	no	—
122	C₄×Q₈	—	94	2	no	—
123	C₂×C₄⋊C₄	—	47	2	no	—
124	C₂×C₄⋊C₄	—	47	1	no	—
125	C₂×C₄²	—	47	2	no	—
126	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
127	C₄⋊C₈	—	2	2	no	—
128	C₄⋊C₈	—	2	2	no	—
129	C₄⋊C₈	—	94	4	no	—
130	C₄⋊C₈	—	94	2	no	—
131	C₂²⋊C₈	—	2	2	no	—
132	C₂²⋊C₈	—	94	2	no	—
133	#97	—	2	2	no	—
134	#97	—	94	2	no	—
135	C₄×C₈	—	47	2	no	—
136	C₄×C₈	—	47	2	no	—
137	C₄×C₈	—	94	2	no	—
138	C₄×C₈	—	94	2	no	—
139	C₂×Q₈	—	94	4	no	—
140	C₂×Q₈	—	94	2	no	—
141	C₂²×C₄	—	47	2	no	—
142	C₂²×C₄	—	47	2	no	—
143	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
144	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
145	C₂²×C₄	Q₈×D₄₇	1	1	yes	characteristic
146	C₂×C₈	—	2	2	no	—
147	C₂×C₈	—	4	2	no	—
148	C₂×C₈	—	4	2	no	—
149	C₂×C₈	—	47	1	no	—
150	C₂×C₈	—	47	1	no	—
151	C₂×C₈	—	47	2	no	—
152	C₂×C₈	—	94	2	no	—
153	C₂×C₈	—	188	1	no	—
154	C₂×C₈	—	188	2	no	—
155	C₄⋊C₄	—	94	4	no	—
156	C₄⋊C₄	—	94	2	no	—
157	C₄²	—	94	2	no	—
158	C₄²	—	94	2	no	—
159	C₄²	#10625350	1	2	yes	—
160	C₄²	D₄×D₄₇	1	2	yes	—
161	C₂³	C₄⋊C₄×D₄₇	1	1	yes	characteristic
162	Q₈	—	94	8	no	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	2	1	no	—
165	C₂×C₄	—	2	1	no	—
166	C₂×C₄	—	4	1	no	—
167	C₂×C₄	—	47	4	no	—
168	C₂×C₄	—	94	2	no	—
169	C₂×C₄	—	94	2	no	—
170	C₂×C₄	—	94	2	no	—
171	C₂×C₄	—	188	2	no	—
172	C₂×C₄	C₂²⋊C₄×D₄₇	1	1	yes	characteristic
173	C₂×C₄	#13093981	1	1	yes	characteristic
174	C₂×C₄	#13093997	1	2	yes	—
175	C₂×C₄	#13094002	1	2	yes	—
176	C₈	—	4	2	no	—
177	C₈	—	8	2	no	—
178	C₈	—	47	2	no	—
179	C₈	—	94	1	no	—
180	C₈	—	94	2	no	—
181	C₈	—	376	2	no	—
182	C₂²	—	2	1	no	—
183	C₂²	—	2	1	no	—
184	C₂²	#15688982	1	1	yes	characteristic
185	C₂²	#108×D₄₇	1	2	yes	—
186	C₄	—	2	1	no	—
187	C₄	—	2	2	no	—
188	C₄	—	2	2	no	—
189	C₄	—	94	4	no	—
190	C₄	—	188	2	no	—
191	C₄	#101×D₄₇	1	1	yes	characteristic
192	C₄	#15689104	1	1	yes	characteristic
193	C₂	—	2	2	no	—
194	C₂	#20243383	1	1	yes	characteristic
195	C₂	#20243425	1	1	yes	characteristic
196	C₂	#20243433	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15689247

C₂

yes

maximal, characteristic

#20242149

C₂

yes

maximal

#20241731

C₂

yes

maximal

#20241258

C₂

yes

maximal, characteristic

C₄₇×#1290

C₂

yes

maximal, characteristic

C₂×C₄×Dic₉₄

C₂²

yes

—

—

C₂²

yes

—

C₂×Dic₄₇⋊C₈

C₄

yes

—

C₂×C₄×C₄₇⋊C₈

C₂²

yes

characteristic

C₂×C₈×Dic₄₇

C₄

yes

—

#15688875

—

#15688873

—

#15688848

—

C₄₇×#432

C₂²

yes

—

C₂×C₄⋊C₈×C₄₇

C₂²

yes

characteristic

C₂²×Dic₉₄

Q₈

yes

characteristic

C₂²×C₄₇⋊C₈

C₂×C₄

yes

characteristic

C₂²×C₄₇⋊C₈

C₂×C₄

yes

characteristic

C₄×Dic₉₄

—

C₄×Dic₉₄

—

C₂×C₄⋊Dic₄₇

Q₈

yes

characteristic

C₂×Dic₄₇⋊C₄

D₄

yes

—

C₂×C₄×Dic₄₇

C₂×C₄

yes

—

C₄⋊(C₄₇⋊C₈)

—

Dic₄₇⋊C₈

—

C₂²⋊(C₄₇⋊C₈)

—

#13093935

—

C₄×C₄₇⋊C₈

D₄

yes

—

C₄×C₄₇⋊C₈

D₄

yes

—

C₈×Dic₄₇

—

C₈×Dic₄₇

—

C₂²×C₈×C₄₇

C₂×C₄

yes

—

C₂×C₄²×C₄₇

C₂³

yes

characteristic

C₄⋊C₈×C₄₇

—

C₄⋊C₈×C₄₇

—

C₂²⋊C₈×C₄₇

—

#97×C₄₇

—

C₂²×C₄×C₄₇

C₂²×C₄

yes

characteristic

C₂²×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₄₇

C₂×Q₈

yes

characteristic

C₂²×Dic₄₇

C₄⋊C₄

yes

—

C₂²×Dic₄₇

C₄⋊C₄

yes

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₄²×C₄₇

C₂×D₄

yes

—

C₄²×C₄₇

#41

yes

—

C₄⋊Dic₄₇

—

Dic₄₇⋊C₄

—

C₄×Dic₄₇

—

C₄×Dic₄₇

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

C₂²⋊C₄

yes

characteristic

C₂×C₄₇⋊C₈

C₂²⋊C₄

yes

—

C₂×C₄₇⋊C₈

C₂²⋊C₄

yes

characteristic

C₂³×C₄₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×C₂²⋊C₄

yes

C₂×C₄×C₄₇

yes

C₂×C₄×C₄₇

yes

—

C₂×C₄×C₄₇

C₂²⋊Q₈

yes

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

#108

yes

—

Dic₉₄

—

C₈×C₄₇

—

C₈×C₄₇

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₄₇⋊C₈

#101

yes

—

#8623

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

#380

yes

characteristic

C₂²×C₄₇

C₂×#108

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

yes

C₄×C₄₇

yes

Dic₄₇

—

Dic₄₇

—

C₂×C₈×Q₈

—

—

—

—

D₄₇

yes

characteristic

100

C₂×C₄₇

—

101

C₂×C₄₇

#1801

yes

characteristic

102

C₂×C₄₇

#1810

yes

characteristic

103

C₂×C₄₇

#1812

yes

characteristic

104

C₂×C₄×Q₈

—

105

C₈×Q₈

—

106

C₈×Q₈

—

107

#432

—

108

#432

C₂×D₄₇

yes

—

109

C₂×C₄⋊C₈

—

110

C₂×C₄⋊C₈

C₂×D₄₇

yes

characteristic

111

C₂×C₄×C₈

—

112

C₂×C₄×C₈

—

113

#335

—

114

#334

—

115

#324

—

116

C₄₇

#8623

yes

characteristic, complemented, Sylow

117

C₂²×Q₈

—

118

C₂²×C₈

—

119

C₂²×C₈

—

120

C₂²×C₈

C₄×D₄₇

yes

—

121

C₄×Q₈

—

122

C₄×Q₈

—

123

C₂×C₄⋊C₄

—

124

C₂×C₄⋊C₄

—

125

C₂×C₄²

—

126

C₂×C₄²

C₂²×D₄₇

yes

characteristic

127

C₄⋊C₈

—

128

C₄⋊C₈

—

129

C₄⋊C₈

—

130

C₄⋊C₈

—

131

C₂²⋊C₈

—

132

C₂²⋊C₈

—

133

#97

—

134

#97

—

135

C₄×C₈

—

136

C₄×C₈

—

137

C₄×C₈

—

138

C₄×C₈

—

139

C₂×Q₈

—

140

C₂×Q₈

—

141

C₂²×C₄

—

142

C₂²×C₄

—

143

C₂²×C₄

C₂×C₄×D₄₇

yes

characteristic

144

C₂²×C₄

D₄×D₄₇

yes

characteristic

145

C₂²×C₄

Q₈×D₄₇

yes

characteristic

146

C₂×C₈

—

147

C₂×C₈

—

148

C₂×C₈

—

149

C₂×C₈

—

150

C₂×C₈

—

151

C₂×C₈

—

152

C₂×C₈

—

153

C₂×C₈

—

188

—

154

C₂×C₈

—

188

—

155

C₄⋊C₄

—

156

C₄⋊C₄

—

157

C₄²

—

158

C₄²

—

159

C₄²

#10625350

yes

—

160

C₄²

D₄×D₄₇

yes

—

161

C₂³

C₄⋊C₄×D₄₇

yes

characteristic

162

Q₈

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

—

188

—

172

C₂×C₄

C₂²⋊C₄×D₄₇

yes

characteristic

173

C₂×C₄

#13093981

yes

characteristic

174

C₂×C₄

#13093997

yes

—

175

C₂×C₄

#13094002

yes

—

176

C₈

—

177

C₈

—

178

C₈

—

179

C₈

—

180

C₈

—

181

C₈

—

376

—

182

C₂²

—

183

C₂²

—

184

C₂²

#15688982

yes

characteristic

185

C₂²

#108×D₄₇

yes

—

186

C₄

—

187

C₄

—

188

C₄

—

189

C₄

—

190

C₄

—

188

—

191

C₄

#101×D₄₇

yes

characteristic

192

C₄

#15689104

yes

characteristic

193

C₂

—

194

C₂

#20243383

yes

characteristic

195

C₂

#20243425

yes

characteristic

196

C₂

#20243433

yes

characteristic

	subgroup	G/H
center	C₂²	#15688982
commutator	C₂²×C₄×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1290	C₂
socle	C₂²×C₄₇	#380
prime radical	C₂³×C₄₇	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#1290	D₄₇
hypocenter	C₄₇	#8623
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1290

subgroup

G/H

center

C₂²

#15688982

commutator

C₂²×C₄×C₄₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₄₇

Fitting

C₄₇×#1290

C₂

socle

C₂²×C₄₇

#380

prime radical

C₂³×C₄₇

C₂×C₄⋊C₄

cosocle

C₂×C₄²×C₄₇

C₂³

D₄₇

C₄₇

G ▷ C₂²×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1290

N	H ≅ G/N	type
116. C₄₇	#8623	semidirect

H ≅ G/N

type

116. C₄₇

#8623

semidirect

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

subgroup	index
C₂×#15689247	2
#20242149	2
#20241731	2
#20241258	2
C₄₇×#1290	2
#8623	47

subgroup

index

C₄₇×#1290

Subgroups — #26109235

Conjugacy classes of subgroups

Complements

Maximal subgroups