Group #26102813 — subgroups

Subgroups — #26102813

Order 12032.

306 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 107 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	#20243531	C₂	1	1	yes	maximal, characteristic
2	C₄×#13093935	C₂	1	1	yes	maximal, characteristic
3	C₄₇×#1839	C₂	1	1	yes	maximal, characteristic
4	C₂×#13093935	—	2	1	no	—
5	C₂×#13093935	—	2	1	no	—
6	C₂×C₄×C₄₇⋊C₈	C₄	1	2	yes	—
7	#15688878	—	2	1	no	—
8	#15688878	—	2	1	no	—
9	#15688878	C₄	1	2	yes	—
10	C₄³×C₄₇	C₂²	1	1	yes	characteristic
11	C₄₇×#336	—	2	1	no	—
12	C₄₇×#336	—	2	1	no	—
13	C₄₇×#336	C₄	1	2	yes	—
14	C₂²×C₄₇⋊C₈	—	2	2	no	—
15	C₂²×C₄₇⋊C₈	—	2	2	no	—
16	#13093935	—	2	4	no	—
17	#13093935	—	2	4	no	—
18	C₄×C₄₇⋊C₈	—	2	2	no	—
19	C₄×C₄₇⋊C₈	C₈	1	4	yes	complemented
20	C₂²×C₈×C₄₇	—	2	2	no	—
21	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
25	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
26	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
27	C₂×C₄²×C₄₇	Q₈	1	1	yes	characteristic
28	C₂²×C₄×C₄₇	C₄²	1	1	yes	characteristic
29	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
30	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
31	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
32	C₂²×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
33	C₂²×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
34	C₂²×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₈×C₄₇	—	4	2	no	—
36	C₂×C₈×C₄₇	—	4	2	no	—
37	C₂×C₈×C₄₇	—	4	2	no	—
38	C₄²×C₄₇	—	2	2	no	—
39	C₄²×C₄₇	—	2	2	no	—
40	C₄²×C₄₇	—	4	1	no	—
41	C₄²×C₄₇	—	4	1	no	—
42	C₄²×C₄₇	—	4	1	no	—
43	C₄²×C₄₇	—	4	1	no	—
44	C₄²×C₄₇	C₂×C₈	1	2	yes	—
45	C₄²×C₄₇	M₁₆	1	2	yes	—
46	C₂×C₄₇⋊C₈	—	2	4	no	—
47	C₂×C₄₇⋊C₈	—	2	4	no	—
48	C₂×C₄₇⋊C₈	—	2	4	no	—
49	C₂×C₄₇⋊C₈	—	4	2	no	—
50	C₂×C₄₇⋊C₈	—	4	2	no	—
51	C₂×C₄₇⋊C₈	—	4	2	no	—
52	C₂³×C₄₇	#95	1	1	yes	characteristic
53	C₂×C₄×C₄₇	—	2	1	no	—
54	C₂×C₄×C₄₇	—	2	1	no	—
55	C₂×C₄×C₄₇	—	2	1	no	—
56	C₂×C₄×C₄₇	—	2	1	no	—
57	C₂×C₄×C₄₇	—	2	1	no	—
58	C₂×C₄×C₄₇	—	2	1	no	—
59	C₂×C₄×C₄₇	—	2	1	no	—
60	C₂×C₄×C₄₇	—	2	1	no	—
61	C₂×C₄×C₄₇	—	2	1	no	—
62	C₂×C₄×C₄₇	—	2	1	no	—
63	C₂×C₄×C₄₇	—	2	1	no	—
64	C₂×C₄×C₄₇	—	2	1	no	—
65	C₂×C₄×C₄₇	—	2	1	no	—
66	C₂×C₄×C₄₇	—	2	1	no	—
67	C₂×C₄×C₄₇	—	2	1	no	—
68	C₂×C₄×C₄₇	—	2	1	no	—
69	C₂×C₄×C₄₇	C₄×C₈	1	1	yes	characteristic
70	C₂×C₄×C₄₇	#97	1	1	yes	characteristic
71	C₂×C₄×C₄₇	C₂²⋊C₈	1	2	yes	—
72	C₂×C₄×C₄₇	#104	1	2	yes	—
73	C₂×C₄×C₄₇	#104	1	2	yes	—
74	C₂×C₄×C₄₇	C₄⋊C₈	1	2	yes	—
75	C₈×C₄₇	—	8	2	no	—
76	C₄₇⋊C₈	—	2	8	no	—
77	C₄₇⋊C₈	—	8	2	no	—
78	#7019	—	47	1	no	maximal, Sylow
79	C₂²×C₄₇	#336	1	1	yes	characteristic
80	C₂²×C₄₇	H(3,ℤ/4)	1	1	yes	characteristic
81	C₂²×C₄₇	#338	1	1	yes	characteristic
82	C₂²×C₄₇	#339	1	1	yes	characteristic
83	C₂²×C₄₇	#339	1	1	yes	characteristic
84	C₂²×C₄₇	#344	1	1	yes	characteristic
85	C₂²×C₄₇	#344	1	1	yes	characteristic
86	C₄×C₄₇	—	2	1	no	—
87	C₄×C₄₇	—	2	1	no	—
88	C₄×C₄₇	—	2	2	no	—
89	C₄×C₄₇	—	2	2	no	—
90	C₄×C₄₇	—	2	2	no	—
91	C₄×C₄₇	—	2	2	no	—
92	C₄×C₄₇	—	4	1	no	—
93	C₄×C₄₇	—	4	1	no	—
94	#1839	—	47	1	no	—
95	#1839	D₄₇	1	1	yes	characteristic
96	C₄×#97	—	47	1	no	—
97	C₂×C₄₇	#1275	1	1	yes	characteristic
98	C₂×C₄₇	#1276	1	1	yes	characteristic
99	C₂×C₄₇	#1281	1	1	yes	characteristic
100	C₂×C₄₇	#1288	1	1	yes	characteristic
101	C₂×C₄₇	#1288	1	1	yes	characteristic
102	C₂×C₄₇	#1291	1	1	yes	characteristic
103	C₂×C₄₇	#1291	1	1	yes	characteristic
104	C₂×#97	—	94	1	no	—
105	C₂×#97	—	94	1	no	—
106	C₂×C₄×C₈	—	47	2	no	—
107	C₄³	C₂×D₄₇	1	1	yes	characteristic
108	#336	—	2	1	no	—
109	#336	—	2	1	no	—
110	#336	—	47	2	no	—
111	#336	—	94	1	no	—
112	#336	—	94	1	no	—
113	#336	Dic₄₇	1	2	yes	—
114	C₄₇	#7019	1	1	yes	characteristic, complemented, Sylow
115	C₂²×C₈	—	2	2	no	—
116	C₂²×C₈	—	94	2	no	—
117	C₂²×C₈	—	94	2	no	—
118	C₂×C₄²	Dic₉₄	1	1	yes	characteristic
119	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
120	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
121	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
122	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
123	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
124	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
125	#97	—	94	4	no	—
126	#97	—	94	4	no	—
127	C₄×C₈	—	47	4	no	—
128	C₄×C₈	—	94	2	no	—
129	C₂²×C₄	C₄×Dic₄₇	1	1	yes	characteristic
130	C₂²×C₄	Dic₄₇⋊C₄	1	1	yes	characteristic
131	C₂²×C₄	Dic₄₇⋊C₄	1	1	yes	characteristic
132	C₂²×C₄	C₄⋊Dic₄₇	1	1	yes	characteristic
133	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
134	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
135	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
136	C₂×C₈	—	4	2	no	—
137	C₂×C₈	—	4	2	no	—
138	C₂×C₈	—	4	2	no	—
139	C₂×C₈	—	94	4	no	—
140	C₂×C₈	—	94	4	no	—
141	C₂×C₈	—	94	4	no	—
142	C₂×C₈	—	188	2	no	—
143	C₂×C₈	—	188	2	no	—
144	C₂×C₈	—	188	2	no	—
145	C₄²	—	2	2	no	—
146	C₄²	—	2	2	no	—
147	C₄²	—	4	1	no	—
148	C₄²	—	4	1	no	—
149	C₄²	—	4	1	no	—
150	C₄²	—	4	1	no	—
151	C₄²	C₈×D₄₇	1	2	yes	—
152	C₄²	#10625324	1	2	yes	—
153	C₂³	C₄₇⋊#95	1	1	yes	characteristic
154	C₂×C₄	—	2	1	no	—
155	C₂×C₄	—	2	1	no	—
156	C₂×C₄	—	2	1	no	—
157	C₂×C₄	—	2	1	no	—
158	C₂×C₄	—	2	1	no	—
159	C₂×C₄	—	2	1	no	—
160	C₂×C₄	—	2	1	no	—
161	C₂×C₄	—	2	1	no	—
162	C₂×C₄	—	2	1	no	—
163	C₂×C₄	—	2	1	no	—
164	C₂×C₄	—	2	1	no	—
165	C₂×C₄	—	2	1	no	—
166	C₂×C₄	—	2	1	no	—
167	C₂×C₄	—	2	1	no	—
168	C₂×C₄	—	2	1	no	—
169	C₂×C₄	—	2	1	no	—
170	C₂×C₄	C₈×Dic₄₇	1	1	yes	characteristic
171	C₂×C₄	#13093934	1	1	yes	characteristic
172	C₂×C₄	(C₂×D₄₇)⋊C₈	1	2	yes	—
173	C₂×C₄	#13093950	1	2	yes	—
174	C₂×C₄	#13093951	1	2	yes	—
175	C₂×C₄	Dic₄₇⋊C₈	1	2	yes	—
176	C₈	—	8	2	no	—
177	C₈	—	94	8	no	—
178	C₈	—	376	2	no	—
179	C₂²	#15688877	1	1	yes	characteristic
180	C₂²	#15688879	1	1	yes	characteristic
181	C₂²	C₄₇⋊#338	1	1	yes	characteristic
182	C₂²	#15688882	1	1	yes	characteristic
183	C₂²	#15688884	1	1	yes	characteristic
184	C₂²	#15688893	1	1	yes	characteristic
185	C₂²	#15688895	1	1	yes	characteristic
186	C₄	—	2	2	no	—
187	C₄	—	2	2	no	—
188	C₄	—	2	1	no	—
189	C₄	—	2	1	no	—
190	C₄	—	2	2	no	—
191	C₄	—	2	2	no	—
192	C₄	—	4	1	no	—
193	C₄	—	4	1	no	—
194	C₂	#20241218	1	1	yes	characteristic
195	C₂	#20241220	1	1	yes	characteristic
196	C₂	#20241232	1	1	yes	characteristic
197	C₂	#20241252	1	1	yes	characteristic
198	C₂	#20241254	1	1	yes	characteristic
199	C₂	#20241259	1	1	yes	characteristic
200	C₂	#20241261	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20243531

C₂

yes

maximal, characteristic

C₄×#13093935

C₂

yes

maximal, characteristic

C₄₇×#1839

C₂

yes

maximal, characteristic

C₂×#13093935

—

C₂×#13093935

—

C₂×C₄×C₄₇⋊C₈

C₄

yes

—

#15688878

—

#15688878

—

#15688878

C₄

yes

—

C₄³×C₄₇

C₂²

yes

characteristic

C₄₇×#336

—

C₄₇×#336

—

C₄₇×#336

C₄

yes

—

C₂²×C₄₇⋊C₈

—

C₂²×C₄₇⋊C₈

—

#13093935

—

#13093935

—

C₄×C₄₇⋊C₈

—

C₄×C₄₇⋊C₈

C₈

yes

complemented

C₂²×C₈×C₄₇

—

C₂×C₄²×C₄₇

C₂×C₄

yes

characteristic

C₂×C₄²×C₄₇

C₂×C₄

yes

characteristic

C₂×C₄²×C₄₇

C₂×C₄

yes

characteristic

C₂×C₄²×C₄₇

D₄

yes

characteristic

C₂×C₄²×C₄₇

D₄

yes

characteristic

C₂×C₄²×C₄₇

D₄

yes

characteristic

C₂×C₄²×C₄₇

Q₈

yes

characteristic

C₂²×C₄×C₄₇

C₄²

yes

characteristic

C₂²×C₄×C₄₇

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₄₇

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₄₇

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₄₇

C₄⋊C₄

yes

characteristic

C₂²×C₄×C₄₇

C₄⋊C₄

yes

characteristic

C₂²×C₄×C₄₇

C₄⋊C₄

yes

characteristic

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

C₂×C₈

yes

—

C₄²×C₄₇

M₁₆

yes

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

#95

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₄×C₈

yes

characteristic

C₂×C₄×C₄₇

#97

yes

characteristic

C₂×C₄×C₄₇

C₂²⋊C₈

yes

—

C₂×C₄×C₄₇

#104

yes

—

C₂×C₄×C₄₇

#104

yes

—

C₂×C₄×C₄₇

C₄⋊C₈

yes

—

C₈×C₄₇

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

—

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

#1839

—

#1839

D₄₇

yes

characteristic

C₄×#97

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

100

C₂×C₄₇

#1288

yes

characteristic

101

C₂×C₄₇

#1288

yes

characteristic

102

C₂×C₄₇

#1291

yes

characteristic

103

C₂×C₄₇

#1291

yes

characteristic

104

C₂×#97

—

105

C₂×#97

—

106

C₂×C₄×C₈

—

107

C₄³

C₂×D₄₇

yes

characteristic

108

#336

—

109

#336

—

110

#336

—

111

#336

—

112

#336

—

113

#336

Dic₄₇

yes

—

114

C₄₇

#7019

yes

characteristic, complemented, Sylow

115

C₂²×C₈

—

116

C₂²×C₈

—

117

C₂²×C₈

—

118

C₂×C₄²

Dic₉₄

yes

characteristic

119

C₂×C₄²

C₄×D₄₇

yes

characteristic

120

C₂×C₄²

C₄×D₄₇

yes

characteristic

121

C₂×C₄²

D₁₈₈

yes

characteristic

122

C₂×C₄²

C₂×Dic₄₇

yes

characteristic

123

C₂×C₄²

C₂²⋊D₄₇

yes

characteristic

124

C₂×C₄²

C₂²⋊D₄₇

yes

characteristic

125

#97

—

126

#97

—

127

C₄×C₈

—

128

C₄×C₈

—

129

C₂²×C₄

C₄×Dic₄₇

yes

characteristic

130

C₂²×C₄

Dic₄₇⋊C₄

yes

characteristic

131

C₂²×C₄

Dic₄₇⋊C₄

yes

characteristic

132

C₂²×C₄

C₄⋊Dic₄₇

yes

characteristic

133

C₂²×C₄

(C₂×D₄₇)⋊C₄

yes

characteristic

134

C₂²×C₄

(C₂×D₄₇)⋊C₄

yes

characteristic

135

C₂²×C₄

C₂²⋊Dic₄₇

yes

characteristic

136

C₂×C₈

—

137

C₂×C₈

—

138

C₂×C₈

—

139

C₂×C₈

—

140

C₂×C₈

—

141

C₂×C₈

—

142

C₂×C₈

—

188

—

143

C₂×C₈

—

188

—

144

C₂×C₈

—

188

—

145

C₄²

—

146

C₄²

—

147

C₄²

—

148

C₄²

—

149

C₄²

—

150

C₄²

—

151

C₄²

C₈×D₄₇

yes

—

152

C₄²

#10625324

yes

—

153

C₂³

C₄₇⋊#95

yes

characteristic

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

C₈×Dic₄₇

yes

characteristic

171

C₂×C₄

#13093934

yes

characteristic

172

C₂×C₄

(C₂×D₄₇)⋊C₈

yes

—

173

C₂×C₄

#13093950

yes

—

174

C₂×C₄

#13093951

yes

—

175

C₂×C₄

Dic₄₇⋊C₈

yes

—

176

C₈

—

177

C₈

—

178

C₈

—

376

—

179

C₂²

#15688877

yes

characteristic

180

C₂²

#15688879

yes

characteristic

181

C₂²

C₄₇⋊#338

yes

characteristic

182

C₂²

#15688882

yes

characteristic

183

C₂²

#15688884

yes

characteristic

184

C₂²

#15688893

yes

characteristic

185

C₂²

#15688895

yes

characteristic

186

C₄

—

187

C₄

—

188

C₄

—

189

C₄

—

190

C₄

—

191

C₄

—

192

C₄

—

193

C₄

—

194

C₂

#20241218

yes

characteristic

195

C₂

#20241220

yes

characteristic

196

C₂

#20241232

yes

characteristic

197

C₂

#20241252

yes

characteristic

198

C₂

#20241254

yes

characteristic

199

C₂

#20241259

yes

characteristic

200

C₂

#20241261

yes

characteristic

	subgroup	G/H
center	C₂³	C₄₇⋊#95
commutator	C₂×C₄×C₄₇	C₄×C₈
Frattini	C₄³	C₂×D₄₇
Fitting	C₄₇×#1839	C₂
socle, prime radical	C₂³×C₄₇	#95
cosocle	C₄³×C₄₇	C₂²
hypercenter	#1839	D₄₇
hypocenter	C₄₇	#7019
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #1839

subgroup

G/H

center

C₂³

C₄₇⋊#95

commutator

C₂×C₄×C₄₇

C₄×C₈

Frattini

C₄³

C₂×D₄₇

Fitting

C₄₇×#1839

C₂

socle, prime radical

C₂³×C₄₇

#95

cosocle

C₄³×C₄₇

C₂²

D₄₇

C₄₇

G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂³ ◁ C₄³ ◁ #1839

N	H ≅ G/N	type
19. C₄×C₄₇⋊C₈	C₈	semidirect
114. C₄₇	#7019	semidirect

H ≅ G/N

type

19. C₄×C₄₇⋊C₈

C₈

semidirect

114. C₄₇

#7019

semidirect

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

subgroup	index
#20243531	2
C₄×#13093935	2
C₄₇×#1839	2
#7019	47

subgroup

index

#20243531

C₄×#13093935

C₄₇×#1839

#7019

Subgroups — #26102813

Conjugacy classes of subgroups

Complements

Maximal subgroups