Group #26102831 — subgroups

Subgroups — #26102831

Order 12032.

442 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 442 conjugacy classes of subgroups — too many to draw. Only its 135 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	C₄×C₂²⋊(C₄₇⋊C₈)	C₂	1	2	yes	maximal
2	C₄₇×#1728	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
4	C₂×C₄×C₄₇⋊C₈	—	2	2	no	—
5	#15688878	C₄	1	4	yes	—
6	C₂²×M₁₆×C₄₇	C₄	1	2	yes	—
7	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
8	C₄₇×#336	—	2	2	no	—
9	C₂²×C₄₇⋊C₈	—	2	4	no	—
10	C₂²⋊(C₄₇⋊C₈)	—	2	8	no	—
11	C₄×C₄₇⋊C₈	—	4	4	no	—
12	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
13	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
14	C₂³×C₄×C₄₇	Q₈	1	1	yes	characteristic
15	C₂×M₁₆×C₄₇	—	2	2	no	—
16	C₂×M₁₆×C₄₇	—	4	2	no	—
17	C₂×M₁₆×C₄₇	—	4	2	no	—
18	C₂×M₁₆×C₄₇	C₈	1	4	yes	complemented
19	C₂²×C₈×C₄₇	—	2	2	no	—
20	C₂×C₄²×C₄₇	—	4	2	no	—
21	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
22	C₂×C₄²×C₄₇	D₄	1	2	yes	—
23	C₂⁴×C₄₇	C₄⋊C₄	1	1	yes	characteristic
24	C₂²×C₄×C₄₇	—	2	2	no	—
25	C₂²×C₄×C₄₇	—	2	1	no	—
26	C₂²×C₄×C₄₇	—	2	1	no	—
27	C₂²×C₄×C₄₇	—	2	4	no	—
28	C₂²×C₄×C₄₇	—	2	4	no	—
29	C₂²×C₄×C₄₇	—	4	1	no	—
30	C₂²×C₄×C₄₇	—	4	1	no	—
31	C₂²×C₄×C₄₇	C₄²	1	1	yes	characteristic
32	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
33	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
34	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
35	C₂²×C₄×C₄₇	C₂×C₈	1	2	yes	—
36	C₂²×C₄×C₄₇	M₁₆	1	2	yes	—
37	M₁₆×C₄₇	—	4	4	no	—
38	M₁₆×C₄₇	—	4	4	no	—
39	C₂×C₈×C₄₇	—	2	4	no	—
40	C₂×C₈×C₄₇	—	4	2	no	—
41	C₂×C₈×C₄₇	—	4	2	no	—
42	C₄²×C₄₇	—	2	4	no	—
43	C₄²×C₄₇	—	4	2	no	—
44	C₂×C₄₇⋊C₈	—	2	8	no	—
45	C₂×C₄₇⋊C₈	—	4	4	no	—
46	C₂×C₄₇⋊C₈	—	4	4	no	—
47	C₂³×C₄₇	—	2	2	no	—
48	C₂³×C₄₇	—	2	2	no	—
49	C₂³×C₄₇	—	2	1	no	—
50	C₂³×C₄₇	—	2	1	no	—
51	C₂³×C₄₇	#95	1	1	yes	characteristic
52	C₂³×C₄₇	C₄⋊C₈	1	2	yes	—
53	C₂×C₄×C₄₇	—	2	2	no	—
54	C₂×C₄×C₄₇	—	2	2	no	—
55	C₂×C₄×C₄₇	—	2	2	no	—
56	C₂×C₄×C₄₇	—	2	2	no	—
57	C₂×C₄×C₄₇	—	2	1	no	—
58	C₂×C₄×C₄₇	—	2	1	no	—
59	C₂×C₄×C₄₇	—	2	2	no	—
60	C₂×C₄×C₄₇	—	2	2	no	—
61	C₂×C₄×C₄₇	—	2	2	no	—
62	C₂×C₄×C₄₇	—	2	2	no	—
63	C₂×C₄×C₄₇	—	4	4	no	—
64	C₂×C₄×C₄₇	—	4	4	no	—
65	C₂×C₄×C₄₇	C₄×C₈	1	1	yes	characteristic
66	C₂×C₄×C₄₇	#97	1	1	yes	characteristic
67	C₂×C₄×C₄₇	C₂²⋊C₈	1	2	yes	—
68	C₂×C₄×C₄₇	#99	1	2	yes	—
69	C₂×C₄×C₄₇	#100	1	1	yes	characteristic
70	C₂×C₄×C₄₇	#101	1	1	yes	characteristic
71	C₂×C₄×C₄₇	#104	1	4	yes	—
72	C₂×C₄×C₄₇	#108	1	4	yes	—
73	C₈×C₄₇	—	4	4	no	—
74	C₄₇⋊C₈	—	4	8	no	—
75	#7026	—	47	1	no	maximal, Sylow
76	C₂²×C₄₇	—	2	2	no	—
77	C₂²×C₄₇	—	2	4	no	—
78	C₂²×C₄₇	—	4	1	no	—
79	C₂²×C₄₇	—	4	1	no	—
80	C₂²×C₄₇	—	4	2	no	—
81	C₂²×C₄₇	#336	1	1	yes	characteristic
82	C₂²×C₄₇	#339	1	2	yes	—
83	C₂²×C₄₇	#341	1	2	yes	—
84	C₂²×C₄₇	#342	1	1	yes	characteristic
85	C₂²×C₄₇	#343	1	1	yes	characteristic
86	C₄×C₄₇	—	2	1	no	—
87	C₄×C₄₇	—	2	1	no	—
88	C₄×C₄₇	—	2	4	no	—
89	C₄×C₄₇	—	2	4	no	—
90	C₄×C₄₇	#323	1	2	yes	—
91	C₄×C₄₇	#324	1	2	yes	—
92	C₄×C₂²⋊C₈	—	47	2	no	—
93	#1728	D₄₇	1	1	yes	characteristic
94	C₂×C₄₇	—	4	2	no	—
95	C₂×C₄₇	#1278	1	2	yes	—
96	C₂×C₄₇	#1280	1	1	yes	characteristic
97	C₂×C₄₇	#1297	1	1	yes	characteristic
98	C₂×C₄₇	#1298	1	1	yes	characteristic
99	C₂×C₄₇	#1299	1	2	yes	—
100	C₂²×M₁₆	Dic₄₇	1	2	yes	—
101	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
102	C₂×C₂²⋊C₈	—	94	2	no	—
103	C₂×C₄×C₈	—	94	2	no	—
104	#336	—	2	2	no	—
105	#336	—	47	4	no	—
106	C₄₇	#7026	1	1	yes	characteristic, complemented, Sylow
107	C₂³×C₄	Dic₉₄	1	1	yes	characteristic
108	C₂³×C₄	D₁₈₈	1	1	yes	characteristic
109	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
110	C₂×M₁₆	—	2	2	no	—
111	C₂×M₁₆	—	4	2	no	—
112	C₂×M₁₆	—	4	2	no	—
113	C₂×M₁₆	C₄₇⋊C₈	1	4	yes	complemented
114	C₂²×C₈	—	2	2	no	—
115	C₂²×C₈	—	94	4	no	—
116	C₂×C₄²	—	4	2	no	—
117	C₂×C₄²	C₄×D₄₇	1	2	yes	—
118	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
119	C₂²⋊C₈	—	94	8	no	—
120	C₄×C₈	—	188	4	no	—
121	C₂⁴	C₄⋊Dic₄₇	1	1	yes	characteristic
122	C₂²×C₄	—	2	2	no	—
123	C₂²×C₄	—	2	1	no	—
124	C₂²×C₄	—	2	1	no	—
125	C₂²×C₄	—	2	4	no	—
126	C₂²×C₄	—	2	4	no	—
127	C₂²×C₄	—	4	1	no	—
128	C₂²×C₄	—	4	1	no	—
129	C₂²×C₄	C₂×C₄₇⋊C₈	1	2	yes	—
130	C₂²×C₄	#10625329	1	2	yes	—
131	C₂²×C₄	C₄×Dic₄₇	1	1	yes	characteristic
132	C₂²×C₄	Dic₄₇⋊C₄	1	2	yes	—
133	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
134	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
135	M₁₆	—	4	4	no	—
136	M₁₆	—	4	4	no	—
137	C₂×C₈	—	2	4	no	—
138	C₂×C₈	—	4	2	no	—
139	C₂×C₈	—	4	2	no	—
140	C₂×C₈	—	94	8	no	—
141	C₂×C₈	—	188	4	no	—
142	C₂×C₈	—	188	4	no	—
143	C₄²	—	2	4	no	—
144	C₄²	—	4	2	no	—
145	C₂³	—	2	2	no	—
146	C₂³	—	2	2	no	—
147	C₂³	—	2	1	no	—
148	C₂³	—	2	1	no	—
149	C₂³	C₄₇⋊#95	1	1	yes	characteristic
150	C₂³	C₄⋊(C₄₇⋊C₈)	1	2	yes	—
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	2	1	no	—
156	C₂×C₄	—	2	1	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	4	4	no	—
162	C₂×C₄	—	4	4	no	—
163	C₂×C₄	C₄×C₄₇⋊C₈	1	1	yes	characteristic
164	C₂×C₄	#13093935	1	1	yes	characteristic
165	C₂×C₄	C₂²⋊(C₄₇⋊C₈)	1	2	yes	—
166	C₂×C₄	#13093939	1	2	yes	—
167	C₂×C₄	#13093941	1	1	yes	characteristic
168	C₂×C₄	#13093943	1	1	yes	characteristic
169	C₂×C₄	#13093950	1	4	yes	—
170	C₂×C₄	#13093959	1	4	yes	—
171	C₈	—	4	4	no	—
172	C₈	—	188	8	no	—
173	C₂²	—	2	4	no	—
174	C₂²	—	2	2	no	—
175	C₂²	—	4	1	no	—
176	C₂²	—	4	1	no	—
177	C₂²	—	4	2	no	—
178	C₂²	#15688878	1	1	yes	characteristic
179	C₂²	#15688884	1	2	yes	—
180	C₂²	#15688887	1	2	yes	—
181	C₂²	#15688890	1	1	yes	characteristic
182	C₂²	#15688892	1	1	yes	characteristic
183	C₄	—	2	1	no	—
184	C₄	—	2	1	no	—
185	C₄	—	2	4	no	—
186	C₄	—	2	4	no	—
187	C₄	#15688847	1	2	yes	—
188	C₄	#15688849	1	2	yes	—
189	C₂	—	4	2	no	—
190	C₂	#20241226	1	2	yes	—
191	C₂	#20241231	1	1	yes	characteristic
192	C₂	#20241273	1	1	yes	characteristic
193	C₂	#20241275	1	1	yes	characteristic
194	C₂	#20241277	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×C₂²⋊(C₄₇⋊C₈)

C₂

yes

maximal

C₄₇×#1728

C₂

yes

maximal, characteristic

C₂×C₂²⋊(C₄₇⋊C₈)

—

C₂×C₄×C₄₇⋊C₈

—

#15688878

C₄

yes

—

C₂²×M₁₆×C₄₇

C₄

yes

—

C₂²×C₄²×C₄₇

C₂²

yes

characteristic

C₄₇×#336

—

C₂²×C₄₇⋊C₈

—

C₂²⋊(C₄₇⋊C₈)

—

C₄×C₄₇⋊C₈

—

C₂³×C₄×C₄₇

C₂×C₄

yes

characteristic

C₂³×C₄×C₄₇

D₄

yes

characteristic

C₂³×C₄×C₄₇

Q₈

yes

characteristic

C₂×M₁₆×C₄₇

—

C₂×M₁₆×C₄₇

—

C₂×M₁₆×C₄₇

—

C₂×M₁₆×C₄₇

C₈

yes

complemented

C₂²×C₈×C₄₇

—

C₂×C₄²×C₄₇

—

C₂×C₄²×C₄₇

C₂×C₄

yes

—

C₂×C₄²×C₄₇

D₄

yes

—

C₂⁴×C₄₇

C₄⋊C₄

yes

characteristic

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₄²

yes

characteristic

C₂²×C₄×C₄₇

C₂²⋊C₄

yes

—

C₂²×C₄×C₄₇

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₄₇

C₄⋊C₄

yes

—

C₂²×C₄×C₄₇

C₂×C₈

yes

—

C₂²×C₄×C₄₇

M₁₆

yes

—

M₁₆×C₄₇

—

M₁₆×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

#95

yes

characteristic

C₂³×C₄₇

C₄⋊C₈

yes

—

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

#99

yes

—

C₂×C₄×C₄₇

yes

C₂×C₄×C₄₇

yes

C₂×C₄×C₄₇

yes

—

C₂×C₄×C₄₇

#108

yes

—

C₈×C₄₇

—

C₄₇⋊C₈

—

#7026

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

#336

yes

characteristic

C₂²×C₄₇

#339

yes

—

C₂²×C₄₇

#341

yes

—

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#323

yes

—

C₄×C₄₇

#324

yes

—

C₄×C₂²⋊C₈

—

#1728

D₄₇

yes

characteristic

C₂×C₄₇

—

C₂×C₄₇

#1278

yes

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

—

100

C₂²×M₁₆

Dic₄₇

yes

—

101

C₂²×C₄²

C₂×D₄₇

yes

characteristic

102

C₂×C₂²⋊C₈

—

103

C₂×C₄×C₈

—

104

#336

—

105

#336

—

106

C₄₇

#7026

yes

characteristic, complemented, Sylow

107

C₂³×C₄

Dic₉₄

yes

characteristic

108

C₂³×C₄

D₁₈₈

yes

characteristic

109

C₂³×C₄

C₂×Dic₄₇

yes

characteristic

110

C₂×M₁₆

—

111

C₂×M₁₆

—

112

C₂×M₁₆

—

113

C₂×M₁₆

C₄₇⋊C₈

yes

complemented

114

C₂²×C₈

—

115

C₂²×C₈

—

116

C₂×C₄²

—

117

C₂×C₄²

C₄×D₄₇

yes

—

118

C₂×C₄²

C₂²⋊D₄₇

yes

—

119

C₂²⋊C₈

—

120

C₄×C₈

—

188

—

121

C₂⁴

C₄⋊Dic₄₇

yes

characteristic

122

C₂²×C₄

—

123

C₂²×C₄

—

124

C₂²×C₄

—

125

C₂²×C₄

—

126

C₂²×C₄

—

127

C₂²×C₄

—

128

C₂²×C₄

—

129

C₂²×C₄

C₂×C₄₇⋊C₈

yes

—

130

C₂²×C₄

#10625329

yes

—

131

C₂²×C₄

C₄×Dic₄₇

yes

characteristic

132

C₂²×C₄

Dic₄₇⋊C₄

yes

—

133

C₂²×C₄

(C₂×D₄₇)⋊C₄

yes

—

134

C₂²×C₄

C₂²⋊Dic₄₇

yes

characteristic

135

M₁₆

—

136

M₁₆

—

137

C₂×C₈

—

138

C₂×C₈

—

139

C₂×C₈

—

140

C₂×C₈

—

141

C₂×C₈

—

188

—

142

C₂×C₈

—

188

—

143

C₄²

—

144

C₄²

—

145

C₂³

—

146

C₂³

—

147

C₂³

—

148

C₂³

—

149

C₂³

C₄₇⋊#95

yes

characteristic

150

C₂³

C₄⋊(C₄₇⋊C₈)

yes

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

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

C₄×C₄₇⋊C₈

yes

characteristic

164

C₂×C₄

#13093935

yes

characteristic

165

C₂×C₄

C₂²⋊(C₄₇⋊C₈)

yes

—

166

C₂×C₄

#13093939

yes

—

167

C₂×C₄

#13093941

yes

characteristic

168

C₂×C₄

#13093943

yes

characteristic

169

C₂×C₄

#13093950

yes

—

170

C₂×C₄

#13093959

yes

—

171

C₈

—

172

C₈

—

188

—

173

C₂²

—

174

C₂²

—

175

C₂²

—

176

C₂²

—

177

C₂²

—

178

C₂²

#15688878

yes

characteristic

179

C₂²

#15688884

yes

—

180

C₂²

#15688887

yes

—

181

C₂²

#15688890

yes

characteristic

182

C₂²

#15688892

yes

characteristic

183

C₄

—

184

C₄

—

185

C₄

—

186

C₄

—

187

C₄

#15688847

yes

—

188

C₄

#15688849

yes

—

189

C₂

—

190

C₂

#20241226

yes

—

191

C₂

#20241231

yes

characteristic

192

C₂

#20241273

yes

characteristic

193

C₂

#20241275

yes

characteristic

194

C₂

#20241277

yes

—

	subgroup	G/H
center	C₂²×C₄	C₂²⋊Dic₄₇
commutator	C₂×C₄×C₄₇	C₄×C₈
Frattini	C₂²×C₄²	C₂×D₄₇
Fitting	C₄₇×#1728	C₂
socle	C₂³×C₄₇	#95
prime radical	C₂⁴×C₄₇	C₄⋊C₄
cosocle	C₂²×C₄²×C₄₇	C₂²
hypercenter	#1728	D₄₇
hypocenter	C₄₇	#7026
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ #1728

subgroup

G/H

center

C₂²×C₄

C₂²⋊Dic₄₇

commutator

C₂×C₄×C₄₇

C₄×C₈

Frattini

C₂²×C₄²

C₂×D₄₇

Fitting

C₄₇×#1728

C₂

socle

C₂³×C₄₇

#95

prime radical

C₂⁴×C₄₇

C₄⋊C₄

cosocle

C₂²×C₄²×C₄₇

C₂²

D₄₇

C₄₇

G ▷ C₂×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ #1728

N	H ≅ G/N	type
18. C₂×M₁₆×C₄₇	C₈	semidirect
106. C₄₇	#7026	semidirect
113. C₂×M₁₆	C₄₇⋊C₈	semidirect

H ≅ G/N

type

18. C₂×M₁₆×C₄₇

C₈

semidirect

106. C₄₇

#7026

semidirect

113. C₂×M₁₆

C₄₇⋊C₈

semidirect

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

subgroup	index
C₄×C₂²⋊(C₄₇⋊C₈)	2
C₄₇×#1728	2
#7026	47

subgroup

index

C₄×C₂²⋊(C₄₇⋊C₈)

C₄₇×#1728

#7026

Subgroups — #26102831

Conjugacy classes of subgroups

Complements

Maximal subgroups