Group #26107418 — subgroups

Subgroups — #26107418

Order 12032.

470 conjugacy classes of subgroups; only the 199 normal subgroups are shown.

normal subgroups only

This group has 470 conjugacy classes of subgroups — too many to draw. Only its 199 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×Dic₄₇⋊C₈	C₂	1	2	yes	maximal
2	C₄×#13093934	C₂	1	1	yes	maximal, characteristic
3	C₂×#15688845	C₂	1	2	yes	maximal
4	C₄₇⋊#1273	C₂	1	1	yes	maximal, characteristic
5	C₄₇×#1273	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₄₇⋊C₈	C₄	1	4	yes	—
7	C₂×#13093934	C₄	1	2	yes	—
8	C₂×#13093934	C₄	1	2	yes	—
9	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
10	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
11	C₄²×Dic₄₇	C₂²	1	1	yes	characteristic
12	#15688877	C₄	1	4	yes	—
13	#15688845	—	2	4	no	—
14	C₂×C₄×C₈×C₄₇	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
16	C₂²×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
17	C₂²×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
18	C₂×C₄×Dic₄₇	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×Dic₄₇	C₂×C₄	1	2	yes	—
20	C₂×C₄×Dic₄₇	C₂×C₄	1	2	yes	—
21	C₂×C₄×Dic₄₇	C₂×C₄	1	1	yes	characteristic
22	Dic₄₇⋊C₈	—	2	8	no	—
23	#13093934	—	2	4	no	—
24	#13093934	—	2	4	no	—
25	C₄×C₄₇⋊C₈	—	2	2	no	—
26	C₄×C₄₇⋊C₈	—	2	2	no	—
27	C₄×C₄₇⋊C₈	D₄	1	2	yes	—
28	C₄×C₄₇⋊C₈	Q₈	1	2	yes	—
29	C₂²×C₈×C₄₇	C₂×C₄	1	4	yes	—
30	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
31	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
32	C₄×C₈×C₄₇	—	2	2	no	—
33	C₄×C₈×C₄₇	—	2	2	no	—
34	C₄×C₈×C₄₇	—	2	2	no	—
35	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
36	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
37	C₂²×Dic₄₇	C₄²	1	1	yes	characteristic
38	C₂²×Dic₄₇	C₄²	1	1	yes	characteristic
39	C₂²×Dic₄₇	C₄²	1	2	yes	—
40	C₂×C₈×C₄₇	—	2	2	no	—
41	C₂×C₈×C₄₇	—	2	2	no	—
42	C₂×C₈×C₄₇	—	2	4	no	—
43	C₂×C₈×C₄₇	—	2	4	no	—
44	C₂×C₈×C₄₇	—	2	4	no	—
45	C₂×C₈×C₄₇	M₁₆	1	4	yes	—
46	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
47	C₄²×C₄₇	C₂×Q₈	1	1	yes	characteristic
48	C₄²×C₄₇	#41	1	2	yes	—
49	C₄×Dic₄₇	—	2	2	no	—
50	C₄×Dic₄₇	—	2	4	no	—
51	C₄×Dic₄₇	—	2	4	no	—
52	C₄×Dic₄₇	M₁₆	1	4	yes	—
53	C₂×C₄₇⋊C₈	—	2	4	no	—
54	C₂×C₄₇⋊C₈	—	2	2	no	—
55	C₂×C₄₇⋊C₈	—	2	2	no	—
56	C₂×C₄₇⋊C₈	—	2	4	no	—
57	C₂×C₄₇⋊C₈	C₄⋊C₄	1	4	yes	—
58	C₂×C₄₇⋊C₈	M₁₆	1	8	yes	—
59	C₂³×C₄₇	C₂×C₄²	1	1	yes	characteristic
60	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
61	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
62	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
63	C₂×C₄×C₄₇	C₄×Q₈	1	2	yes	—
64	C₂×C₄×C₄₇	C₂×M₁₆	1	2	yes	—
65	C₂×C₄×C₄₇	C₂×M₁₆	1	4	yes	—
66	C₂×C₄×C₄₇	C₂×M₁₆	1	2	yes	—
67	C₂×C₄×C₄₇	#131	1	4	yes	—
68	C₂×Dic₄₇	—	2	2	no	—
69	C₂×Dic₄₇	—	2	2	no	—
70	C₂×Dic₄₇	—	2	1	no	—
71	C₂×Dic₄₇	—	2	1	no	—
72	C₂×Dic₄₇	—	2	2	no	—
73	C₂×Dic₄₇	—	2	2	no	—
74	C₂×Dic₄₇	#97	1	4	yes	—
75	C₈×C₄₇	—	2	4	no	—
76	C₈×C₄₇	—	4	4	no	—
77	C₄₇⋊C₈	—	2	4	no	—
78	C₄₇⋊C₈	—	2	8	no	—
79	#8263	—	47	1	no	maximal, Sylow
80	C₂²×C₄₇	C₄×C₄⋊C₄	1	1	yes	characteristic
81	C₂²×C₄₇	C₂×#97	1	1	yes	characteristic
82	C₂²×C₄₇	C₄×M₁₆	1	1	yes	characteristic
83	C₂²×C₄₇	C₄×M₁₆	1	2	yes	—
84	C₂²×C₄₇	#405	1	2	yes	—
85	C₄×C₄₇	#423	1	2	yes	—
86	C₄×C₄₇	#432	1	2	yes	—
87	C₄×C₄₇	#435	1	4	yes	—
88	C₄×C₄₇	#446	1	4	yes	—
89	Dic₄₇	—	2	4	no	—
90	Dic₄₇	—	4	1	no	—
91	Dic₄₇	—	4	1	no	—
92	C₄×C₄⋊C₈	—	47	2	no	—
93	C₄×#97	—	47	1	no	—
94	C₂×#322	—	47	2	no	—
95	#1273	—	47	1	no	—
96	#1273	D₄₇	1	1	yes	characteristic
97	C₂×C₄₇	#1451	1	2	yes	—
98	C₂×C₄₇	#1453	1	1	yes	characteristic
99	C₂×C₄₇	#1455	1	1	yes	characteristic
100	C₂×C₄₇	#1767	1	1	yes	characteristic
101	C₂×C₄₇	#1771	1	2	yes	—
102	C₂×C₄⋊C₈	—	47	4	no	—
103	C₂×#97	—	47	2	no	—
104	C₂×#97	—	47	2	no	—
105	C₂×C₄×C₈	—	47	1	no	—
106	C₂×C₄×C₈	—	47	2	no	—
107	C₂×C₄×C₈	C₂×D₄₇	1	2	yes	—
108	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
109	C₄³	—	47	1	no	—
110	#336	—	47	4	no	—
111	#322	—	94	4	no	—
112	C₄₇	#8263	1	1	yes	characteristic, complemented, Sylow
113	C₂²×C₈	—	47	4	no	—
114	C₂²×C₈	—	47	2	no	—
115	C₂²×C₈	C₄×D₄₇	1	4	yes	—
116	C₂²×C₈	C₄×D₄₇	1	2	yes	—
117	C₂×C₄²	—	47	1	no	—
118	C₂×C₄²	—	47	2	no	—
119	C₂×C₄²	—	47	1	no	—
120	C₂×C₄²	—	47	2	no	—
121	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
122	C₄⋊C₈	—	94	8	no	—
123	#97	—	94	4	no	—
124	#97	—	94	4	no	—
125	C₄×C₈	—	2	2	no	—
126	C₄×C₈	—	2	2	no	—
127	C₄×C₈	—	2	2	no	—
128	C₄×C₈	—	47	2	no	—
129	C₄×C₈	—	47	2	no	—
130	C₄×C₈	—	94	2	no	—
131	C₄×C₈	—	94	2	no	—
132	C₂²×C₄	—	47	2	no	—
133	C₂²×C₄	—	47	1	no	—
134	C₂²×C₄	—	47	1	no	—
135	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
136	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
137	C₂×C₈	—	2	2	no	—
138	C₂×C₈	—	2	2	no	—
139	C₂×C₈	—	2	4	no	—
140	C₂×C₈	—	2	4	no	—
141	C₂×C₈	—	2	4	no	—
142	C₂×C₈	—	47	4	no	—
143	C₂×C₈	—	47	8	no	—
144	C₂×C₈	—	94	4	no	—
145	C₂×C₈	—	94	2	no	—
146	C₂×C₈	—	94	2	no	—
147	C₂×C₈	—	94	4	no	—
148	C₂×C₈	#10625324	1	4	yes	—
149	C₄²	—	47	4	no	—
150	C₄²	—	94	4	no	—
151	C₄²	—	94	2	no	—
152	C₄²	—	94	4	no	—
153	C₄²	D₄×D₄₇	1	1	yes	characteristic
154	C₄²	#10625352	1	2	yes	—
155	C₄²	Q₈×D₄₇	1	1	yes	characteristic
156	C₂³	C₄²×D₄₇	1	1	yes	characteristic
157	C₂×C₄	—	47	4	no	—
158	C₂×C₄	—	94	2	no	—
159	C₂×C₄	—	94	2	no	—
160	C₂×C₄	—	94	1	no	—
161	C₂×C₄	—	94	1	no	—
162	C₂×C₄	—	94	2	no	—
163	C₂×C₄	—	94	2	no	—
164	C₂×C₄	C₄⋊C₄×D₄₇	1	1	yes	characteristic
165	C₂×C₄	#13093980	1	1	yes	characteristic
166	C₂×C₄	(C₂²⋊D₄₇)⋊C₄	1	2	yes	—
167	C₂×C₄	#13093989	1	2	yes	—
168	C₂×C₄	M₁₆×D₄₇	1	4	yes	—
169	C₂×C₄	M₁₆×D₄₇	1	2	yes	—
170	C₂×C₄	C₂×#10625324	1	2	yes	—
171	C₂×C₄	#13094030	1	4	yes	—
172	C₈	—	2	4	no	—
173	C₈	—	4	4	no	—
174	C₈	—	94	4	no	—
175	C₈	—	94	8	no	—
176	C₂²	#15688974	1	1	yes	characteristic
177	C₂²	#97×D₄₇	1	1	yes	characteristic
178	C₂²	C₄×#10625324	1	1	yes	characteristic
179	C₂²	#15689075	1	2	yes	—
180	C₂²	#15689077	1	2	yes	—
181	C₄	—	94	4	no	—
182	C₄	—	188	1	no	—
183	C₄	—	188	1	no	—
184	C₄	#15689145	1	2	yes	—
185	C₄	#15689177	1	2	yes	—
186	C₄	#15689197	1	4	yes	—
187	C₄	#15689252	1	4	yes	—
188	C₂	#20241676	1	2	yes	—
189	C₂	#20241682	1	1	yes	characteristic
190	C₂	#20241692	1	1	yes	characteristic
191	C₂	#20243240	1	1	yes	characteristic
192	C₂	#20243261	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×Dic₄₇⋊C₈

C₂

yes

maximal

C₄×#13093934

C₂

yes

maximal, characteristic

C₂×#15688845

C₂

yes

maximal

C₄₇⋊#1273

C₂

yes

maximal, characteristic

C₄₇×#1273

C₂

yes

maximal, characteristic

C₂×Dic₄₇⋊C₈

C₄

yes

—

C₂×#13093934

C₄

yes

—

C₂×#13093934

C₄

yes

—

C₂×C₄×C₄₇⋊C₈

C₂²

yes

—

C₂×C₄×C₄₇⋊C₈

C₂²

yes

characteristic

C₄²×Dic₄₇

C₂²

yes

characteristic

#15688877

C₄

yes

—

#15688845

—

C₂×C₄×C₈×C₄₇

C₂²

yes

—

C₂×C₄×C₈×C₄₇

C₂²

yes

characteristic

C₂²×C₄₇⋊C₈

C₂×C₄

yes

—

C₂²×C₄₇⋊C₈

C₂×C₄

yes

—

C₂×C₄×Dic₄₇

C₂×C₄

yes

characteristic

C₂×C₄×Dic₄₇

C₂×C₄

yes

—

C₂×C₄×Dic₄₇

C₂×C₄

yes

—

C₂×C₄×Dic₄₇

C₂×C₄

yes

characteristic

Dic₄₇⋊C₈

—

#13093934

—

#13093934

—

C₄×C₄₇⋊C₈

—

C₄×C₄₇⋊C₈

—

C₄×C₄₇⋊C₈

D₄

yes

—

C₄×C₄₇⋊C₈

Q₈

yes

—

C₂²×C₈×C₄₇

C₂×C₄

yes

—

C₂²×C₈×C₄₇

C₂×C₄

yes

—

C₂×C₄²×C₄₇

C₂³

yes

characteristic

C₄×C₈×C₄₇

—

C₄×C₈×C₄₇

—

C₄×C₈×C₄₇

—

C₂²×C₄×C₄₇

C₂²×C₄

yes

characteristic

C₂²×C₄×C₄₇

C₂²×C₄

yes

—

C₂²×Dic₄₇

C₄²

yes

characteristic

C₂²×Dic₄₇

C₄²

yes

characteristic

C₂²×Dic₄₇

C₄²

yes

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

M₁₆

yes

—

C₄²×C₄₇

C₂×D₄

yes

characteristic

C₄²×C₄₇

C₂×Q₈

yes

characteristic

C₄²×C₄₇

#41

yes

—

C₄×Dic₄₇

—

C₄×Dic₄₇

—

C₄×Dic₄₇

—

C₄×Dic₄₇

M₁₆

yes

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

C₄⋊C₄

yes

—

C₂×C₄₇⋊C₈

M₁₆

yes

—

C₂³×C₄₇

C₂×C₄²

yes

characteristic

C₂×C₄×C₄₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₄₇

#117

yes

characteristic

C₂×C₄×C₄₇

C₄×D₄

yes

—

C₂×C₄×C₄₇

C₄×Q₈

yes

—

C₂×C₄×C₄₇

C₂×M₁₆

yes

—

C₂×C₄×C₄₇

C₂×M₁₆

yes

—

C₂×C₄×C₄₇

C₂×M₁₆

yes

—

C₂×C₄×C₄₇

#131

yes

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

#97

yes

—

C₈×C₄₇

—

C₈×C₄₇

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

#8263

—

maximal, Sylow

C₂²×C₄₇

C₄×C₄⋊C₄

yes

characteristic

C₂²×C₄₇

C₂×#97

yes

characteristic

C₂²×C₄₇

C₄×M₁₆

yes

characteristic

C₂²×C₄₇

C₄×M₁₆

yes

—

C₂²×C₄₇

#405

yes

—

C₄×C₄₇

#423

yes

—

C₄×C₄₇

#432

yes

—

C₄×C₄₇

#435

yes

—

C₄×C₄₇

#446

yes

—

Dic₄₇

—

Dic₄₇

—

Dic₄₇

—

C₄×C₄⋊C₈

—

C₄×#97

—

C₂×#322

—

#1273

—

#1273

D₄₇

yes

characteristic

C₂×C₄₇

#1451

yes

—

C₂×C₄₇

yes

C₂×C₄₇

yes

100

C₂×C₄₇

#1767

yes

characteristic

101

C₂×C₄₇

#1771

yes

—

102

C₂×C₄⋊C₈

—

103

C₂×#97

—

104

C₂×#97

—

105

C₂×C₄×C₈

—

106

C₂×C₄×C₈

—

107

C₂×C₄×C₈

C₂×D₄₇

yes

—

108

C₂×C₄×C₈

C₂×D₄₇

yes

characteristic

109

C₄³

—

110

#336

—

111

#322

—

112

C₄₇

#8263

yes

characteristic, complemented, Sylow

113

C₂²×C₈

—

114

C₂²×C₈

—

115

C₂²×C₈

C₄×D₄₇

yes

—

116

C₂²×C₈

C₄×D₄₇

yes

—

117

C₂×C₄²

—

118

C₂×C₄²

—

119

C₂×C₄²

—

120

C₂×C₄²

—

121

C₂×C₄²

C₂²×D₄₇

yes

characteristic

122

C₄⋊C₈

—

123

#97

—

124

#97

—

125

C₄×C₈

—

126

C₄×C₈

—

127

C₄×C₈

—

128

C₄×C₈

—

129

C₄×C₈

—

130

C₄×C₈

—

131

C₄×C₈

—

132

C₂²×C₄

—

133

C₂²×C₄

—

134

C₂²×C₄

—

135

C₂²×C₄

C₂×C₄×D₄₇

yes

characteristic

136

C₂²×C₄

C₂×C₄×D₄₇

yes

—

137

C₂×C₈

—

138

C₂×C₈

—

139

C₂×C₈

—

140

C₂×C₈

—

141

C₂×C₈

—

142

C₂×C₈

—

143

C₂×C₈

—

144

C₂×C₈

—

145

C₂×C₈

—

146

C₂×C₈

—

147

C₂×C₈

—

148

C₂×C₈

#10625324

yes

—

149

C₄²

—

150

C₄²

—

151

C₄²

—

152

C₄²

—

153

C₄²

D₄×D₄₇

yes

characteristic

154

C₄²

#10625352

yes

—

155

C₄²

Q₈×D₄₇

yes

characteristic

156

C₂³

C₄²×D₄₇

yes

characteristic

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

C₄⋊C₄×D₄₇

yes

characteristic

165

C₂×C₄

#13093980

yes

characteristic

166

C₂×C₄

(C₂²⋊D₄₇)⋊C₄

yes

—

167

C₂×C₄

#13093989

yes

—

168

C₂×C₄

M₁₆×D₄₇

yes

—

169

C₂×C₄

M₁₆×D₄₇

yes

—

170

C₂×C₄

C₂×#10625324

yes

—

171

C₂×C₄

#13094030

yes

—

172

C₈

—

173

C₈

—

174

C₈

—

175

C₈

—

176

C₂²

#15688974

yes

characteristic

177

C₂²

#97×D₄₇

yes

characteristic

178

C₂²

C₄×#10625324

yes

characteristic

179

C₂²

#15689075

yes

—

180

C₂²

#15689077

yes

—

181

C₄

—

182

C₄

—

188

—

183

C₄

—

188

—

184

C₄

#15689145

yes

—

185

C₄

#15689177

yes

—

186

C₄

#15689197

yes

—

187

C₄

#15689252

yes

—

188

C₂

#20241676

yes

—

189

C₂

#20241682

yes

characteristic

190

C₂

#20241692

yes

characteristic

191

C₂

#20243240

yes

characteristic

192

C₂

#20243261

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂²×D₄₇
commutator, socle, prime radical	C₂³×C₄₇	C₂×C₄²
Fitting	C₄₇×#1273	C₂
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#1273	D₄₇
hypocenter	C₄₇	#8263
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄² ◁ #1273

subgroup

G/H

center, Frattini

C₂×C₄²

C₂²×D₄₇

commutator, socle, prime radical

C₂³×C₄₇

C₂×C₄²

Fitting

C₄₇×#1273

C₂

cosocle

C₂×C₄²×C₄₇

C₂³

D₄₇

C₄₇

G ▷ C₂³×C₄₇ ▷ C₄₇

C₁ ◁ C₂×C₄² ◁ #1273

N	H ≅ G/N	type
112. C₄₇	#8263	semidirect

H ≅ G/N

type

112. C₄₇

#8263

semidirect

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

subgroup	index
C₄×Dic₄₇⋊C₈	2
C₄×#13093934	2
C₂×#15688845	2
C₄₇⋊#1273	2
C₄₇×#1273	2
#8263	47

subgroup

index

C₄×Dic₄₇⋊C₈

C₄×#13093934

C₂×#15688845

C₄₇⋊#1273

C₄₇×#1273

#8263

Subgroups — #26107418

Conjugacy classes of subgroups

Complements

Maximal subgroups