Group #26116421 — subgroups

Subgroups — #26116421

Order 12032.

438 conjugacy classes of subgroups; only the 163 normal subgroups are shown.

normal subgroups only

This group has 438 conjugacy classes of subgroups — too many to draw. Only its 163 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	#20251751	C₂	1	1	yes	maximal, characteristic
2	#20243540	C₂	1	1	yes	maximal, characteristic
3	#20241272	C₂	1	4	yes	maximal
4	C₄₇×#1767	C₂	1	1	yes	maximal, characteristic
5	C₂×#13094016	—	2	1	no	—
6	C₂×C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
7	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
8	#15689021	—	2	2	no	—
9	#15689021	C₄	1	4	yes	—
10	#15688993	C₂²	1	2	yes	—
11	#15688989	—	2	1	no	—
12	C₂×C₄⋊C₈×C₄₇	C₂²	1	2	yes	—
13	C₂×#97×C₄₇	—	2	1	no	—
14	C₄³×C₄₇	C₂²	1	1	yes	characteristic
15	C₄₇×#336	—	2	1	no	—
16	C₂²×C₄₇⋊C₈	—	2	2	no	—
17	#13094016	—	4	4	no	—
18	C₂×C₄⋊Dic₄₇	—	2	2	no	—
19	C₂×C₄⋊Dic₄₇	—	2	2	no	—
20	C₂×C₄⋊Dic₄₇	C₂×C₄	1	4	yes	—
21	C₂×Dic₄₇⋊C₄	—	2	4	no	—
22	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
23	C₄⋊(C₄₇⋊C₈)	—	4	4	no	—
24	C₄₇⋊#95	—	2	4	no	—
25	C₂²×C₈×C₄₇	—	2	2	no	—
26	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
27	C₂×C₄²×C₄₇	D₄	1	2	yes	—
28	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
29	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
30	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
31	C₄⋊C₈×C₄₇	—	2	2	no	—
32	C₄⋊C₈×C₄₇	Q₈	1	4	yes	—
33	#97×C₄₇	—	4	2	no	—
34	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
35	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
36	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
37	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
38	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
39	C₂²×Dic₄₇	—	2	4	no	—
40	C₂×C₈×C₄₇	—	2	4	no	—
41	C₂×C₈×C₄₇	—	4	2	no	—
42	C₂×C₈×C₄₇	—	4	2	no	—
43	C₄²×C₄₇	—	2	2	no	—
44	C₄²×C₄₇	—	2	2	no	—
45	C₄²×C₄₇	—	2	2	no	—
46	C₄²×C₄₇	—	4	2	no	—
47	C₄²×C₄₇	SD₁₆	1	2	yes	—
48	C₄²×C₄₇	Q₁₆	1	2	yes	—
49	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
50	C₄²×C₄₇	#41	1	2	yes	—
51	C₄⋊Dic₄₇	—	4	2	no	—
52	C₄⋊Dic₄₇	—	4	4	no	—
53	C₄⋊Dic₄₇	—	4	2	no	—
54	C₄⋊Dic₄₇	—	4	4	no	—
55	Dic₄₇⋊C₄	—	4	8	no	—
56	C₂×C₄₇⋊C₈	—	4	2	no	—
57	C₂×C₄₇⋊C₈	—	4	4	no	—
58	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
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	—	2	2	no	—
64	C₂×C₄×C₄₇	—	2	2	no	—
65	C₂×C₄×C₄₇	#103	1	4	yes	—
66	C₂×C₄×C₄₇	C₄×Q₈	1	2	yes	—
67	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
68	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
69	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
70	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
71	C₂×C₄×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
72	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
73	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
74	C₂×C₄×C₄₇	#137	1	2	yes	—
75	C₂×Dic₄₇	—	4	4	no	—
76	C₂×Dic₄₇	—	4	4	no	—
77	C₂×Dic₄₇	—	4	4	no	—
78	C₈×C₄₇	—	4	4	no	—
79	C₄₇⋊C₈	—	8	2	no	—
80	#9902	—	47	1	no	maximal, Sylow
81	C₂²×C₄₇	#391	1	1	yes	characteristic
82	C₂²×C₄₇	#413	1	2	yes	—
83	C₂²×C₄₇	C₂×#103	1	1	yes	characteristic
84	C₂²×C₄₇	#417	1	2	yes	—
85	C₂²×C₄₇	#418	1	1	yes	characteristic
86	C₄×C₄₇	—	2	4	no	—
87	C₄×C₄₇	—	2	2	no	—
88	C₄×C₄₇	—	4	2	no	—
89	C₄×C₄₇	#476	1	2	yes	—
90	C₄×C₄₇	#477	1	2	yes	—
91	C₄×C₄₇	#490	1	1	yes	characteristic
92	C₄×C₄₇	#491	1	1	yes	characteristic
93	C₄×C₄₇	#501	1	2	yes	—
94	Dic₄₇	—	8	4	no	—
95	#2859	—	47	1	no	—
96	#1842	—	47	1	no	—
97	#1767	D₄₇	1	1	yes	characteristic
98	#1296	—	47	4	no	—
99	C₂×C₄₇	#1550	1	2	yes	—
100	C₂×C₄₇	#1557	1	2	yes	—
101	C₂×C₄₇	#1987	1	1	yes	characteristic
102	C₂×C₄₇	#1992	1	1	yes	characteristic
103	C₂×C₄₇	#1998	1	1	yes	characteristic
104	C₂×#125	—	94	1	no	—
105	C₂×C₄⋊C₈	—	47	2	no	—
106	C₂×C₄⋊C₈	—	94	2	no	—
107	C₂×C₄⋊C₈	C₂×D₄₇	1	2	yes	—
108	C₂×#97	—	2	1	no	—
109	#389	—	47	4	no	—
110	#389	—	94	2	no	—
111	#384	—	47	2	no	—
112	#382	—	94	1	no	—
113	C₄³	C₂×D₄₇	1	1	yes	characteristic
114	#336	—	2	1	no	—
115	C₄₇	#9902	1	1	yes	characteristic, complemented, Sylow
116	C₂²×C₈	—	2	2	no	—
117	C₂²×C₈	—	94	2	no	—
118	#125	—	188	4	no	—
119	C₂×C₄⋊C₄	—	47	4	no	—
120	C₂×C₄⋊C₄	—	94	4	no	—
121	C₂×C₄⋊C₄	—	94	2	no	—
122	C₂×C₄⋊C₄	—	94	2	no	—
123	C₂×C₄²	C₄×D₄₇	1	2	yes	—
124	C₂×C₄²	D₁₈₈	1	2	yes	—
125	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
126	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
127	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
128	C₄⋊C₈	—	2	2	no	—
129	C₄⋊C₈	—	94	4	no	—
130	C₄⋊C₈	—	188	4	no	—
131	C₄⋊C₈	Dic₉₄	1	4	yes	—
132	#97	—	4	2	no	—
133	#95	—	94	4	no	—
134	C₂²×C₄	—	94	4	no	—
135	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
136	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
137	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
138	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
139	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
140	C₂×C₈	—	2	4	no	—
141	C₂×C₈	—	4	2	no	—
142	C₂×C₈	—	4	2	no	—
143	C₂×C₈	—	188	2	no	—
144	C₂×C₈	—	188	4	no	—
145	C₄⋊C₄	—	188	4	no	—
146	C₄⋊C₄	—	188	8	no	—
147	C₄⋊C₄	—	188	4	no	—
148	C₄⋊C₄	—	188	2	no	—
149	C₄⋊C₄	—	188	2	no	—
150	C₄²	—	2	2	no	—
151	C₄²	—	2	2	no	—
152	C₄²	—	2	2	no	—
153	C₄²	—	4	2	no	—
154	C₄²	#10625335	1	2	yes	—
155	C₄²	#10625337	1	2	yes	—
156	C₄²	C₂×Dic₉₄	1	2	yes	—
157	C₄²	#10625350	1	2	yes	—
158	C₂³	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	188	4	no	—
166	C₂×C₄	—	188	4	no	—
167	C₂×C₄	—	188	4	no	—
168	C₂×C₄	#13093947	1	4	yes	—
169	C₂×C₄	C₄×Dic₉₄	1	2	yes	—
170	C₂×C₄	#13094003	1	2	yes	—
171	C₂×C₄	#13094012	1	1	yes	characteristic
172	C₂×C₄	#13094024	1	1	yes	characteristic
173	C₂×C₄	C₂×#10625335	1	1	yes	characteristic
174	C₂×C₄	C₂×#10625337	1	1	yes	characteristic
175	C₂×C₄	#13094049	1	2	yes	—
176	C₂×C₄	#13094052	1	2	yes	—
177	C₂×C₄	#13094057	1	2	yes	—
178	C₈	—	4	4	no	—
179	C₈	—	376	2	no	—
180	C₂²	#15689029	1	1	yes	characteristic
181	C₂²	#15689105	1	2	yes	—
182	C₂²	C₂×#13093947	1	1	yes	characteristic
183	C₂²	#15689125	1	2	yes	—
184	C₂²	#15689127	1	1	yes	characteristic
185	C₄	—	2	2	no	—
186	C₄	—	2	4	no	—
187	C₄	—	4	2	no	—
188	C₄	—	376	4	no	—
189	C₄	#15689413	1	2	yes	—
190	C₄	#15689418	1	2	yes	—
191	C₄	#15689488	1	1	yes	characteristic
192	C₄	#15689492	1	1	yes	characteristic
193	C₄	#15689532	1	2	yes	—
194	C₂	#20242079	1	2	yes	—
195	C₂	#20242111	1	2	yes	—
196	C₂	#20244336	1	1	yes	characteristic
197	C₂	#20244357	1	1	yes	characteristic
198	C₂	#20244381	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20251751

C₂

yes

maximal, characteristic

#20243540

C₂

yes

maximal, characteristic

#20241272

C₂

yes

maximal

C₄₇×#1767

C₂

yes

maximal, characteristic

C₂×#13094016

—

C₂×C₄⋊(C₄₇⋊C₈)

—

C₂×C₄⋊(C₄₇⋊C₈)

C₂²

yes

—

#15689021

—

#15689021

C₄

yes

—

#15688993

C₂²

yes

—

#15688989

—

C₂×C₄⋊C₈×C₄₇

C₂²

yes

—

C₂×#97×C₄₇

—

C₄³×C₄₇

C₂²

yes

characteristic

C₄₇×#336

—

C₂²×C₄₇⋊C₈

—

#13094016

—

C₂×C₄⋊Dic₄₇

—

C₂×C₄⋊Dic₄₇

—

C₂×C₄⋊Dic₄₇

C₂×C₄

yes

—

C₂×Dic₄₇⋊C₄

—

C₄⋊(C₄₇⋊C₈)

—

C₄⋊(C₄₇⋊C₈)

—

C₄₇⋊#95

—

C₂²×C₈×C₄₇

—

C₂×C₄²×C₄₇

C₂×C₄

yes

—

C₂×C₄²×C₄₇

D₄

yes

—

C₂×C₄²×C₄₇

D₄

yes

characteristic

C₂×C₄²×C₄₇

D₄

yes

characteristic

C₂×C₄²×C₄₇

C₂³

yes

characteristic

C₄⋊C₈×C₄₇

—

C₄⋊C₈×C₄₇

Q₈

yes

—

#97×C₄₇

—

C₂²×C₄×C₄₇

C₂²⋊C₄

yes

—

C₂²×C₄×C₄₇

C₂²⋊C₄

yes

—

C₂²×C₄×C₄₇

C₂²×C₄

yes

characteristic

C₂²×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂²×Dic₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

SD₁₆

yes

—

C₄²×C₄₇

Q₁₆

yes

—

C₄²×C₄₇

C₂×Q₈

yes

—

C₄²×C₄₇

#41

yes

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

Dic₄₇⋊C₄

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

#103

yes

—

C₂×C₄×C₄₇

C₄×Q₈

yes

—

C₂×C₄×C₄₇

C₂²⋊Q₈

yes

—

C₂×C₄×C₄₇

#124

yes

characteristic

C₂×C₄×C₄₇

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₄₇

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₄₇

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₄₇

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₄₇

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₄₇

#137

yes

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₈×C₄₇

—

C₄₇⋊C₈

—

—

C₂²×C₄₇

yes

C₂²×C₄₇

yes

—

C₂²×C₄₇

C₂×#103

yes

characteristic

C₂²×C₄₇

#417

yes

—

C₂²×C₄₇

#418

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#476

yes

—

C₄×C₄₇

#477

yes

—

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

—

Dic₄₇

—

#2859

—

#1842

—

#1767

D₄₇

yes

characteristic

#1296

—

C₂×C₄₇

#1550

yes

—

100

C₂×C₄₇

#1557

yes

—

101

C₂×C₄₇

#1987

yes

characteristic

102

C₂×C₄₇

#1992

yes

characteristic

103

C₂×C₄₇

#1998

yes

characteristic

104

C₂×#125

—

105

C₂×C₄⋊C₈

—

106

C₂×C₄⋊C₈

—

107

C₂×C₄⋊C₈

C₂×D₄₇

yes

—

108

C₂×#97

—

109

#389

—

110

#389

—

111

#384

—

112

#382

—

113

C₄³

C₂×D₄₇

yes

characteristic

114

#336

—

115

C₄₇

#9902

yes

characteristic, complemented, Sylow

116

C₂²×C₈

—

117

C₂²×C₈

—

118

#125

—

188

—

119

C₂×C₄⋊C₄

—

120

C₂×C₄⋊C₄

—

121

C₂×C₄⋊C₄

—

122

C₂×C₄⋊C₄

—

123

C₂×C₄²

C₄×D₄₇

yes

—

124

C₂×C₄²

D₁₈₈

yes

—

125

C₂×C₄²

C₂²⋊D₄₇

yes

characteristic

126

C₂×C₄²

C₂²⋊D₄₇

yes

characteristic

127

C₂×C₄²

C₂²×D₄₇

yes

characteristic

128

C₄⋊C₈

—

129

C₄⋊C₈

—

130

C₄⋊C₈

—

188

—

131

C₄⋊C₈

Dic₉₄

yes

—

132

#97

—

133

#95

—

134

C₂²×C₄

—

135

C₂²×C₄

(C₂×D₄₇)⋊C₄

yes

—

136

C₂²×C₄

(C₂×D₄₇)⋊C₄

yes

—

137

C₂²×C₄

C₂×C₄×D₄₇

yes

characteristic

138

C₂²×C₄

C₂×D₁₈₈

yes

characteristic

139

C₂²×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

140

C₂×C₈

—

141

C₂×C₈

—

142

C₂×C₈

—

143

C₂×C₈

—

188

—

144

C₂×C₈

—

188

—

145

C₄⋊C₄

—

188

—

146

C₄⋊C₄

—

188

—

147

C₄⋊C₄

—

188

—

148

C₄⋊C₄

—

188

—

149

C₄⋊C₄

—

188

—

150

C₄²

—

151

C₄²

—

152

C₄²

—

153

C₄²

—

154

C₄²

#10625335

yes

—

155

C₄²

#10625337

yes

—

156

C₄²

C₂×Dic₉₄

yes

—

157

C₄²

#10625350

yes

—

158

C₂³

C₂×(C₂×D₄₇)⋊C₄

yes

characteristic

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

188

—

166

C₂×C₄

—

188

—

167

C₂×C₄

—

188

—

168

C₂×C₄

#13093947

yes

—

169

C₂×C₄

C₄×Dic₉₄

yes

—

170

C₂×C₄

#13094003

yes

—

171

C₂×C₄

#13094012

yes

characteristic

172

C₂×C₄

#13094024

yes

characteristic

173

C₂×C₄

C₂×#10625335

yes

characteristic

174

C₂×C₄

C₂×#10625337

yes

characteristic

175

C₂×C₄

#13094049

yes

—

176

C₂×C₄

#13094052

yes

—

177

C₂×C₄

#13094057

yes

—

178

C₈

—

179

C₈

—

376

—

180

C₂²

#15689029

yes

characteristic

181

C₂²

#15689105

yes

—

182

C₂²

C₂×#13093947

yes

characteristic

183

C₂²

#15689125

yes

—

184

C₂²

#15689127

yes

characteristic

185

C₄

—

186

C₄

—

187

C₄

—

188

C₄

—

376

—

189

C₄

#15689413

yes

—

190

C₄

#15689418

yes

—

191

C₄

#15689488

yes

characteristic

192

C₄

#15689492

yes

characteristic

193

C₄

#15689532

yes

—

194

C₂

#20242079

yes

—

195

C₂

#20242111

yes

—

196

C₂

#20244336

yes

characteristic

197

C₂

#20244357

yes

characteristic

198

C₂

#20244381

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×(C₂×D₄₇)⋊C₄
commutator	C₂²×C₄×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1767	C₂
socle, prime radical	C₂³×C₄₇	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#1767	D₄₇
hypocenter	C₄₇	#9902
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #1767

subgroup

G/H

center

C₂³

C₂×(C₂×D₄₇)⋊C₄

commutator

C₂²×C₄×C₄₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₄₇

Fitting

C₄₇×#1767

C₂

socle, prime radical

C₂³×C₄₇

C₂×C₂²⋊C₄

cosocle

C₂×C₄²×C₄₇

C₂³

D₄₇

C₄₇

G ▷ C₂²×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂³ ◁ C₄³ ◁ #1767

N	H ≅ G/N	type
115. C₄₇	#9902	semidirect

H ≅ G/N

type

115. C₄₇

#9902

semidirect

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

subgroup	index
#20251751	2
#20243540	2
#20241272	2
C₄₇×#1767	2
#9902	47

subgroup

index

C₄₇×#1767

Subgroups — #26116421

Conjugacy classes of subgroups

Complements

Maximal subgroups