Group #26116422 — subgroups

Subgroups — #26116422

Order 12032.

438 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 438 conjugacy classes of subgroups — too many to draw. Only its 167 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	#20243540	C₂	1	1	yes	maximal, characteristic
2	#20243241	C₂	1	1	yes	maximal, characteristic
3	#20241271	C₂	1	4	yes	maximal
4	C₄₇×#2859	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
6	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
7	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
8	C₂×#13093935	—	2	1	no	—
9	#15688878	—	2	1	no	—
10	C₂×#125×C₄₇	—	2	1	no	—
11	C₄₇×#389	—	2	2	no	—
12	C₄₇×#389	C₄	1	4	yes	—
13	C₄₇×#384	C₂²	1	2	yes	—
14	C₄₇×#382	—	2	1	no	—
15	C₄³×C₄₇	C₂²	1	1	yes	characteristic
16	C₂²×C₄₇⋊C₈	—	2	2	no	—
17	C₂²×C₄₇⋊C₈	—	2	2	no	—
18	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
19	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
20	C₄⋊(C₄₇⋊C₈)	—	4	4	no	—
21	C₄⋊(C₄₇⋊C₈)	Q₈	1	4	yes	—
22	#13093935	—	4	2	no	—
23	#125×C₄₇	—	4	4	no	—
24	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
25	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
26	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
27	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
28	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
29	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
30	C₂×C₄²×C₄₇	D₄	1	2	yes	—
31	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
32	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
33	#95×C₄₇	—	2	4	no	—
34	C₂²×C₄×C₄₇	—	2	4	no	—
35	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
36	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
37	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
38	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
39	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
40	C₄⋊C₄×C₄₇	—	4	8	no	—
41	C₄⋊C₄×C₄₇	—	4	4	no	—
42	C₄⋊C₄×C₄₇	—	4	2	no	—
43	C₄⋊C₄×C₄₇	—	4	4	no	—
44	C₄⋊C₄×C₄₇	—	4	2	no	—
45	C₄²×C₄₇	—	2	2	no	—
46	C₄²×C₄₇	—	2	2	no	—
47	C₄²×C₄₇	—	2	2	no	—
48	C₄²×C₄₇	—	4	2	no	—
49	C₄²×C₄₇	SD₁₆	1	2	yes	—
50	C₄²×C₄₇	Q₁₆	1	2	yes	—
51	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
52	C₄²×C₄₇	#41	1	2	yes	—
53	C₂×C₄₇⋊C₈	—	2	4	no	—
54	C₂×C₄₇⋊C₈	—	4	2	no	—
55	C₂×C₄₇⋊C₈	—	4	2	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	—	4	4	no	—
66	C₂×C₄×C₄₇	—	4	4	no	—
67	C₂×C₄×C₄₇	—	4	4	no	—
68	C₂×C₄×C₄₇	#103	1	4	yes	—
69	C₂×C₄×C₄₇	C₄×Q₈	1	2	yes	—
70	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
71	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
72	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
73	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
74	C₂×C₄×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
75	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
76	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
77	C₂×C₄×C₄₇	#137	1	2	yes	—
78	C₄₇⋊C₈	—	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	—	8	4	no	—
90	C₄×C₄₇	#476	1	2	yes	—
91	C₄×C₄₇	#477	1	2	yes	—
92	C₄×C₄₇	#490	1	1	yes	characteristic
93	C₄×C₄₇	#491	1	1	yes	characteristic
94	C₄×C₄₇	#501	1	2	yes	—
95	#2859	D₄₇	1	1	yes	characteristic
96	#1842	—	47	1	no	—
97	#1767	—	47	1	no	—
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	—	2	1	no	—
105	C₂×C₄⋊C₈	—	47	2	no	—
106	C₂×C₄⋊C₈	—	47	2	no	—
107	C₂×C₄⋊C₈	—	94	2	no	—
108	C₂×#97	—	94	1	no	—
109	#389	—	2	2	no	—
110	#389	Dic₄₇	1	4	yes	—
111	#384	C₂×D₄₇	1	2	yes	—
112	#382	—	2	1	no	—
113	C₄³	C₂×D₄₇	1	1	yes	characteristic
114	#336	—	94	1	no	—
115	C₄₇	#9902	1	1	yes	characteristic, complemented, Sylow
116	C₂²×C₈	—	94	2	no	—
117	C₂²×C₈	—	94	2	no	—
118	#125	—	4	4	no	—
119	C₂×C₄⋊C₄	—	2	4	no	—
120	C₂×C₄⋊C₄	—	2	2	no	—
121	C₂×C₄⋊C₄	—	2	2	no	—
122	C₂×C₄⋊C₄	C₂×Dic₄₇	1	4	yes	—
123	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
124	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
125	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
126	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
127	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
128	C₄⋊C₈	—	47	4	no	—
129	C₄⋊C₈	—	94	2	no	—
130	C₄⋊C₈	—	94	4	no	—
131	C₄⋊C₈	—	188	4	no	—
132	#97	—	188	2	no	—
133	#95	—	2	4	no	—
134	C₂²×C₄	—	2	4	no	—
135	C₂²×C₄	C₂²⋊Dic₄₇	1	2	yes	—
136	C₂²×C₄	C₂²⋊Dic₄₇	1	2	yes	—
137	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
138	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
139	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
140	C₂×C₈	—	94	4	no	—
141	C₂×C₈	—	188	2	no	—
142	C₂×C₈	—	188	4	no	—
143	C₂×C₈	—	188	2	no	—
144	C₂×C₈	—	188	2	no	—
145	C₄⋊C₄	—	4	4	no	—
146	C₄⋊C₄	—	4	8	no	—
147	C₄⋊C₄	—	4	2	no	—
148	C₄⋊C₄	—	4	4	no	—
149	C₄⋊C₄	—	4	2	no	—
150	C₄²	—	2	2	no	—
151	C₄²	—	2	2	no	—
152	C₄²	—	2	2	no	—
153	C₄²	—	4	2	no	—
154	C₄²	#10625336	1	2	yes	—
155	C₄²	#10625337	1	2	yes	—
156	C₄²	Q₈×D₄₇	1	2	yes	—
157	C₄²	#10625354	1	2	yes	—
158	C₂³	C₂×C₂²⋊Dic₄₇	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	—	4	4	no	—
166	C₂×C₄	—	4	4	no	—
167	C₂×C₄	—	4	4	no	—
168	C₂×C₄	#13093948	1	4	yes	—
169	C₂×C₄	Q₈×Dic₄₇	1	2	yes	—
170	C₂×C₄	#13094000	1	2	yes	—
171	C₂×C₄	#13094013	1	1	yes	characteristic
172	C₂×C₄	#13094022	1	1	yes	characteristic
173	C₂×C₄	C₂×#10625336	1	1	yes	characteristic
174	C₂×C₄	C₂×#10625337	1	1	yes	characteristic
175	C₂×C₄	#13094049	1	2	yes	—
176	C₂×C₄	#13094050	1	2	yes	—
177	C₂×C₄	#13094054	1	2	yes	—
178	C₈	—	376	2	no	—
179	C₈	—	188	4	no	—
180	C₂²	#15689030	1	1	yes	characteristic
181	C₂²	#15689107	1	2	yes	—
182	C₂²	C₂×#13093948	1	1	yes	characteristic
183	C₂²	#15689124	1	2	yes	—
184	C₂²	#15689128	1	1	yes	characteristic
185	C₄	—	2	4	no	—
186	C₄	—	2	2	no	—
187	C₄	—	4	2	no	—
188	C₄	—	8	4	no	—
189	C₄	#15689410	1	2	yes	—
190	C₄	#15689415	1	2	yes	—
191	C₄	#15689489	1	1	yes	characteristic
192	C₄	#15689493	1	1	yes	characteristic
193	C₄	#15689533	1	2	yes	—
194	C₂	#20242078	1	2	yes	—
195	C₂	#20242110	1	2	yes	—
196	C₂	#20244337	1	1	yes	characteristic
197	C₂	#20244358	1	1	yes	characteristic
198	C₂	#20244382	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20243540

C₂

yes

maximal, characteristic

#20243241

C₂

yes

maximal, characteristic

#20241271

C₂

yes

maximal

C₄₇×#2859

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₄₇⋊C₈)

—

C₂×C₄⋊(C₄₇⋊C₈)

C₂²

yes

—

C₂×C₄⋊(C₄₇⋊C₈)

C₂²

yes

—

C₂×#13093935

—

#15688878

—

C₂×#125×C₄₇

—

C₄₇×#389

—

C₄₇×#389

C₄

yes

—

C₄₇×#384

C₂²

yes

—

C₄₇×#382

—

C₄³×C₄₇

C₂²

yes

characteristic

C₂²×C₄₇⋊C₈

—

C₂²×C₄₇⋊C₈

—

C₄⋊(C₄₇⋊C₈)

—

C₄⋊(C₄₇⋊C₈)

—

C₄⋊(C₄₇⋊C₈)

—

C₄⋊(C₄₇⋊C₈)

Q₈

yes

—

#13093935

—

#125×C₄₇

—

C₂×C₄⋊C₄×C₄₇

—

C₂×C₄⋊C₄×C₄₇

—

C₂×C₄⋊C₄×C₄₇

—

C₂×C₄⋊C₄×C₄₇

C₂×C₄

yes

—

C₂×C₄²×C₄₇

C₂×C₄

yes

—

C₂×C₄²×C₄₇

D₄

yes

characteristic

C₂×C₄²×C₄₇

D₄

yes

—

C₂×C₄²×C₄₇

D₄

yes

characteristic

C₂×C₄²×C₄₇

C₂³

yes

characteristic

#95×C₄₇

—

C₂²×C₄×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

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×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

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

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

—

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

—

C₄₇⋊C₈

—

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

—

C₄×C₄₇

#476

yes

—

C₄×C₄₇

#477

yes

—

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

—

#2859

D₄₇

yes

—

—

—

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

—

108

C₂×#97

—

109

#389

—

110

#389

Dic₄₇

yes

—

111

#384

C₂×D₄₇

yes

—

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

—

119

C₂×C₄⋊C₄

—

120

C₂×C₄⋊C₄

—

121

C₂×C₄⋊C₄

—

122

C₂×C₄⋊C₄

C₂×Dic₄₇

yes

—

123

C₂×C₄²

C₂×Dic₄₇

yes

—

124

C₂×C₄²

C₂²⋊D₄₇

yes

characteristic

125

C₂×C₄²

C₂²⋊D₄₇

yes

—

126

C₂×C₄²

C₂²⋊D₄₇

yes

characteristic

127

C₂×C₄²

C₂²×D₄₇

yes

characteristic

128

C₄⋊C₈

—

129

C₄⋊C₈

—

130

C₄⋊C₈

—

131

C₄⋊C₈

—

188

—

132

#97

—

188

—

133

#95

—

134

C₂²×C₄

—

135

C₂²×C₄

C₂²⋊Dic₄₇

yes

—

136

C₂²×C₄

C₂²⋊Dic₄₇

yes

—

137

C₂²×C₄

C₂²×Dic₄₇

yes

characteristic

138

C₂²×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

139

C₂²×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

140

C₂×C₈

—

141

C₂×C₈

—

188

—

142

C₂×C₈

—

188

—

143

C₂×C₈

—

188

—

144

C₂×C₈

—

188

—

145

C₄⋊C₄

—

146

C₄⋊C₄

—

147

C₄⋊C₄

—

148

C₄⋊C₄

—

149

C₄⋊C₄

—

150

C₄²

—

151

C₄²

—

152

C₄²

—

153

C₄²

—

154

C₄²

#10625336

yes

—

155

C₄²

#10625337

yes

—

156

C₄²

Q₈×D₄₇

yes

—

157

C₄²

#10625354

yes

—

158

C₂³

C₂×C₂²⋊Dic₄₇

yes

characteristic

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

#13093948

yes

—

169

C₂×C₄

Q₈×Dic₄₇

yes

—

170

C₂×C₄

#13094000

yes

—

171

C₂×C₄

#13094013

yes

characteristic

172

C₂×C₄

#13094022

yes

characteristic

173

C₂×C₄

C₂×#10625336

yes

characteristic

174

C₂×C₄

C₂×#10625337

yes

characteristic

175

C₂×C₄

#13094049

yes

—

176

C₂×C₄

#13094050

yes

—

177

C₂×C₄

#13094054

yes

—

178

C₈

—

376

—

179

C₈

—

188

—

180

C₂²

#15689030

yes

characteristic

181

C₂²

#15689107

yes

—

182

C₂²

C₂×#13093948

yes

characteristic

183

C₂²

#15689124

yes

—

184

C₂²

#15689128

yes

characteristic

185

C₄

—

186

C₄

—

187

C₄

—

188

C₄

—

189

C₄

#15689410

yes

—

190

C₄

#15689415

yes

—

191

C₄

#15689489

yes

characteristic

192

C₄

#15689493

yes

characteristic

193

C₄

#15689533

yes

—

194

C₂

#20242078

yes

—

195

C₂

#20242110

yes

—

196

C₂

#20244337

yes

characteristic

197

C₂

#20244358

yes

characteristic

198

C₂

#20244382

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₄₇
commutator	C₂²×C₄×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#2859	C₂
socle, prime radical	C₂³×C₄₇	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#2859	D₄₇
hypocenter	C₄₇	#9902
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #2859

subgroup

G/H

center

C₂³

C₂×C₂²⋊Dic₄₇

commutator

C₂²×C₄×C₄₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₄₇

Fitting

C₄₇×#2859

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 ◁ #2859

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
#20243540	2
#20243241	2
#20241271	2
C₄₇×#2859	2
#9902	47

subgroup

index

C₄₇×#2859

Subgroups — #26116422

Conjugacy classes of subgroups

Complements

Maximal subgroups