Group #26216026 — subgroups

Subgroups — #26216026

Order 12032.

1546 conjugacy classes of subgroups; only the 395 normal subgroups are shown.

normal subgroups only

This group has 1546 conjugacy classes of subgroups — too many to draw. Only its 395 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	#20245970	C₂	1	2	yes	maximal
2	#20243462	C₂	1	8	yes	maximal, complemented
3	#20243121	C₂	1	4	yes	maximal, complemented
4	C₂×C₄₇×#518	C₂	1	1	yes	maximal, characteristic
5	C₂×#13093983	C₂²	1	4	yes	—
6	#15689026	—	2	8	no	—
7	#15688979	C₂²	1	8	yes	—
8	#15688890	C₂²	1	16	yes	—
9	C₂³×D₄×C₄₇	C₂²	1	1	yes	characteristic
10	C₄₇×#518	—	2	8	no	—
11	C₂×C₄×D₄×C₄₇	C₄	1	8	yes	complemented
12	C₂×#117×C₄₇	C₂²	1	2	yes	—
13	C₂²×C₂²⋊C₄×C₄₇	C₂²	1	4	yes	—
14	#13093983	—	2	16	no	—
15	C₂×C₄⋊Dic₄₇	—	2	4	no	—
16	C₂×C₂²⋊Dic₄₇	—	2	16	no	—
17	C₂×C₂²⋊Dic₄₇	C₂³	1	8	yes	—
18	C₂×C₄×Dic₄₇	—	2	4	no	—
19	C₂⁵×C₄₇	C₂³	1	2	yes	—
20	C₂²×D₄×C₄₇	—	2	8	no	—
21	C₂²×D₄×C₄₇	C₂×C₄	1	4	yes	—
22	C₂²×D₄×C₄₇	D₄	1	8	yes	—
23	C₂³×C₄×C₄₇	C₂×C₄	1	8	yes	—
24	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
25	C₄×D₄×C₄₇	—	2	32	no	—
26	#117×C₄₇	—	2	8	no	—
27	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
28	C₂×C₂²⋊C₄×C₄₇	—	2	16	no	—
29	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	8	yes	—
30	C₂×C₂²⋊C₄×C₄₇	C₂³	1	4	yes	—
31	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
32	C₂⁴×C₄₇	—	2	8	no	—
33	C₂⁴×C₄₇	—	8	4	no	—
34	C₂⁴×C₄₇	C₂²×C₄	1	4	yes	—
35	C₂⁴×C₄₇	C₂×D₄	1	8	yes	—
36	C₂⁴×C₄₇	C₂⁴	1	1	yes	characteristic
37	C₂×D₄×C₄₇	—	2	16	no	—
38	C₂×D₄×C₄₇	—	2	16	no	—
39	C₂×D₄×C₄₇	—	2	16	no	—
40	C₂×D₄×C₄₇	C₂²⋊C₄	1	16	yes	—
41	C₂²×C₄×C₄₇	—	2	16	no	—
42	C₂²×C₄×C₄₇	—	8	8	no	—
43	C₂²×C₄×C₄₇	C₂²×C₄	1	8	yes	—
44	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
45	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
46	C₂²×C₄×C₄₇	#41	1	8	yes	—
47	C₂²×Dic₄₇	—	2	8	no	—
48	C₄⋊C₄×C₄₇	—	2	8	no	—
49	C₂²⋊C₄×C₄₇	—	2	16	no	—
50	C₂²⋊C₄×C₄₇	—	2	8	no	—
51	C₄²×C₄₇	—	2	8	no	—
52	C₂²⋊Dic₄₇	—	2	16	no	—
53	C₂²⋊Dic₄₇	—	8	16	no	—
54	C₄⋊Dic₄₇	—	4	8	no	—
55	C₄×Dic₄₇	—	4	8	no	—
56	C₂³×C₄₇	—	2	8	no	—
57	C₂³×C₄₇	—	2	16	no	—
58	C₂³×C₄₇	—	4	8	no	—
59	C₂³×C₄₇	—	4	2	no	—
60	C₂³×C₄₇	—	4	16	no	—
61	C₂³×C₄₇	—	8	16	no	—
62	C₂³×C₄₇	C₂×C₂²⋊C₄	1	8	yes	—
63	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
64	C₂³×C₄₇	C₂²×D₄	1	2	yes	—
65	C₂³×C₄₇	C₂×#41	1	4	yes	—
66	D₄×C₄₇	—	2	32	no	—
67	C₂×C₄×C₄₇	—	2	8	no	—
68	C₂×C₄×C₄₇	—	4	8	no	—
69	C₂×C₄×C₄₇	—	8	16	no	—
70	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	4	yes	—
71	C₂×C₄×C₄₇	#117	1	8	yes	—
72	C₂×C₄×C₄₇	#123	1	8	yes	—
73	C₂×C₄×C₄₇	#123	1	8	yes	—
74	C₂×Dic₄₇	—	4	16	no	—
75	C₂×Dic₄₇	—	4	8	no	—
76	#20690	—	47	1	no	maximal, Sylow
77	C₂²×C₄₇	—	2	4	no	—
78	C₂²×C₄₇	—	2	2	no	—
79	C₂²×C₄₇	—	2	16	no	—
80	C₂²×C₄₇	—	2	8	no	—
81	C₂²×C₄₇	—	4	4	no	—
82	C₂²×C₄₇	—	4	16	no	—
83	C₂²×C₄₇	—	8	16	no	—
84	C₂²×C₄₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
85	C₂²×C₄₇	C₂×#117	1	2	yes	—
86	C₂²×C₄₇	C₂×#123	1	4	yes	—
87	C₄×C₄₇	—	4	8	no	—
88	C₄×C₄₇	#381	1	8	yes	—
89	Dic₄₇	—	8	8	no	—
90	C₂×#518	D₄₇	1	1	yes	characteristic
91	#2309	—	47	2	no	—
92	#1819	—	47	8	no	—
93	#1740	—	47	4	no	—
94	C₂×C₄₇	—	2	4	no	—
95	C₂×C₄₇	—	2	2	no	—
96	C₂×C₄₇	—	4	8	no	—
97	C₂×C₄₇	C₂×#381	1	1	yes	characteristic
98	C₂×C₄₇	#2883	1	2	yes	—
99	C₂³×D₄	C₂×D₄₇	1	1	yes	characteristic
100	#518	—	2	8	no	—
101	C₂×C₄×D₄	Dic₄₇	1	8	yes	complemented
102	C₂×#117	—	47	4	no	—
103	C₂×#117	C₂×D₄₇	1	2	yes	—
104	C₂²×C₂²⋊C₄	C₂×D₄₇	1	4	yes	—
105	#390	—	94	8	no	—
106	#379	—	47	8	no	—
107	#342	—	47	16	no	—
108	C₄₇	#20690	1	1	yes	characteristic, complemented, Sylow
109	C₂⁵	C₂²×D₄₇	1	2	yes	—
110	C₂²×D₄	—	2	8	no	—
111	C₂²×D₄	C₂×Dic₄₇	1	4	yes	—
112	C₂²×D₄	C₂²⋊D₄₇	1	8	yes	—
113	C₂³×C₄	C₂×Dic₄₇	1	8	yes	—
114	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
115	C₄×D₄	—	2	32	no	—
116	#117	—	2	8	no	—
117	#117	—	94	16	no	—
118	C₂×C₄⋊C₄	—	94	4	no	—
119	C₂×C₄⋊C₄	C₂×Dic₄₇	1	4	yes	—
120	C₂×C₂²⋊C₄	—	2	16	no	—
121	C₂×C₂²⋊C₄	—	47	8	no	—
122	C₂×C₂²⋊C₄	—	94	16	no	—
123	C₂×C₂²⋊C₄	C₂×Dic₄₇	1	8	yes	—
124	C₂×C₂²⋊C₄	C₂²×D₄₇	1	4	yes	—
125	C₂×C₄²	—	94	4	no	—
126	C₂×C₄²	C₂×Dic₄₇	1	4	yes	—
127	C₂⁴	—	2	8	no	—
128	C₂⁴	—	8	4	no	—
129	C₂⁴	C₂²×Dic₄₇	1	4	yes	—
130	C₂⁴	C₂×C₂²⋊D₄₇	1	8	yes	—
131	C₂⁴	C₂³×D₄₇	1	1	yes	characteristic
132	C₂×D₄	—	2	16	no	—
133	C₂×D₄	—	2	16	no	—
134	C₂×D₄	—	2	16	no	—
135	C₂×D₄	C₂²⋊Dic₄₇	1	16	yes	—
136	C₂²×C₄	—	2	16	no	—
137	C₂²×C₄	—	8	8	no	—
138	C₂²×C₄	—	94	8	no	—
139	C₂²×C₄	#10625350	1	8	yes	—
140	C₂²×C₄	C₂²×Dic₄₇	1	8	yes	—
141	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
142	C₂²×C₄	C₂×C₂²⋊D₄₇	1	4	yes	—
143	C₄⋊C₄	—	2	8	no	—
144	C₄⋊C₄	—	188	8	no	—
145	C₂²⋊C₄	—	2	16	no	—
146	C₂²⋊C₄	—	2	8	no	—
147	C₂²⋊C₄	—	94	16	no	—
148	C₂²⋊C₄	—	376	16	no	—
149	C₄²	—	2	8	no	—
150	C₄²	—	188	8	no	—
151	C₂³	—	2	8	no	—
152	C₂³	—	2	16	no	—
153	C₂³	—	4	8	no	—
154	C₂³	—	4	2	no	—
155	C₂³	—	4	16	no	—
156	C₂³	—	8	16	no	—
157	C₂³	C₂×C₂²⋊Dic₄₇	1	8	yes	—
158	C₂³	C₂³×Dic₄₇	1	1	yes	characteristic
159	C₂³	C₂²×C₂²⋊D₄₇	1	2	yes	—
160	C₂³	C₂×#10625350	1	4	yes	—
161	D₄	—	2	32	no	—
162	C₂×C₄	—	2	8	no	—
163	C₂×C₄	—	4	8	no	—
164	C₂×C₄	—	8	16	no	—
165	C₂×C₄	—	188	16	no	—
166	C₂×C₄	—	188	8	no	—
167	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	4	yes	—
168	C₂×C₄	#13093983	1	8	yes	—
169	C₂×C₄	#13094007	1	8	yes	—
170	C₂×C₄	#13094007	1	8	yes	—
171	C₂²	—	2	2	no	—
172	C₂²	—	2	4	no	—
173	C₂²	—	2	8	no	—
174	C₂²	—	2	16	no	—
175	C₂²	—	4	4	no	—
176	C₂²	—	4	16	no	—
177	C₂²	—	8	16	no	—
178	C₂²	C₂²×C₂²⋊Dic₄₇	1	1	yes	characteristic
179	C₂²	C₂×#13093983	1	2	yes	—
180	C₂²	C₂×#13094007	1	4	yes	—
181	C₄	—	4	8	no	—
182	C₄	—	376	8	no	—
183	C₄	#15688986	1	8	yes	—
184	C₂	—	2	4	no	—
185	C₂	—	2	2	no	—
186	C₂	—	4	8	no	—
187	C₂	C₂×#15688986	1	1	yes	characteristic
188	C₂	#20251849	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20245970

C₂

yes

maximal

#20243462

C₂

yes

maximal, complemented

#20243121

C₂

yes

maximal, complemented

C₂×C₄₇×#518

C₂

yes

maximal, characteristic

C₂×#13093983

C₂²

yes

—

#15689026

—

#15688979

C₂²

yes

—

#15688890

C₂²

yes

—

C₂³×D₄×C₄₇

C₂²

yes

characteristic

C₄₇×#518

—

C₂×C₄×D₄×C₄₇

C₄

yes

complemented

C₂×#117×C₄₇

C₂²

yes

—

C₂²×C₂²⋊C₄×C₄₇

C₂²

yes

—

#13093983

—

C₂×C₄⋊Dic₄₇

—

C₂×C₂²⋊Dic₄₇

—

C₂×C₂²⋊Dic₄₇

C₂³

yes

—

C₂×C₄×Dic₄₇

—

C₂⁵×C₄₇

C₂³

yes

—

C₂²×D₄×C₄₇

—

C₂²×D₄×C₄₇

C₂×C₄

yes

—

C₂²×D₄×C₄₇

D₄

yes

—

C₂³×C₄×C₄₇

C₂×C₄

yes

—

C₂³×C₄×C₄₇

C₂³

yes

characteristic

C₄×D₄×C₄₇

—

#117×C₄₇

—

C₂×C₄⋊C₄×C₄₇

C₂×C₄

yes

—

C₂×C₂²⋊C₄×C₄₇

—

C₂×C₂²⋊C₄×C₄₇

C₂×C₄

yes

—

C₂×C₂²⋊C₄×C₄₇

C₂³

yes

—

C₂×C₄²×C₄₇

C₂×C₄

yes

—

C₂⁴×C₄₇

—

C₂⁴×C₄₇

—

C₂⁴×C₄₇

C₂²×C₄

yes

—

C₂⁴×C₄₇

C₂×D₄

yes

—

C₂⁴×C₄₇

C₂⁴

yes

characteristic

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

C₂²⋊C₄

yes

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂²×C₄

yes

—

C₂²×C₄×C₄₇

C₂²×C₄

yes

—

C₂²×C₄×C₄₇

C₂×D₄

yes

—

C₂²×C₄×C₄₇

#41

yes

—

C₂²×Dic₄₇

—

C₄⋊C₄×C₄₇

—

C₂²⋊C₄×C₄₇

—

C₂²⋊C₄×C₄₇

—

C₄²×C₄₇

—

C₂²⋊Dic₄₇

—

C₂²⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄×Dic₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

C₂×C₂²⋊C₄

yes

—

C₂³×C₄₇

C₂³×C₄

yes

characteristic

C₂³×C₄₇

C₂²×D₄

yes

—

C₂³×C₄₇

C₂×#41

yes

—

D₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×C₂²⋊C₄

yes

—

C₂×C₄×C₄₇

#117

yes

—

C₂×C₄×C₄₇

#123

yes

—

C₂×C₄×C₄₇

#123

yes

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

#20690

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

C₂²×C₂²⋊C₄

yes

characteristic

C₂²×C₄₇

C₂×#117

yes

—

C₂²×C₄₇

C₂×#123

yes

—

C₄×C₄₇

—

C₄×C₄₇

#381

yes

—

Dic₄₇

—

C₂×#518

D₄₇

yes

—

—

—

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

C₂×#381

yes

characteristic

C₂×C₄₇

#2883

yes

—

C₂³×D₄

C₂×D₄₇

yes

characteristic

100

#518

—

101

C₂×C₄×D₄

Dic₄₇

yes

complemented

102

C₂×#117

—

103

C₂×#117

C₂×D₄₇

yes

—

104

C₂²×C₂²⋊C₄

C₂×D₄₇

yes

—

105

#390

—

106

#379

—

107

#342

—

108

C₄₇

#20690

yes

characteristic, complemented, Sylow

109

C₂⁵

C₂²×D₄₇

yes

—

110

C₂²×D₄

—

111

C₂²×D₄

C₂×Dic₄₇

yes

—

112

C₂²×D₄

C₂²⋊D₄₇

yes

—

113

C₂³×C₄

C₂×Dic₄₇

yes

—

114

C₂³×C₄

C₂²×D₄₇

yes

characteristic

115

C₄×D₄

—

116

#117

—

117

#117

—

118

C₂×C₄⋊C₄

—

119

C₂×C₄⋊C₄

C₂×Dic₄₇

yes

—

120

C₂×C₂²⋊C₄

—

121

C₂×C₂²⋊C₄

—

122

C₂×C₂²⋊C₄

—

123

C₂×C₂²⋊C₄

C₂×Dic₄₇

yes

—

124

C₂×C₂²⋊C₄

C₂²×D₄₇

yes

—

125

C₂×C₄²

—

126

C₂×C₄²

C₂×Dic₄₇

yes

—

127

C₂⁴

—

128

C₂⁴

—

129

C₂⁴

C₂²×Dic₄₇

yes

—

130

C₂⁴

C₂×C₂²⋊D₄₇

yes

—

131

C₂⁴

C₂³×D₄₇

yes

characteristic

132

C₂×D₄

—

133

C₂×D₄

—

134

C₂×D₄

—

135

C₂×D₄

C₂²⋊Dic₄₇

yes

—

136

C₂²×C₄

—

137

C₂²×C₄

—

138

C₂²×C₄

—

139

C₂²×C₄

#10625350

yes

—

140

C₂²×C₄

C₂²×Dic₄₇

yes

—

141

C₂²×C₄

C₂²×Dic₄₇

yes

—

142

C₂²×C₄

C₂×C₂²⋊D₄₇

yes

—

143

C₄⋊C₄

—

144

C₄⋊C₄

—

188

—

145

C₂²⋊C₄

—

146

C₂²⋊C₄

—

147

C₂²⋊C₄

—

148

C₂²⋊C₄

—

376

—

149

C₄²

—

150

C₄²

—

188

—

151

C₂³

—

152

C₂³

—

153

C₂³

—

154

C₂³

—

155

C₂³

—

156

C₂³

—

157

C₂³

C₂×C₂²⋊Dic₄₇

yes

—

158

C₂³

C₂³×Dic₄₇

yes

characteristic

159

C₂³

C₂²×C₂²⋊D₄₇

yes

—

160

C₂³

C₂×#10625350

yes

—

161

D₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

188

—

166

C₂×C₄

—

188

—

167

C₂×C₄

C₂×C₂²⋊Dic₄₇

yes

—

168

C₂×C₄

#13093983

yes

—

169

C₂×C₄

#13094007

yes

—

170

C₂×C₄

#13094007

yes

—

171

C₂²

—

172

C₂²

—

173

C₂²

—

174

C₂²

—

175

C₂²

—

176

C₂²

—

177

C₂²

—

178

C₂²

C₂²×C₂²⋊Dic₄₇

yes

characteristic

179

C₂²

C₂×#13093983

yes

—

180

C₂²

C₂×#13094007

yes

—

181

C₄

—

182

C₄

—

376

—

183

C₄

#15688986

yes

—

184

C₂

—

185

C₂

—

186

C₂

—

187

C₂

C₂×#15688986

yes

characteristic

188

C₂

#20251849

yes

—

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊Dic₄₇
commutator	C₂³×C₄₇	C₂³×C₄
Frattini	C₂⁴	C₂³×D₄₇
Fitting	C₂×C₄₇×#518	C₂
socle	C₂²×C₄₇	C₂²×C₂²⋊C₄
prime radical	C₂³×D₄×C₄₇	C₂²
cosocle	C₂⁴×C₄₇	C₂⁴
hypercenter	C₂×#518	D₄₇
hypocenter	C₄₇	#20690
LCS	G ▷ C₂³×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂³×D₄ ◁ C₂×#518

subgroup

G/H

center

C₂²

C₂²×C₂²⋊Dic₄₇

commutator

C₂³×C₄₇

C₂³×C₄

Frattini

C₂⁴

C₂³×D₄₇

Fitting

C₂×C₄₇×#518

C₂

socle

C₂²×C₄₇

C₂²×C₂²⋊C₄

prime radical

C₂³×D₄×C₄₇

C₂²

cosocle

C₂⁴×C₄₇

C₂⁴

D₄₇

C₄₇

G ▷ C₂³×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₂³×D₄ ◁ C₂×#518

N	H ≅ G/N	type
2. #20243462	C₂	semidirect
3. #20243121	C₂	semidirect
11. C₂×C₄×D₄×C₄₇	C₄	semidirect
101. C₂×C₄×D₄	Dic₄₇	semidirect
108. C₄₇	#20690	semidirect

H ≅ G/N

type

2. #20243462

C₂

semidirect

3. #20243121

C₂

semidirect

11. C₂×C₄×D₄×C₄₇

C₄

semidirect

101. C₂×C₄×D₄

Dic₄₇

semidirect

108. C₄₇

#20690

semidirect

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

subgroup	index
#20245970	2
#20243462	2
#20243121	2
C₂×C₄₇×#518	2
#20690	47

subgroup

index

C₂×C₄₇×#518

Subgroups — #26216026

Conjugacy classes of subgroups

Complements

Maximal subgroups