Group #26132457 — subgroups

Subgroups — #26132457

Order 12032.

394 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 139 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	C₂³×C₄₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#15688952	C₂	1	2	yes	maximal, complemented
3	#20243426	C₂	1	1	yes	maximal, characteristic
4	#20241481	C₂	1	2	yes	maximal, complemented
5	C₄₇×#1810	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₄₇⋊C₁₆	—	2	4	no	—
7	C₂²×C₄₇⋊C₁₆	C₄	1	4	yes	—
8	C₂²×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
9	C₂²×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
10	C₂×#13093960	C₂²	1	2	yes	—
11	#15689087	—	2	1	no	—
12	#15688952	—	2	4	no	—
13	#15688888	—	2	1	no	—
14	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
15	C₂×#108×C₄₇	C₂²	1	2	yes	—
16	C₄₇×#407	—	2	1	no	—
17	C₄₇×#341	—	2	1	no	—
18	C₂×#10625329	—	2	2	no	—
19	C₂×C₄₇⋊C₁₆	—	2	2	no	—
20	C₂×C₄₇⋊C₁₆	—	2	8	no	—
21	C₂×C₄₇⋊C₁₆	C₂×C₄	1	2	yes	—
22	C₂×C₄₇⋊C₁₆	C₂×C₄	1	2	yes	—
23	C₂×C₄₇⋊C₁₆	D₄	1	4	yes	—
24	#13093960	—	2	2	no	—
25	#13093960	—	2	2	no	—
26	C₂²⋊(C₄₇⋊C₈)	—	4	2	no	—
27	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
28	C₂×M₁₆×C₄₇	—	2	2	no	—
29	C₂²×C₈×C₄₇	—	2	2	no	—
30	C₂²×C₈×C₄₇	—	2	2	no	—
31	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
32	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₄₇	Q₈	1	2	yes	—
34	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
35	#108×C₄₇	—	2	2	no	—
36	#108×C₄₇	—	2	2	no	—
37	C₂²⋊C₈×C₄₇	—	4	2	no	—
38	C₂⁴×C₄₇	Q₁₆	1	1	yes	characteristic
39	C₂²×C₄×C₄₇	—	2	2	no	—
40	C₂²×C₄×C₄₇	—	2	2	no	—
41	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
42	C₂²×C₄×C₄₇	D₈	1	2	yes	—
43	C₂²×C₄×C₄₇	Q₁₆	1	1	yes	characteristic
44	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
45	M₁₆×C₄₇	—	4	2	no	—
46	M₁₆×C₄₇	—	4	2	no	—
47	C₂×C₈×C₄₇	—	2	2	no	—
48	C₂×C₈×C₄₇	—	2	2	no	—
49	C₂×C₈×C₄₇	—	2	2	no	—
50	C₂×C₈×C₄₇	—	2	4	no	—
51	C₂×C₈×C₄₇	—	4	2	no	—
52	C₂×C₈×C₄₇	C₄⋊C₄	1	2	yes	—
53	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
55	C₂×C₈×C₄₇	C₂×Q₈	1	1	yes	characteristic
56	C₂×C₈×C₄₇	#41	1	2	yes	—
57	#10625329	—	4	2	no	—
58	#10625329	—	4	2	no	—
59	C₂×C₄₇⋊C₈	—	4	2	no	—
60	C₄₇⋊C₁₆	—	2	2	no	—
61	C₄₇⋊C₁₆	C₂²⋊C₄	1	4	yes	—
62	C₂³×C₄₇	—	2	4	no	—
63	C₂³×C₄₇	—	2	2	no	—
64	C₂³×C₄₇	#107	1	2	yes	—
65	C₂³×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
66	C₂×C₄×C₄₇	—	2	2	no	—
67	C₂×C₄×C₄₇	—	2	2	no	—
68	C₂×C₄×C₄₇	—	2	2	no	—
69	C₂×C₄×C₄₇	—	2	4	no	—
70	C₂×C₄×C₄₇	#107	1	2	yes	—
71	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
72	C₂×C₄×C₄₇	#121	1	2	yes	—
73	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
74	C₂×C₄×C₄₇	#135	1	2	yes	—
75	C₈×C₄₇	—	2	2	no	—
76	C₈×C₄₇	—	8	2	no	—
77	C₈×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
78	C₈×C₄₇	#117	1	1	yes	characteristic
79	C₈×C₄₇	C₂²⋊Q₈	1	2	yes	—
80	C₄₇⋊C₈	—	8	2	no	—
81	#12837	—	47	1	no	maximal, Sylow
82	C₂²×C₄₇	—	2	8	no	—
83	C₂²×C₄₇	—	2	2	no	—
84	C₂²×C₄₇	—	2	2	no	—
85	C₂²×C₄₇	—	2	2	no	—
86	C₂²×C₄₇	#368	1	4	yes	complemented
87	C₂²×C₄₇	C₂×#107	1	1	yes	characteristic
88	C₂²×C₄₇	#467	1	2	yes	—
89	C₄×C₄₇	—	2	2	no	—
90	C₄×C₄₇	#380	1	1	yes	characteristic
91	C₄×C₄₇	#427	1	1	yes	characteristic
92	C₄×C₄₇	#466	1	2	yes	—
93	C₂³×C₁₆	—	47	1	no	—
94	C₂×#368	—	47	2	no	—
95	#1810	—	47	1	no	—
96	#1810	D₄₇	1	1	yes	characteristic
97	#1381	—	47	2	no	—
98	C₂×C₄₇	—	2	4	no	—
99	C₂×C₄₇	—	2	2	no	—
100	C₂×C₄₇	#1808	1	1	yes	characteristic
101	C₂×C₄₇	C₂×#368	1	2	yes	—
102	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
103	C₂²×C₁₆	—	47	1	no	—
104	C₂²×C₁₆	—	47	4	no	—
105	C₂²×C₁₆	—	47	1	no	—
106	C₂²×C₁₆	—	94	4	no	—
107	C₂×#108	—	47	2	no	—
108	C₂×#108	C₂×D₄₇	1	2	yes	—
109	#407	—	2	1	no	—
110	#407	—	94	1	no	—
111	#368	—	94	4	no	—
112	#341	—	2	1	no	—
113	#341	—	94	1	no	—
114	C₄₇	#12837	1	1	yes	characteristic, complemented, Sylow
115	C₂³×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
116	C₂×M₁₆	—	2	2	no	—
117	C₂×M₁₆	—	94	2	no	—
118	C₂²×C₈	—	2	2	no	—
119	C₂²×C₈	—	2	2	no	—
120	C₂²×C₈	Dic₉₄	1	2	yes	—
121	C₂²×C₈	C₄×D₄₇	1	2	yes	—
122	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
123	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
124	C₂×C₁₆	—	47	2	no	—
125	C₂×C₁₆	—	47	4	no	—
126	C₂×C₁₆	—	47	2	no	—
127	C₂×C₁₆	—	94	8	no	—
128	C₂×C₁₆	—	94	2	no	—
129	#108	—	2	2	no	—
130	#108	—	2	2	no	—
131	#108	—	94	2	no	—
132	#108	—	94	2	no	—
133	C₂²⋊C₈	—	4	2	no	—
134	C₂²⋊C₈	—	188	2	no	—
135	C₂⁴	#10625337	1	1	yes	characteristic
136	C₂²×C₄	—	2	2	no	—
137	C₂²×C₄	—	2	2	no	—
138	C₂²×C₄	Dic₄₇⋊C₄	1	2	yes	—
139	C₂²×C₄	#10625334	1	2	yes	—
140	C₂²×C₄	#10625337	1	1	yes	characteristic
141	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
142	M₁₆	—	4	2	no	—
143	M₁₆	—	4	2	no	—
144	M₁₆	—	188	2	no	—
145	M₁₆	—	188	2	no	—
146	C₂×C₈	—	2	2	no	—
147	C₂×C₈	—	2	2	no	—
148	C₂×C₈	—	2	2	no	—
149	C₂×C₈	—	2	4	no	—
150	C₂×C₈	—	4	2	no	—
151	C₂×C₈	—	188	2	no	—
152	C₂×C₈	Dic₄₇⋊C₄	1	2	yes	—
153	C₂×C₈	C₂×Dic₉₄	1	1	yes	characteristic
154	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
155	C₂×C₈	D₄×D₄₇	1	2	yes	—
156	C₂×C₈	#10625352	1	2	yes	—
157	C₁₆	—	47	4	no	—
158	C₁₆	—	94	2	no	—
159	C₂³	—	2	2	no	—
160	C₂³	—	2	4	no	—
161	C₂³	#13093957	1	2	yes	—
162	C₂³	C₂×#10625337	1	1	yes	characteristic
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	2	4	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	#13093957	1	2	yes	—
168	C₂×C₄	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
169	C₂×C₄	#13093996	1	2	yes	—
170	C₂×C₄	C₂×#10625334	1	1	yes	characteristic
171	C₂×C₄	#13094044	1	2	yes	—
172	C₈	—	2	2	no	—
173	C₈	—	8	2	no	—
174	C₈	—	376	2	no	—
175	C₈	C₂²⋊C₄×D₄₇	1	1	yes	characteristic
176	C₈	#13093980	1	1	yes	characteristic
177	C₈	#13094001	1	2	yes	—
178	C₂²	—	2	8	no	—
179	C₂²	—	2	2	no	—
180	C₂²	—	2	2	no	—
181	C₂²	—	2	2	no	—
182	C₂²	#15688952	1	4	yes	complemented
183	C₂²	C₂×#13093957	1	1	yes	characteristic
184	C₂²	#15689362	1	2	yes	—
185	C₄	—	2	2	no	—
186	C₄	#15688981	1	1	yes	characteristic
187	C₄	#15689159	1	1	yes	characteristic
188	C₄	#15689357	1	2	yes	—
189	C₂	—	2	2	no	—
190	C₂	—	2	4	no	—
191	C₂	#20243415	1	1	yes	characteristic
192	C₂	C₂×#15688952	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂³×C₄₇⋊C₁₆

C₂

yes

maximal, characteristic

C₂×#15688952

C₂

yes

maximal, complemented

#20243426

C₂

yes

maximal, characteristic

#20241481

C₂

yes

maximal, complemented

C₄₇×#1810

C₂

yes

maximal, characteristic

C₂²×C₄₇⋊C₁₆

—

C₂²×C₄₇⋊C₁₆

C₄

yes

—

C₂²×C₄₇⋊C₁₆

C₂²

yes

characteristic

C₂²×C₄₇⋊C₁₆

C₂²

yes

characteristic

C₂×#13093960

C₂²

yes

—

#15689087

—

#15688952

—

#15688888

—

C₂³×C₈×C₄₇

C₂²

yes

characteristic

C₂×#108×C₄₇

C₂²

yes

—

C₄₇×#407

—

C₄₇×#341

—

C₂×#10625329

—

C₂×C₄₇⋊C₁₆

—

C₂×C₄₇⋊C₁₆

—

C₂×C₄₇⋊C₁₆

C₂×C₄

yes

—

C₂×C₄₇⋊C₁₆

C₂×C₄

yes

—

C₂×C₄₇⋊C₁₆

D₄

yes

—

#13093960

—

#13093960

—

C₂²⋊(C₄₇⋊C₈)

—

C₂³×C₄×C₄₇

D₄

yes

characteristic

C₂×M₁₆×C₄₇

—

C₂²×C₈×C₄₇

—

C₂²×C₈×C₄₇

—

C₂²×C₈×C₄₇

C₂×C₄

yes

—

C₂²×C₈×C₄₇

D₄

yes

characteristic

C₂²×C₈×C₄₇

Q₈

yes

—

C₂²×C₈×C₄₇

C₂³

yes

characteristic

#108×C₄₇

—

#108×C₄₇

—

C₂²⋊C₈×C₄₇

—

C₂⁴×C₄₇

Q₁₆

yes

characteristic

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₄⋊C₄

yes

—

C₂²×C₄×C₄₇

D₈

yes

—

C₂²×C₄×C₄₇

Q₁₆

yes

characteristic

C₂²×C₄×C₄₇

C₂×D₄

yes

characteristic

M₁₆×C₄₇

—

M₁₆×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

C₄⋊C₄

yes

—

C₂×C₈×C₄₇

C₂²×C₄

yes

characteristic

C₂×C₈×C₄₇

C₂×D₄

yes

—

C₂×C₈×C₄₇

C₂×Q₈

yes

characteristic

C₂×C₈×C₄₇

#41

yes

—

#10625329

—

#10625329

—

C₂×C₄₇⋊C₈

—

C₄₇⋊C₁₆

—

C₄₇⋊C₁₆

C₂²⋊C₄

yes

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

#107

yes

—

C₂³×C₄₇

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

#107

yes

—

C₂×C₄×C₄₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₄₇

#121

yes

—

C₂×C₄×C₄₇

C₂×D₈

yes

characteristic

C₂×C₄×C₄₇

#135

yes

—

C₈×C₄₇

—

C₈×C₄₇

—

C₈×C₄₇

C₂×C₂²⋊C₄

yes

characteristic

C₈×C₄₇

#117

yes

characteristic

C₈×C₄₇

C₂²⋊Q₈

yes

—

C₄₇⋊C₈

—

#12837

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

—

C₄×C₄₇

—

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

—

C₂³×C₁₆

—

C₂×#368

—

#1810

—

#1810

D₄₇

yes

characteristic

#1381

—

C₂×C₄₇

—

C₂×C₄₇

—

100

C₂×C₄₇

#1808

yes

characteristic

101

C₂×C₄₇

C₂×#368

yes

—

102

C₂³×C₈

C₂×D₄₇

yes

characteristic

103

C₂²×C₁₆

—

104

C₂²×C₁₆

—

105

C₂²×C₁₆

—

106

C₂²×C₁₆

—

107

C₂×#108

—

108

C₂×#108

C₂×D₄₇

yes

—

109

#407

—

110

#407

—

111

#368

—

112

#341

—

113

#341

—

114

C₄₇

#12837

yes

characteristic, complemented, Sylow

115

C₂³×C₄

C₂²⋊D₄₇

yes

characteristic

116

C₂×M₁₆

—

117

C₂×M₁₆

—

118

C₂²×C₈

—

119

C₂²×C₈

—

120

C₂²×C₈

Dic₉₄

yes

—

121

C₂²×C₈

C₄×D₄₇

yes

—

122

C₂²×C₈

C₂²⋊D₄₇

yes

characteristic

123

C₂²×C₈

C₂²×D₄₇

yes

characteristic

124

C₂×C₁₆

—

125

C₂×C₁₆

—

126

C₂×C₁₆

—

127

C₂×C₁₆

—

128

C₂×C₁₆

—

129

#108

—

130

#108

—

131

#108

—

132

#108

—

133

C₂²⋊C₈

—

134

C₂²⋊C₈

—

188

—

135

C₂⁴

#10625337

yes

characteristic

136

C₂²×C₄

—

137

C₂²×C₄

—

138

C₂²×C₄

Dic₄₇⋊C₄

yes

—

139

C₂²×C₄

#10625334

yes

—

140

C₂²×C₄

#10625337

yes

characteristic

141

C₂²×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

142

M₁₆

—

143

M₁₆

—

144

M₁₆

—

188

—

145

M₁₆

—

188

—

146

C₂×C₈

—

147

C₂×C₈

—

148

C₂×C₈

—

149

C₂×C₈

—

150

C₂×C₈

—

151

C₂×C₈

—

188

—

152

C₂×C₈

Dic₄₇⋊C₄

yes

—

153

C₂×C₈

C₂×Dic₉₄

yes

characteristic

154

C₂×C₈

C₂×C₄×D₄₇

yes

characteristic

155

C₂×C₈

D₄×D₄₇

yes

—

156

C₂×C₈

#10625352

yes

—

157

C₁₆

—

158

C₁₆

—

159

C₂³

—

160

C₂³

—

161

C₂³

#13093957

yes

—

162

C₂³

C₂×#10625337

yes

characteristic

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

#13093957

yes

—

168

C₂×C₄

C₂×Dic₄₇⋊C₄

yes

characteristic

169

C₂×C₄

#13093996

yes

—

170

C₂×C₄

C₂×#10625334

yes

characteristic

171

C₂×C₄

#13094044

yes

—

172

C₈

—

173

C₈

—

174

C₈

—

376

—

175

C₈

C₂²⋊C₄×D₄₇

yes

characteristic

176

C₈

#13093980

yes

characteristic

177

C₈

#13094001

yes

—

178

C₂²

—

179

C₂²

—

180

C₂²

—

181

C₂²

—

182

C₂²

#15688952

yes

complemented

183

C₂²

C₂×#13093957

yes

characteristic

184

C₂²

#15689362

yes

—

185

C₄

—

186

C₄

#15688981

yes

characteristic

187

C₄

#15689159

yes

characteristic

188

C₄

#15689357

yes

—

189

C₂

—

190

C₂

—

191

C₂

#20243415

yes

characteristic

192

C₂

C₂×#15688952

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×#10625334
commutator	C₂×C₈×C₄₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₄₇
Fitting	C₄₇×#1810	C₂
socle	C₂²×C₄₇	C₂×#107
prime radical	C₂⁴×C₄₇	Q₁₆
cosocle	C₂²×C₈×C₄₇	C₂³
hypercenter	#1810	D₄₇
hypocenter	C₄₇	#12837
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ #1810

subgroup

G/H

center

C₂×C₄

C₂×#10625334

commutator

C₂×C₈×C₄₇

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₄₇

Fitting

C₄₇×#1810

C₂

socle

C₂²×C₄₇

C₂×#107

prime radical

C₂⁴×C₄₇

Q₁₆

cosocle

C₂²×C₈×C₄₇

C₂³

D₄₇

C₄₇

G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ #1810

N	H ≅ G/N	type
2. C₂×#15688952	C₂	semidirect
4. #20241481	C₂	semidirect
86. C₂²×C₄₇	#368	semidirect
114. C₄₇	#12837	semidirect
182. C₂²	#15688952	semidirect

H ≅ G/N

type

2. C₂×#15688952

C₂

semidirect

4. #20241481

C₂

semidirect

86. C₂²×C₄₇

#368

semidirect

114. C₄₇

#12837

semidirect

182. C₂²

#15688952

semidirect

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

subgroup	index
C₂³×C₄₇⋊C₁₆	2
C₂×#15688952	2
#20243426	2
#20241481	2
C₄₇×#1810	2
#12837	47

subgroup

index

C₂³×C₄₇⋊C₁₆

C₄₇×#1810

Subgroups — #26132457

Conjugacy classes of subgroups

Complements

Maximal subgroups