Group #26132753 — subgroups

Subgroups — #26132753

Order 12032.

374 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 135 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	#20245148	C₂	1	2	yes	maximal, complemented
2	#20245002	C₂	1	1	yes	maximal, characteristic
3	#20243188	C₂	1	1	yes	maximal, characteristic
4	#20241473	C₂	1	2	yes	maximal, complemented
5	C₄₇×#3407	C₂	1	1	yes	maximal, characteristic
6	C₂×#13093964	C₂²	1	1	yes	characteristic
7	C₂²×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
8	#15689107	C₄	1	1	yes	characteristic
9	#15689097	C₄	1	1	yes	characteristic
10	#15689078	C₂²	1	2	yes	—
11	Dic₄₇⋊C₁₆	—	2	4	no	—
12	C₂²⋊(C₄₇⋊C₁₆)	—	2	2	no	—
13	#15688894	C₄	1	2	yes	—
14	C₂×#131×C₄₇	C₂²	1	1	yes	characteristic
15	C₄₇×#504	—	2	4	no	—
16	C₂×M₃₂×C₄₇	—	2	2	no	—
17	C₂×M₃₂×C₄₇	C₂²	1	2	yes	—
18	C₂×#10625329	C₂×C₄	1	2	yes	—
19	#13093983	C₂×C₄	1	2	yes	—
20	#13093964	—	2	2	no	—
21	C₂×C₄₇⋊C₁₆	—	2	1	no	—
22	C₂×C₄₇⋊C₁₆	—	2	2	no	—
23	C₂×C₄₇⋊C₁₆	—	2	1	no	—
24	#13093943	—	2	1	no	—
25	#13093941	—	2	1	no	—
26	#13093939	C₈	1	4	yes	—
27	#13093934	—	2	2	no	—
28	C₈×Dic₄₇	—	2	2	no	—
29	C₂×#41×C₄₇	C₂×C₄	1	1	yes	characteristic
30	#131×C₄₇	—	2	2	no	—
31	#131×C₄₇	—	2	2	no	—
32	C₂×M₁₆×C₄₇	C₂×C₄	1	1	yes	characteristic
33	C₂×M₁₆×C₄₇	Q₈	1	2	yes	—
34	C₂²×C₈×C₄₇	D₄	1	2	yes	—
35	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
36	M₃₂×C₄₇	—	2	4	no	—
37	M₃₂×C₄₇	—	2	2	no	—
38	M₃₂×C₄₇	—	2	2	no	—
39	C₂×C₁₆×C₄₇	—	2	4	no	—
40	C₂×C₁₆×C₄₇	D₄	1	4	yes	—
41	#41×C₄₇	—	2	2	no	—
42	#41×C₄₇	—	2	2	no	—
43	C₂×Q₈×C₄₇	M₁₆	1	1	yes	characteristic
44	C₂×D₄×C₄₇	C₂×C₈	1	2	yes	—
45	C₂×D₄×C₄₇	M₁₆	1	1	yes	characteristic
46	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
47	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
48	M₁₆×C₄₇	—	2	2	no	—
49	M₁₆×C₄₇	—	2	4	no	—
50	C₂×C₈×C₄₇	—	2	2	no	—
51	C₂×C₈×C₄₇	—	2	2	no	—
52	C₂×C₈×C₄₇	C₄⋊C₄	1	2	yes	—
53	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
54	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
55	C₂×C₈×C₄₇	C₂×Q₈	1	1	yes	characteristic
56	C₂×C₈×C₄₇	#41	1	2	yes	—
57	C₂²⋊Dic₄₇	C₂×C₈	1	4	yes	—
58	C₄⋊Dic₄₇	—	2	2	no	—
59	C₄×Dic₄₇	—	2	2	no	—
60	#10625329	—	2	2	no	—
61	#10625329	—	2	2	no	—
62	C₂×C₄₇⋊C₈	—	2	2	no	—
63	C₁₆×C₄₇	—	2	4	no	—
64	C₄₇⋊C₁₆	—	4	1	no	—
65	C₄₇⋊C₁₆	—	4	1	no	—
66	C₂³×C₄₇	C₄⋊C₈	1	2	yes	—
67	C₂³×C₄₇	C₂²×C₈	1	1	yes	characteristic
68	Q₈×C₄₇	—	2	2	no	—
69	D₄×C₄₇	—	2	4	no	—
70	D₄×C₄₇	—	2	2	no	—
71	C₂×C₄×C₄₇	—	2	2	no	—
72	C₂×C₄×C₄₇	—	2	2	no	—
73	C₂×C₄×C₄₇	C₄⋊C₈	1	2	yes	—
74	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
75	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
76	C₂×C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
77	C₂×C₄×C₄₇	#131	1	2	yes	—
78	C₂×Dic₄₇	—	2	4	no	—
79	C₈×C₄₇	—	2	2	no	—
80	C₈×C₄₇	#120	1	1	yes	characteristic
81	C₈×C₄₇	C₂²⋊Q₈	1	2	yes	—
82	C₈×C₄₇	#123	1	1	yes	characteristic
83	C₄₇⋊C₈	—	4	2	no	—
84	#12915	—	47	1	no	maximal, Sylow
85	C₂²×C₄₇	—	2	2	no	—
86	C₂²×C₄₇	—	4	1	no	—
87	C₂²×C₄₇	—	4	2	no	—
88	C₂²×C₄₇	C₂×C₄⋊C₈	1	1	yes	characteristic
89	C₂²×C₄₇	C₈×D₄	1	2	yes	—
90	C₄×C₄₇	—	2	2	no	—
91	C₄×C₄₇	#385	1	1	yes	characteristic
92	C₄×C₄₇	#424	1	1	yes	characteristic
93	C₄×C₄₇	#435	1	2	yes	—
94	Dic₄₇	—	4	4	no	—
95	#3407	D₄₇	1	1	yes	characteristic
96	#2151	—	47	2	no	—
97	#2113	—	47	1	no	—
98	#1757	—	47	1	no	—
99	#1378	—	47	2	no	—
100	C₂×C₄₇	—	2	2	no	—
101	C₂×C₄₇	—	2	1	no	—
102	C₂×C₄₇	—	4	2	no	—
103	C₂×C₄₇	#1850	1	1	yes	characteristic
104	C₂×#131	C₂×D₄₇	1	1	yes	characteristic
105	#504	—	2	4	no	—
106	C₂×M₃₂	—	2	2	no	—
107	C₂×M₃₂	—	47	1	no	—
108	C₂×M₃₂	C₂×D₄₇	1	2	yes	—
109	C₂²×C₁₆	—	47	1	no	—
110	#413	—	47	1	no	—
111	#410	—	47	1	no	—
112	#405	—	47	2	no	—
113	C₄⋊C₁₆	—	94	4	no	—
114	C₂²⋊C₁₆	—	94	2	no	—
115	#344	—	47	2	no	—
116	C₄₇	#12915	1	1	yes	characteristic, complemented, Sylow
117	C₂×#41	C₄×D₄₇	1	1	yes	characteristic
118	#131	—	2	2	no	—
119	#131	—	2	2	no	—
120	C₂×M₁₆	—	47	2	no	—
121	C₂×M₁₆	Dic₉₄	1	2	yes	—
122	C₂×M₁₆	C₄×D₄₇	1	1	yes	characteristic
123	C₂²×C₈	C₂²⋊D₄₇	1	2	yes	—
124	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
125	#117	—	47	2	no	—
126	M₃₂	—	2	4	no	—
127	M₃₂	—	2	2	no	—
128	M₃₂	—	2	2	no	—
129	M₃₂	—	94	2	no	—
130	C₂×C₁₆	—	2	4	no	—
131	C₂×C₁₆	—	94	1	no	—
132	C₂×C₁₆	—	94	1	no	—
133	C₂×C₁₆	—	94	2	no	—
134	C₂×C₁₆	C₂²⋊D₄₇	1	4	yes	—
135	#101	—	94	1	no	—
136	#100	—	94	1	no	—
137	#99	—	47	4	no	—
138	#97	—	94	2	no	—
139	C₄×C₈	—	94	2	no	—
140	#41	—	2	2	no	—
141	#41	—	2	2	no	—
142	C₂×Q₈	#10625324	1	1	yes	characteristic
143	C₂×D₄	C₈×D₄₇	1	2	yes	—
144	C₂×D₄	#10625324	1	1	yes	characteristic
145	C₂²×C₄	Dic₄₇⋊C₄	1	2	yes	—
146	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
147	M₁₆	—	2	2	no	—
148	M₁₆	—	2	4	no	—
149	M₁₆	—	94	2	no	—
150	M₁₆	—	94	2	no	—
151	C₂×C₈	—	2	2	no	—
152	C₂×C₈	—	2	2	no	—
153	C₂×C₈	—	94	2	no	—
154	C₂×C₈	Dic₄₇⋊C₄	1	2	yes	—
155	C₂×C₈	C₂×Dic₉₄	1	1	yes	characteristic
156	C₂×C₈	#10625350	1	2	yes	—
157	C₂×C₈	C₂×C₂²⋊D₄₇	1	2	yes	—
158	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
159	C₄⋊C₄	—	94	2	no	—
160	C₂²⋊C₄	—	47	4	no	—
161	C₄²	—	94	2	no	—
162	C₁₆	—	2	4	no	—
163	C₁₆	—	188	1	no	—
164	C₁₆	—	188	1	no	—
165	C₂³	Dic₄₇⋊C₈	1	2	yes	—
166	C₂³	C₂×C₈×D₄₇	1	1	yes	characteristic
167	Q₈	—	2	2	no	—
168	D₄	—	2	4	no	—
169	D₄	—	2	2	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	94	4	no	—
173	C₂×C₄	Dic₄₇⋊C₈	1	2	yes	—
174	C₂×C₄	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
175	C₂×C₄	C₄×C₂²⋊D₄₇	1	2	yes	—
176	C₂×C₄	C₂×#10625324	1	1	yes	characteristic
177	C₂×C₄	#13094030	1	2	yes	—
178	C₈	—	2	2	no	—
179	C₈	—	188	2	no	—
180	C₈	#13093993	1	1	yes	characteristic
181	C₈	#13094003	1	2	yes	—
182	C₈	#13094007	1	1	yes	characteristic
183	C₂²	—	2	2	no	—
184	C₂²	—	4	1	no	—
185	C₂²	—	4	2	no	—
186	C₂²	C₂×Dic₄₇⋊C₈	1	1	yes	characteristic
187	C₂²	C₈×C₂²⋊D₄₇	1	2	yes	—
188	C₄	—	2	2	no	—
189	C₄	—	188	4	no	—
190	C₄	#15688996	1	1	yes	characteristic
191	C₄	#15689149	1	1	yes	characteristic
192	C₄	#15689194	1	2	yes	—
193	C₂	—	2	2	no	—
194	C₂	—	2	1	no	—
195	C₂	—	4	2	no	—
196	C₂	#20243566	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20245148

C₂

yes

maximal, complemented

#20245002

C₂

yes

maximal, characteristic

#20243188

C₂

yes

maximal, characteristic

#20241473

C₂

yes

maximal, complemented

C₄₇×#3407

C₂

yes

maximal, characteristic

C₂×#13093964

C₂²

yes

characteristic

C₂²×C₄₇⋊C₁₆

C₂²

yes

characteristic

#15689107

C₄

yes

characteristic

#15689097

C₄

yes

characteristic

#15689078

C₂²

yes

—

Dic₄₇⋊C₁₆

—

C₂²⋊(C₄₇⋊C₁₆)

—

#15688894

C₄

yes

—

C₂×#131×C₄₇

C₂²

yes

characteristic

C₄₇×#504

—

C₂×M₃₂×C₄₇

—

C₂×M₃₂×C₄₇

C₂²

yes

—

C₂×#10625329

C₂×C₄

yes

—

#13093983

C₂×C₄

yes

—

#13093964

—

C₂×C₄₇⋊C₁₆

—

C₂×C₄₇⋊C₁₆

—

C₂×C₄₇⋊C₁₆

—

#13093943

—

#13093941

—

#13093939

C₈

yes

—

#13093934

—

C₈×Dic₄₇

—

C₂×#41×C₄₇

C₂×C₄

yes

characteristic

#131×C₄₇

—

#131×C₄₇

—

C₂×M₁₆×C₄₇

C₂×C₄

yes

characteristic

C₂×M₁₆×C₄₇

Q₈

yes

—

C₂²×C₈×C₄₇

D₄

yes

—

C₂²×C₈×C₄₇

C₂³

yes

characteristic

M₃₂×C₄₇

—

M₃₂×C₄₇

—

M₃₂×C₄₇

—

C₂×C₁₆×C₄₇

—

C₂×C₁₆×C₄₇

D₄

yes

—

#41×C₄₇

—

#41×C₄₇

—

C₂×Q₈×C₄₇

M₁₆

yes

characteristic

C₂×D₄×C₄₇

C₂×C₈

yes

—

C₂×D₄×C₄₇

M₁₆

yes

characteristic

C₂²×C₄×C₄₇

C₄⋊C₄

yes

—

C₂²×C₄×C₄₇

C₂²×C₄

yes

characteristic

M₁₆×C₄₇

—

M₁₆×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

C₄⋊C₄

yes

—

C₂×C₈×C₄₇

C₂×D₄

yes

—

C₂×C₈×C₄₇

C₂×D₄

yes

characteristic

C₂×C₈×C₄₇

C₂×Q₈

yes

characteristic

C₂×C₈×C₄₇

#41

yes

—

C₂²⋊Dic₄₇

C₂×C₈

yes

—

C₄⋊Dic₄₇

—

C₄×Dic₄₇

—

#10625329

—

#10625329

—

C₂×C₄₇⋊C₈

—

C₁₆×C₄₇

—

C₄₇⋊C₁₆

—

C₄₇⋊C₁₆

—

C₂³×C₄₇

C₄⋊C₈

yes

—

C₂³×C₄₇

C₂²×C₈

yes

characteristic

Q₈×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₄⋊C₈

yes

—

C₂×C₄×C₄₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₄₇

C₄×D₄

yes

—

C₂×C₄×C₄₇

C₂×M₁₆

yes

characteristic

C₂×C₄×C₄₇

#131

yes

—

C₂×Dic₄₇

—

C₈×C₄₇

—

C₈×C₄₇

#120

yes

characteristic

C₈×C₄₇

C₂²⋊Q₈

yes

—

C₈×C₄₇

#123

yes

characteristic

C₄₇⋊C₈

—

#12915

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

C₂×C₄⋊C₈

yes

characteristic

C₂²×C₄₇

C₈×D₄

yes

—

C₄×C₄₇

—

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

—

Dic₄₇

—

#3407

D₄₇

yes

—

—

—

—

100

C₂×C₄₇

—

101

C₂×C₄₇

—

102

C₂×C₄₇

—

103

C₂×C₄₇

#1850

yes

characteristic

104

C₂×#131

C₂×D₄₇

yes

characteristic

105

#504

—

106

C₂×M₃₂

—

107

C₂×M₃₂

—

108

C₂×M₃₂

C₂×D₄₇

yes

—

109

C₂²×C₁₆

—

110

#413

—

111

#410

—

112

#405

—

113

C₄⋊C₁₆

—

114

C₂²⋊C₁₆

—

115

#344

—

116

C₄₇

#12915

yes

characteristic, complemented, Sylow

117

C₂×#41

C₄×D₄₇

yes

characteristic

118

#131

—

119

#131

—

120

C₂×M₁₆

—

121

C₂×M₁₆

Dic₉₄

yes

—

122

C₂×M₁₆

C₄×D₄₇

yes

characteristic

123

C₂²×C₈

C₂²⋊D₄₇

yes

—

124

C₂²×C₈

C₂²×D₄₇

yes

characteristic

125

#117

—

126

M₃₂

—

127

M₃₂

—

128

M₃₂

—

129

M₃₂

—

130

C₂×C₁₆

—

131

C₂×C₁₆

—

132

C₂×C₁₆

—

133

C₂×C₁₆

—

134

C₂×C₁₆

C₂²⋊D₄₇

yes

—

135

#101

—

136

#100

—

137

#99

—

138

#97

—

139

C₄×C₈

—

140

#41

—

141

#41

—

142

C₂×Q₈

#10625324

yes

characteristic

143

C₂×D₄

C₈×D₄₇

yes

—

144

C₂×D₄

#10625324

yes

characteristic

145

C₂²×C₄

Dic₄₇⋊C₄

yes

—

146

C₂²×C₄

C₂×C₄×D₄₇

yes

characteristic

147

M₁₆

—

148

M₁₆

—

149

M₁₆

—

150

M₁₆

—

151

C₂×C₈

—

152

C₂×C₈

—

153

C₂×C₈

—

154

C₂×C₈

Dic₄₇⋊C₄

yes

—

155

C₂×C₈

C₂×Dic₉₄

yes

characteristic

156

C₂×C₈

#10625350

yes

—

157

C₂×C₈

C₂×C₂²⋊D₄₇

yes

—

158

C₂×C₈

C₂×C₂²⋊D₄₇

yes

characteristic

159

C₄⋊C₄

—

160

C₂²⋊C₄

—

161

C₄²

—

162

C₁₆

—

163

C₁₆

—

188

—

164

C₁₆

—

188

—

165

C₂³

Dic₄₇⋊C₈

yes

—

166

C₂³

C₂×C₈×D₄₇

yes

characteristic

167

Q₈

—

168

D₄

—

169

D₄

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

Dic₄₇⋊C₈

yes

—

174

C₂×C₄

C₂×Dic₄₇⋊C₄

yes

characteristic

175

C₂×C₄

C₄×C₂²⋊D₄₇

yes

—

176

C₂×C₄

C₂×#10625324

yes

characteristic

177

C₂×C₄

#13094030

yes

—

178

C₈

—

179

C₈

—

188

—

180

C₈

#13093993

yes

characteristic

181

C₈

#13094003

yes

—

182

C₈

#13094007

yes

characteristic

183

C₂²

—

184

C₂²

—

185

C₂²

—

186

C₂²

C₂×Dic₄₇⋊C₈

yes

characteristic

187

C₂²

C₈×C₂²⋊D₄₇

yes

—

188

C₄

—

189

C₄

—

188

—

190

C₄

#15688996

yes

characteristic

191

C₄

#15689149

yes

characteristic

192

C₄

#15689194

yes

—

193

C₂

—

194

C₂

—

195

C₂

—

196

C₂

#20243566

yes

characteristic

	subgroup	G/H
center	C₈	#13093993
commutator	C₂³×C₄₇	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₄₇
Fitting	C₄₇×#3407	C₂
socle	C₂×C₄₇	#1850
prime radical	C₂×#41×C₄₇	C₂×C₄
cosocle	C₂²×C₈×C₄₇	C₂³
hypercenter	#3407	D₄₇
hypocenter	C₄₇	#12915
LCS	G ▷ C₂³×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #3407

subgroup

G/H

center

C₈

#13093993

commutator

C₂³×C₄₇

C₂²×C₈

Frattini

C₂²×C₈

C₂²×D₄₇

Fitting

C₄₇×#3407

C₂

socle

C₂×C₄₇

#1850

prime radical

C₂×#41×C₄₇

C₂×C₄

cosocle

C₂²×C₈×C₄₇

C₂³

D₄₇

C₄₇

G ▷ C₂³×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #3407

N	H ≅ G/N	type
1. #20245148	C₂	semidirect
4. #20241473	C₂	semidirect
116. C₄₇	#12915	semidirect

H ≅ G/N

type

1. #20245148

C₂

semidirect

4. #20241473

C₂

semidirect

116. C₄₇

#12915

semidirect

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

subgroup	index
#20245148	2
#20245002	2
#20243188	2
#20241473	2
C₄₇×#3407	2
#12915	47

subgroup

index

C₄₇×#3407

Subgroups — #26132753

Conjugacy classes of subgroups

Complements

Maximal subgroups