Group #26132465 — subgroups

Subgroups — #26132465

Order 12032.

362 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 127 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	C₂×#15689540	C₂	1	1	yes	maximal, characteristic
2	#20245170	C₂	1	2	yes	maximal, complemented
3	#20243434	C₂	1	1	yes	maximal, characteristic
4	#20241489	C₂	1	2	yes	maximal, complemented
5	C₄₇×#1812	C₂	1	1	yes	maximal, characteristic
6	#15689540	—	2	2	no	—
7	#15689540	—	2	2	no	—
8	C₂×#13093964	C₄	1	2	yes	complemented
9	C₂×#13093964	C₂²	1	1	yes	characteristic
10	C₂²×C₄₇⋊C₁₆	C₄	1	2	yes	complemented
11	C₂²×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
12	#15689161	C₂²	1	2	yes	—
13	#15689097	—	2	1	no	—
14	#15688950	—	2	2	no	—
15	#15688948	—	2	2	no	—
16	#15688880	—	2	1	no	—
17	C₂×#131×C₄₇	C₂²	1	1	yes	characteristic
18	C₄₇×#427	C₂²	1	2	yes	—
19	C₄₇×#410	—	2	1	no	—
20	C₄₇×H(3,ℤ/4)	—	2	1	no	—
21	#13093983	—	2	2	no	—
22	#13093964	—	2	2	no	—
23	#13093964	—	2	4	no	—
24	C₂×C₄₇⋊C₁₆	—	2	2	no	—
25	C₂×C₄₇⋊C₁₆	—	2	2	no	—
26	C₂×C₄₇⋊C₁₆	C₂×C₄	1	2	yes	—
27	C₂×C₄₇⋊C₁₆	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄₇⋊C₁₆	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄₇⋊C₁₆	D₄	1	2	yes	—
30	C₂×C₄₇⋊C₁₆	D₄	1	2	yes	—
31	#13093958	—	2	2	no	—
32	#13093956	—	2	2	no	—
33	#13093939	—	4	2	no	—
34	C₂×#41×C₄₇	D₄	1	1	yes	characteristic
35	#131×C₄₇	—	2	2	no	—
36	#131×C₄₇	—	2	2	no	—
37	C₂×M₁₆×C₄₇	D₄	1	1	yes	characteristic
38	C₂×M₁₆×C₄₇	Q₈	1	2	yes	—
39	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
40	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
41	#117×C₄₇	—	2	2	no	—
42	#107×C₄₇	—	2	2	no	—
43	#106×C₄₇	—	2	2	no	—
44	#99×C₄₇	—	4	2	no	—
45	#41×C₄₇	—	2	2	no	—
46	#41×C₄₇	—	2	2	no	—
47	C₂×Q₈×C₄₇	Q₁₆	1	1	yes	characteristic
48	C₂×D₄×C₄₇	D₈	1	2	yes	—
49	C₂×D₄×C₄₇	Q₁₆	1	1	yes	characteristic
50	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
51	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
52	M₁₆×C₄₇	—	2	2	no	—
53	M₁₆×C₄₇	—	2	4	no	—
54	C₂×C₈×C₄₇	—	2	2	no	—
55	C₂×C₈×C₄₇	—	2	2	no	—
56	C₂×C₈×C₄₇	C₄⋊C₄	1	2	yes	—
57	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
59	C₂×C₈×C₄₇	C₂×Q₈	1	1	yes	characteristic
60	C₂×C₈×C₄₇	#41	1	2	yes	—
61	C₄⋊C₄×C₄₇	—	4	2	no	—
62	C₂²⋊C₄×C₄₇	—	4	2	no	—
63	C₄²×C₄₇	—	4	2	no	—
64	C₂²⋊Dic₄₇	—	4	2	no	—
65	C₄⋊Dic₄₇	—	4	2	no	—
66	C₄×Dic₄₇	—	4	2	no	—
67	C₄₇⋊C₁₆	—	2	2	no	—
68	C₄₇⋊C₁₆	C₂²⋊C₄	1	1	yes	characteristic
69	C₄₇⋊C₁₆	C₂²⋊C₄	1	1	yes	characteristic
70	C₄₇⋊C₁₆	C₂²⋊C₄	1	2	yes	—
71	C₂³×C₄₇	#107	1	2	yes	—
72	C₂³×C₄₇	C₂×D₈	1	1	yes	characteristic
73	Q₈×C₄₇	—	2	2	no	—
74	D₄×C₄₇	—	2	4	no	—
75	D₄×C₄₇	—	2	2	no	—
76	C₂×C₄×C₄₇	—	2	2	no	—
77	C₂×C₄×C₄₇	—	2	2	no	—
78	C₂×C₄×C₄₇	—	8	2	no	—
79	C₂×C₄×C₄₇	#107	1	2	yes	—
80	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
81	C₂×C₄×C₄₇	#121	1	2	yes	—
82	C₂×C₄×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
83	C₂×C₄×C₄₇	#135	1	2	yes	—
84	C₂×Dic₄₇	—	8	2	no	—
85	C₈×C₄₇	—	2	2	no	—
86	C₈×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
87	C₈×C₄₇	#117	1	1	yes	characteristic
88	C₈×C₄₇	C₂²⋊Q₈	1	2	yes	—
89	#12839	—	47	1	no	maximal, Sylow
90	C₂²×C₄₇	—	2	2	no	—
91	C₂²×C₄₇	—	4	1	no	—
92	C₂²×C₄₇	—	4	2	no	—
93	C₂²×C₄₇	C₂×#107	1	1	yes	characteristic
94	C₂²×C₄₇	#466	1	2	yes	—
95	C₄×C₄₇	—	2	2	no	—
96	C₄×C₄₇	—	16	2	no	—
97	C₄×C₄₇	#380	1	1	yes	characteristic
98	C₄×C₄₇	#427	1	1	yes	characteristic
99	C₄×C₄₇	#467	1	2	yes	—
100	Dic₄₇	—	16	2	no	—
101	C₂×#504	—	47	1	no	—
102	#2158	—	47	2	no	—
103	#1812	—	47	1	no	—
104	#1812	D₄₇	1	1	yes	characteristic
105	#1384	—	47	2	no	—
106	C₂×C₄₇	—	2	2	no	—
107	C₂×C₄₇	—	2	1	no	—
108	C₂×C₄₇	—	4	2	no	—
109	C₂×C₄₇	#1808	1	1	yes	characteristic
110	C₂×#131	C₂×D₄₇	1	1	yes	characteristic
111	#504	—	94	2	no	—
112	#504	—	94	2	no	—
113	C₂×M₃₂	—	47	2	no	—
114	C₂×M₃₂	—	47	1	no	—
115	C₂²×C₁₆	—	47	1	no	—
116	C₂²×C₁₆	—	47	2	no	—
117	#427	—	47	2	no	—
118	#427	C₂×D₄₇	1	2	yes	—
119	#410	—	2	1	no	—
120	#410	—	94	1	no	—
121	#367	—	94	2	no	—
122	#366	—	94	2	no	—
123	H(3,ℤ/4)	—	2	1	no	—
124	H(3,ℤ/4)	—	94	1	no	—
125	C₄₇	#12839	1	1	yes	characteristic, complemented, Sylow
126	C₂×#41	C₂²⋊D₄₇	1	1	yes	characteristic
127	#131	—	2	2	no	—
128	#131	—	2	2	no	—
129	C₂×M₁₆	Dic₉₄	1	2	yes	—
130	C₂×M₁₆	C₂²⋊D₄₇	1	1	yes	characteristic
131	C₂²×C₈	C₄×D₄₇	1	2	yes	—
132	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
133	#117	—	2	2	no	—
134	#117	—	94	2	no	—
135	M₃₂	—	94	4	no	—
136	M₃₂	—	94	2	no	—
137	C₂×C₁₆	—	47	1	no	—
138	C₂×C₁₆	—	47	1	no	—
139	C₂×C₁₆	—	47	2	no	—
140	C₂×C₁₆	—	47	2	no	—
141	C₂×C₁₆	—	47	2	no	—
142	C₂×C₁₆	—	94	2	no	—
143	C₂×C₁₆	—	94	2	no	—
144	#107	—	2	2	no	—
145	#107	—	94	2	no	—
146	#106	—	2	2	no	—
147	#106	—	94	2	no	—
148	#99	—	4	2	no	—
149	#99	—	188	2	no	—
150	#41	—	2	2	no	—
151	#41	—	2	2	no	—
152	C₂×Q₈	#10625337	1	1	yes	characteristic
153	C₂×D₄	#10625334	1	2	yes	—
154	C₂×D₄	#10625337	1	1	yes	characteristic
155	C₂²×C₄	Dic₄₇⋊C₄	1	2	yes	—
156	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
157	M₁₆	—	2	2	no	—
158	M₁₆	—	2	4	no	—
159	C₂×C₈	—	2	2	no	—
160	C₂×C₈	—	2	2	no	—
161	C₂×C₈	Dic₄₇⋊C₄	1	2	yes	—
162	C₂×C₈	C₂×Dic₉₄	1	1	yes	characteristic
163	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
164	C₂×C₈	D₄×D₄₇	1	2	yes	—
165	C₂×C₈	#10625352	1	2	yes	—
166	C₄⋊C₄	—	4	2	no	—
167	C₄⋊C₄	—	188	2	no	—
168	C₂²⋊C₄	—	4	2	no	—
169	C₂²⋊C₄	—	188	2	no	—
170	C₄²	—	4	2	no	—
171	C₄²	—	188	2	no	—
172	C₁₆	—	47	1	no	—
173	C₁₆	—	47	1	no	—
174	C₁₆	—	47	2	no	—
175	C₁₆	—	94	2	no	—
176	C₂³	#13093957	1	2	yes	—
177	C₂³	C₂×#10625334	1	1	yes	characteristic
178	Q₈	—	2	2	no	—
179	D₄	—	2	4	no	—
180	D₄	—	2	2	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	8	2	no	—
184	C₂×C₄	—	376	2	no	—
185	C₂×C₄	#13093957	1	2	yes	—
186	C₂×C₄	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
187	C₂×C₄	#13093996	1	2	yes	—
188	C₂×C₄	C₂×#10625337	1	1	yes	characteristic
189	C₂×C₄	#13094044	1	2	yes	—
190	C₈	—	2	2	no	—
191	C₈	C₂²⋊C₄×D₄₇	1	1	yes	characteristic
192	C₈	#13093980	1	1	yes	characteristic
193	C₈	#13094001	1	2	yes	—
194	C₂²	—	2	2	no	—
195	C₂²	—	4	1	no	—
196	C₂²	—	4	2	no	—
197	C₂²	C₂×#13093957	1	1	yes	characteristic
198	C₂²	#15689357	1	2	yes	—
199	C₄	—	2	2	no	—
200	C₄	—	16	2	no	—
201	C₄	—	752	2	no	—
202	C₄	#15688981	1	1	yes	characteristic
203	C₄	#15689159	1	1	yes	characteristic
204	C₄	#15689362	1	2	yes	—
205	C₂	—	2	2	no	—
206	C₂	—	2	1	no	—
207	C₂	—	4	2	no	—
208	C₂	#20243415	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15689540

C₂

yes

maximal, characteristic

#20245170

C₂

yes

maximal, complemented

#20243434

C₂

yes

maximal, characteristic

#20241489

C₂

yes

maximal, complemented

C₄₇×#1812

C₂

yes

maximal, characteristic

#15689540

—

#15689540

—

C₂×#13093964

C₄

yes

complemented

C₂×#13093964

C₂²

yes

characteristic

C₂²×C₄₇⋊C₁₆

C₄

yes

complemented

C₂²×C₄₇⋊C₁₆

C₂²

yes

characteristic

#15689161

C₂²

yes

—

—

—

—

—

C₂×#131×C₄₇

C₂²

yes

characteristic

C₄₇×#427

C₂²

yes

—

C₄₇×#410

—

C₄₇×H(3,ℤ/4)

—

#13093983

—

#13093964

—

#13093964

—

C₂×C₄₇⋊C₁₆

—

C₂×C₄₇⋊C₁₆

—

C₂×C₄₇⋊C₁₆

C₂×C₄

yes

—

C₂×C₄₇⋊C₁₆

C₂×C₄

yes

characteristic

C₂×C₄₇⋊C₁₆

C₂×C₄

yes

characteristic

C₂×C₄₇⋊C₁₆

D₄

yes

—

C₂×C₄₇⋊C₁₆

D₄

yes

—

#13093958

—

#13093956

—

#13093939

—

C₂×#41×C₄₇

D₄

yes

characteristic

#131×C₄₇

—

#131×C₄₇

—

C₂×M₁₆×C₄₇

D₄

yes

characteristic

C₂×M₁₆×C₄₇

Q₈

yes

—

C₂²×C₈×C₄₇

C₂×C₄

yes

—

C₂²×C₈×C₄₇

C₂³

yes

characteristic

#117×C₄₇

—

#107×C₄₇

—

#106×C₄₇

—

#99×C₄₇

—

#41×C₄₇

—

#41×C₄₇

—

C₂×Q₈×C₄₇

Q₁₆

yes

characteristic

C₂×D₄×C₄₇

D₈

yes

—

C₂×D₄×C₄₇

Q₁₆

yes

characteristic

C₂²×C₄×C₄₇

C₄⋊C₄

yes

—

C₂²×C₄×C₄₇

C₂×D₄

yes

characteristic

M₁₆×C₄₇

—

M₁₆×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

—

C₄⋊C₄×C₄₇

—

C₂²⋊C₄×C₄₇

—

C₄²×C₄₇

—

C₂²⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄×Dic₄₇

—

C₄₇⋊C₁₆

—

C₄₇⋊C₁₆

C₂²⋊C₄

yes

characteristic

C₄₇⋊C₁₆

C₂²⋊C₄

yes

characteristic

C₄₇⋊C₁₆

C₂²⋊C₄

yes

—

C₂³×C₄₇

#107

yes

—

C₂³×C₄₇

C₂×D₈

yes

characteristic

Q₈×C₄₇

—

D₄×C₄₇

—

D₄×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×Q₁₆

yes

characteristic

C₂×C₄×C₄₇

#135

yes

—

C₂×Dic₄₇

—

C₈×C₄₇

—

C₈×C₄₇

C₂×C₂²⋊C₄

yes

characteristic

C₈×C₄₇

#117

yes

characteristic

C₈×C₄₇

C₂²⋊Q₈

yes

—

#12839

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

C₂×#107

yes

characteristic

C₂²×C₄₇

#466

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

—

100

Dic₄₇

—

101

C₂×#504

—

102

#2158

—

103

#1812

—

104

#1812

D₄₇

yes

characteristic

105

#1384

—

106

C₂×C₄₇

—

107

C₂×C₄₇

—

108

C₂×C₄₇

—

109

C₂×C₄₇

#1808

yes

characteristic

110

C₂×#131

C₂×D₄₇

yes

characteristic

111

#504

—

112

#504

—

113

C₂×M₃₂

—

114

C₂×M₃₂

—

115

C₂²×C₁₆

—

116

C₂²×C₁₆

—

117

#427

—

118

#427

C₂×D₄₇

yes

—

119

#410

—

120

#410

—

121

#367

—

122

#366

—

123

H(3,ℤ/4)

—

124

H(3,ℤ/4)

—

125

C₄₇

#12839

yes

characteristic, complemented, Sylow

126

C₂×#41

C₂²⋊D₄₇

yes

characteristic

127

#131

—

128

#131

—

129

C₂×M₁₆

Dic₉₄

yes

—

130

C₂×M₁₆

C₂²⋊D₄₇

yes

characteristic

131

C₂²×C₈

C₄×D₄₇

yes

—

132

C₂²×C₈

C₂²×D₄₇

yes

characteristic

133

#117

—

134

#117

—

135

M₃₂

—

136

M₃₂

—

137

C₂×C₁₆

—

138

C₂×C₁₆

—

139

C₂×C₁₆

—

140

C₂×C₁₆

—

141

C₂×C₁₆

—

142

C₂×C₁₆

—

143

C₂×C₁₆

—

144

#107

—

145

#107

—

146

#106

—

147

#106

—

148

#99

—

149

#99

—

188

—

150

#41

—

151

#41

—

152

C₂×Q₈

#10625337

yes

characteristic

153

C₂×D₄

#10625334

yes

—

154

C₂×D₄

#10625337

yes

characteristic

155

C₂²×C₄

Dic₄₇⋊C₄

yes

—

156

C₂²×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

157

M₁₆

—

158

M₁₆

—

159

C₂×C₈

—

160

C₂×C₈

—

161

C₂×C₈

Dic₄₇⋊C₄

yes

—

162

C₂×C₈

C₂×Dic₉₄

yes

characteristic

163

C₂×C₈

C₂×C₄×D₄₇

yes

characteristic

164

C₂×C₈

D₄×D₄₇

yes

—

165

C₂×C₈

#10625352

yes

—

166

C₄⋊C₄

—

167

C₄⋊C₄

—

188

—

168

C₂²⋊C₄

—

169

C₂²⋊C₄

—

188

—

170

C₄²

—

171

C₄²

—

188

—

172

C₁₆

—

173

C₁₆

—

174

C₁₆

—

175

C₁₆

—

176

C₂³

#13093957

yes

—

177

C₂³

C₂×#10625334

yes

characteristic

178

Q₈

—

179

D₄

—

180

D₄

—

181

C₂×C₄

—

182

C₂×C₄

—

183

C₂×C₄

—

184

C₂×C₄

—

376

—

185

C₂×C₄

#13093957

yes

—

186

C₂×C₄

C₂×Dic₄₇⋊C₄

yes

characteristic

187

C₂×C₄

#13093996

yes

—

188

C₂×C₄

C₂×#10625337

yes

characteristic

189

C₂×C₄

#13094044

yes

—

190

C₈

—

191

C₈

C₂²⋊C₄×D₄₇

yes

characteristic

192

C₈

#13093980

yes

characteristic

193

C₈

#13094001

yes

—

194

C₂²

—

195

C₂²

—

196

C₂²

—

197

C₂²

C₂×#13093957

yes

characteristic

198

C₂²

#15689357

yes

—

199

C₄

—

200

C₄

—

201

C₄

—

752

—

202

C₄

#15688981

yes

characteristic

203

C₄

#15689159

yes

characteristic

204

C₄

#15689362

yes

—

205

C₂

—

206

C₂

—

207

C₂

—

208

C₂

#20243415

yes

characteristic

	subgroup	G/H
center	C₄	#15689159
commutator	C₂×C₈×C₄₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₄₇
Fitting	C₄₇×#1812	C₂
socle	C₂×C₄₇	#1808
prime radical	C₂×#41×C₄₇	D₄
cosocle	C₂²×C₈×C₄₇	C₂³
hypercenter	#1812	D₄₇
hypocenter	C₄₇	#12839
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ C₂×#131 ◁ #1812

subgroup

G/H

center

C₄

#15689159

commutator

C₂×C₈×C₄₇

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₄₇

Fitting

C₄₇×#1812

C₂

socle

C₂×C₄₇

#1808

prime radical

C₂×#41×C₄₇

D₄

cosocle

C₂²×C₈×C₄₇

C₂³

D₄₇

C₄₇

G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ C₂×#131 ◁ #1812

N	H ≅ G/N	type
2. #20245170	C₂	semidirect
4. #20241489	C₂	semidirect
8. C₂×#13093964	C₄	semidirect
10. C₂²×C₄₇⋊C₁₆	C₄	semidirect
125. C₄₇	#12839	semidirect

H ≅ G/N

type

2. #20245170

C₂

semidirect

4. #20241489

C₂

semidirect

8. C₂×#13093964

C₄

semidirect

10. C₂²×C₄₇⋊C₁₆

C₄

semidirect

125. C₄₇

#12839

semidirect

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

subgroup	index
C₂×#15689540	2
#20245170	2
#20243434	2
#20241489	2
C₄₇×#1812	2
#12839	47

subgroup

index

C₄₇×#1812

Subgroups — #26132465

Conjugacy classes of subgroups

Complements

Maximal subgroups