Group #26119076 — subgroups

Subgroups — #26119076

Order 12032.

618 conjugacy classes of subgroups; only the 169 normal subgroups are shown.

normal subgroups only

This group has 618 conjugacy classes of subgroups — too many to draw. Only its 169 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×#15689528	C₂	1	2	yes	maximal
2	C₂×#15689503	C₂	1	1	yes	maximal, characteristic
3	#20244349	C₂	1	2	yes	maximal
4	#20244190	C₂	1	1	yes	maximal, characteristic
5	C₄₇×#1273	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₈₈	—	2	2	no	—
7	C₂×#13094024	C₂²	1	2	yes	—
8	C₂×#13094024	C₂²	1	1	yes	characteristic
9	#15689528	—	2	4	no	—
10	#15689528	—	2	4	no	—
11	#15689503	—	2	4	no	—
12	C₂×#13093958	—	2	4	no	—
13	C₂×#13093949	—	2	2	no	—
14	#15688993	C₂²	1	1	yes	characteristic
15	#15688886	—	2	4	no	—
16	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₈×C₄₇	C₂²	1	2	yes	—
18	C₂²×Dic₉₄	—	2	1	no	—
19	C₂²×Dic₉₄	—	2	2	no	—
20	C₂×Dic₁₈₈	—	2	8	no	—
21	C₂×Dic₁₈₈	—	4	4	no	—
22	C₂×Dic₁₈₈	—	4	4	no	—
23	#13094024	—	2	2	no	—
24	#13094024	—	2	2	no	—
25	#13094024	—	2	2	no	—
26	#13094024	—	2	2	no	—
27	#13094024	—	2	1	no	—
28	#13094024	D₄	1	4	yes	—
29	#13094024	D₄	1	2	yes	—
30	C₂×C₄⋊Dic₄₇	—	2	2	no	—
31	C₂×C₄⋊Dic₄₇	—	2	2	no	—
32	C₂×C₄⋊Dic₄₇	—	2	2	no	—
33	C₂×C₄⋊Dic₄₇	—	2	1	no	—
34	C₂×C₄⋊Dic₄₇	—	2	2	no	—
35	#13093958	—	2	16	no	—
36	#13093949	—	4	4	no	—
37	C₂²×C₈×C₄₇	D₄	1	2	yes	—
38	C₂²×C₈×C₄₇	D₄	1	4	yes	—
39	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
40	C₄×C₈×C₄₇	—	2	2	no	—
41	C₄×C₈×C₄₇	—	2	2	no	—
42	C₄×C₈×C₄₇	—	2	2	no	—
43	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
44	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
45	C₂²×Dic₄₇	—	4	2	no	—
46	C₂²×Dic₄₇	—	4	1	no	—
47	C₂²×Dic₄₇	—	4	1	no	—
48	C₂×Dic₉₄	—	2	4	no	—
49	C₂×Dic₉₄	—	2	2	no	—
50	C₂×Dic₉₄	—	4	2	no	—
51	C₂×Dic₉₄	—	4	4	no	—
52	C₂×Dic₉₄	—	4	4	no	—
53	C₂×Dic₉₄	—	4	2	no	—
54	C₂×Dic₉₄	—	4	1	no	—
55	C₂×Dic₉₄	—	4	2	no	—
56	C₂×C₈×C₄₇	—	2	4	no	—
57	C₂×C₈×C₄₇	—	2	2	no	—
58	C₂×C₈×C₄₇	—	2	2	no	—
59	C₂×C₈×C₄₇	—	2	4	no	—
60	C₂×C₈×C₄₇	D₈	1	4	yes	—
61	C₂×C₈×C₄₇	Q₁₆	1	8	yes	complemented
62	C₄²×C₄₇	C₂×D₄	1	2	yes	—
63	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
64	C₄²×C₄₇	#41	1	1	yes	characteristic
65	C₄⋊Dic₄₇	—	2	4	no	—
66	C₄⋊Dic₄₇	—	2	4	no	—
67	C₄⋊Dic₄₇	—	2	4	no	—
68	C₄⋊Dic₄₇	—	4	2	no	—
69	C₄⋊Dic₄₇	—	4	2	no	—
70	C₄⋊Dic₄₇	—	4	2	no	—
71	C₄⋊Dic₄₇	—	4	2	no	—
72	C₄⋊Dic₄₇	—	4	4	no	—
73	Dic₁₈₈	—	4	16	no	—
74	C₂³×C₄₇	#127	1	1	yes	characteristic
75	C₂×C₄×C₄₇	#120	1	1	yes	characteristic
76	C₂×C₄×C₄₇	#120	1	2	yes	—
77	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
78	C₂×C₄×C₄₇	#121	1	2	yes	—
79	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
80	C₂×C₄×C₄₇	C₂×Q₁₆	1	4	yes	—
81	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
82	C₂×C₄×C₄₇	#137	1	4	yes	—
83	C₂×Dic₄₇	—	4	2	no	—
84	C₂×Dic₄₇	—	4	4	no	—
85	C₂×Dic₄₇	—	8	1	no	—
86	C₂×Dic₄₇	—	8	2	no	—
87	C₂×Dic₄₇	—	8	2	no	—
88	C₂×Dic₄₇	—	8	1	no	—
89	C₂×Dic₄₇	—	8	1	no	—
90	C₂×Dic₄₇	—	8	2	no	—
91	Dic₉₄	—	4	8	no	—
92	Dic₉₄	—	4	4	no	—
93	Dic₉₄	—	8	2	no	—
94	Dic₉₄	—	8	4	no	—
95	C₈×C₄₇	—	2	4	no	—
96	C₈×C₄₇	—	2	8	no	—
97	#10309	—	47	1	no	maximal, Sylow
98	C₂²×C₄₇	#392	1	1	yes	characteristic
99	C₂²×C₄₇	#493	1	1	yes	characteristic
100	C₂²×C₄₇	#494	1	2	yes	—
101	C₂²×C₄₇	#496	1	1	yes	characteristic
102	C₂²×C₄₇	#497	1	2	yes	—
103	C₄×C₄₇	#447	1	2	yes	—
104	C₄×C₄₇	#451	1	4	yes	—
105	C₄×C₄₇	#459	1	2	yes	—
106	C₄×C₄₇	#462	1	4	yes	—
107	Dic₄₇	—	8	4	no	—
108	Dic₄₇	—	8	2	no	—
109	Dic₄₇	—	16	1	no	—
110	C₂×#500	—	47	2	no	—
111	C₂×#494	—	47	1	no	—
112	#1990	—	47	2	no	—
113	#1956	—	47	1	no	—
114	#1273	D₄₇	1	1	yes	characteristic
115	C₂×C₄₇	#1713	1	1	yes	characteristic
116	C₂×C₄₇	#1722	1	1	yes	characteristic
117	C₂×C₄₇	#1723	1	2	yes	—
118	C₂×C₄₇	#2011	1	1	yes	characteristic
119	C₂×C₄₇	#2016	1	2	yes	—
120	C₂²×Q₁₆	—	94	2	no	—
121	C₂×C₄⋊Q₈	—	47	2	no	—
122	C₂×C₄⋊Q₈	—	47	1	no	—
123	#500	—	94	4	no	—
124	#500	—	94	4	no	—
125	#494	—	94	4	no	—
126	C₂×#107	—	94	4	no	—
127	C₂×#103	—	94	2	no	—
128	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
129	C₂×C₄×C₈	C₂×D₄₇	1	2	yes	—
130	#384	—	47	1	no	—
131	#340	—	94	4	no	—
132	C₄₇	#10309	1	1	yes	characteristic, complemented, Sylow
133	C₂²×Q₈	—	94	2	no	—
134	C₂²×Q₈	—	94	1	no	—
135	C₂×Q₁₆	—	94	8	no	—
136	C₂×Q₁₆	—	188	4	no	—
137	C₂×Q₁₆	—	188	4	no	—
138	C₂²×C₈	D₁₈₈	1	2	yes	—
139	C₂²×C₈	D₁₈₈	1	4	yes	—
140	C₄⋊Q₈	—	47	4	no	—
141	C₄⋊Q₈	—	47	2	no	—
142	C₄⋊Q₈	—	94	2	no	—
143	C₄⋊Q₈	—	94	2	no	—
144	C₄⋊Q₈	—	94	2	no	—
145	C₄⋊Q₈	—	94	2	no	—
146	C₄⋊Q₈	—	94	1	no	—
147	C₂×C₄⋊C₄	—	94	2	no	—
148	C₂×C₄⋊C₄	—	94	2	no	—
149	C₂×C₄⋊C₄	—	94	2	no	—
150	C₂×C₄⋊C₄	—	94	2	no	—
151	C₂×C₄⋊C₄	—	94	1	no	—
152	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
153	#107	—	94	16	no	—
154	#103	—	188	4	no	—
155	C₄×C₈	—	2	2	no	—
156	C₄×C₈	—	2	2	no	—
157	C₄×C₈	—	2	2	no	—
158	C₂×Q₈	—	94	4	no	—
159	C₂×Q₈	—	94	2	no	—
160	C₂×Q₈	—	188	2	no	—
161	C₂×Q₈	—	188	4	no	—
162	C₂×Q₈	—	188	4	no	—
163	C₂×Q₈	—	188	2	no	—
164	C₂×Q₈	—	188	1	no	—
165	C₂×Q₈	—	188	2	no	—
166	C₂²×C₄	—	188	1	no	—
167	C₂²×C₄	—	188	2	no	—
168	C₂²×C₄	—	188	1	no	—
169	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
170	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
171	Q₁₆	—	188	16	no	—
172	C₂×C₈	—	2	4	no	—
173	C₂×C₈	—	2	2	no	—
174	C₂×C₈	—	2	2	no	—
175	C₂×C₈	—	2	4	no	—
176	C₂×C₈	D₃₇₆	1	4	yes	—
177	C₂×C₈	Dic₁₈₈	1	8	yes	complemented
178	C₄⋊C₄	—	94	4	no	—
179	C₄⋊C₄	—	94	4	no	—
180	C₄⋊C₄	—	94	4	no	—
181	C₄⋊C₄	—	188	2	no	—
182	C₄⋊C₄	—	188	4	no	—
183	C₄⋊C₄	—	188	2	no	—
184	C₄⋊C₄	—	188	2	no	—
185	C₄⋊C₄	—	188	2	no	—
186	C₄²	D₄×D₄₇	1	2	yes	—
187	C₄²	D₄×D₄₇	1	1	yes	characteristic
188	C₄²	#10625354	1	1	yes	characteristic
189	C₂³	#13094021	1	1	yes	characteristic
190	Q₈	—	376	2	no	—
191	Q₈	—	376	4	no	—
192	Q₈	—	188	8	no	—
193	Q₈	—	188	4	no	—
194	C₂×C₄	—	376	1	no	—
195	C₂×C₄	—	376	2	no	—
196	C₂×C₄	—	376	2	no	—
197	C₂×C₄	—	376	1	no	—
198	C₂×C₄	—	376	1	no	—
199	C₂×C₄	—	376	2	no	—
200	C₂×C₄	—	188	2	no	—
201	C₂×C₄	—	188	4	no	—
202	C₂×C₄	#13093994	1	1	yes	characteristic
203	C₂×C₄	#13093994	1	2	yes	—
204	C₂×C₄	#13093999	1	1	yes	characteristic
205	C₂×C₄	#13093999	1	2	yes	—
206	C₂×C₄	C₂×D₃₇₆	1	2	yes	—
207	C₂×C₄	C₂×Dic₁₈₈	1	4	yes	—
208	C₂×C₄	#13094052	1	2	yes	—
209	C₂×C₄	#13094057	1	4	yes	—
210	C₈	—	2	4	no	—
211	C₈	—	2	8	no	—
212	C₂²	#15689033	1	1	yes	characteristic
213	C₂²	#15689500	1	1	yes	characteristic
214	C₂²	#15689503	1	2	yes	—
215	C₂²	#15689512	1	1	yes	characteristic
216	C₂²	#15689517	1	2	yes	—
217	C₄	—	376	2	no	—
218	C₄	—	376	4	no	—
219	C₄	—	752	1	no	—
220	C₄	#15689259	1	2	yes	—
221	C₄	#15689281	1	4	yes	—
222	C₄	#15689319	1	2	yes	—
223	C₄	#15689334	1	4	yes	—
224	C₂	#20243006	1	1	yes	characteristic
225	C₂	#20243059	1	1	yes	characteristic
226	C₂	#20243065	1	2	yes	—
227	C₂	#20244461	1	1	yes	characteristic
228	C₂	#20244490	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15689528

C₂

yes

maximal

C₂×#15689503

C₂

yes

maximal, characteristic

#20244349

C₂

yes

maximal

#20244190

C₂

yes

maximal, characteristic

C₄₇×#1273

C₂

yes

maximal, characteristic

C₂²×Dic₁₈₈

—

C₂×#13094024

C₂²

yes

—

C₂×#13094024

C₂²

yes

—

—

—

—

—

C₂²

yes

characteristic

#15688886

—

C₂×C₄×C₈×C₄₇

C₂²

yes

characteristic

C₂×C₄×C₈×C₄₇

C₂²

yes

—

C₂²×Dic₉₄

—

C₂²×Dic₉₄

—

C₂×Dic₁₈₈

—

C₂×Dic₁₈₈

—

C₂×Dic₁₈₈

—

—

—

—

—

—

D₄

yes

—

#13094024

D₄

yes

—

C₂×C₄⋊Dic₄₇

—

C₂×C₄⋊Dic₄₇

—

C₂×C₄⋊Dic₄₇

—

C₂×C₄⋊Dic₄₇

—

C₂×C₄⋊Dic₄₇

—

#13093958

—

#13093949

—

C₂²×C₈×C₄₇

D₄

yes

—

C₂²×C₈×C₄₇

D₄

yes

—

C₂×C₄²×C₄₇

C₂³

yes

characteristic

C₄×C₈×C₄₇

—

C₄×C₈×C₄₇

—

C₄×C₈×C₄₇

—

C₂²×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₄₇

C₂×D₄

yes

—

C₂²×Dic₄₇

—

C₂²×Dic₄₇

—

C₂²×Dic₄₇

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

D₈

yes

—

C₂×C₈×C₄₇

Q₁₆

yes

complemented

C₄²×C₄₇

C₂×D₄

yes

—

C₄²×C₄₇

C₂×D₄

yes

characteristic

C₄²×C₄₇

#41

yes

characteristic

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

Dic₁₈₈

—

C₂³×C₄₇

yes

C₂×C₄×C₄₇

yes

C₂×C₄×C₄₇

yes

—

C₂×C₄×C₄₇

#121

yes

characteristic

C₂×C₄×C₄₇

#121

yes

—

C₂×C₄×C₄₇

C₂×D₈

yes

—

C₂×C₄×C₄₇

C₂×Q₁₆

yes

—

C₂×C₄×C₄₇

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₄₇

#137

yes

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

Dic₉₄

—

Dic₉₄

—

Dic₉₄

—

Dic₉₄

—

C₈×C₄₇

—

C₈×C₄₇

—

—

C₂²×C₄₇

yes

C₂²×C₄₇

yes

100

C₂²×C₄₇

#494

yes

—

101

C₂²×C₄₇

#496

yes

characteristic

102

C₂²×C₄₇

#497

yes

—

103

C₄×C₄₇

#447

yes

—

104

C₄×C₄₇

#451

yes

—

105

C₄×C₄₇

#459

yes

—

106

C₄×C₄₇

#462

yes

—

107

Dic₄₇

—

108

Dic₄₇

—

109

Dic₄₇

—

110

C₂×#500

—

111

C₂×#494

—

112

#1990

—

113

#1956

—

114

#1273

D₄₇

yes

characteristic

115

C₂×C₄₇

#1713

yes

characteristic

116

C₂×C₄₇

#1722

yes

characteristic

117

C₂×C₄₇

#1723

yes

—

118

C₂×C₄₇

#2011

yes

characteristic

119

C₂×C₄₇

#2016

yes

—

120

C₂²×Q₁₆

—

121

C₂×C₄⋊Q₈

—

122

C₂×C₄⋊Q₈

—

123

#500

—

124

#500

—

125

#494

—

126

C₂×#107

—

127

C₂×#103

—

128

C₂×C₄×C₈

C₂×D₄₇

yes

characteristic

129

C₂×C₄×C₈

C₂×D₄₇

yes

—

130

#384

—

131

#340

—

132

C₄₇

#10309

yes

characteristic, complemented, Sylow

133

C₂²×Q₈

—

134

C₂²×Q₈

—

135

C₂×Q₁₆

—

136

C₂×Q₁₆

—

188

—

137

C₂×Q₁₆

—

188

—

138

C₂²×C₈

D₁₈₈

yes

—

139

C₂²×C₈

D₁₈₈

yes

—

140

C₄⋊Q₈

—

141

C₄⋊Q₈

—

142

C₄⋊Q₈

—

143

C₄⋊Q₈

—

144

C₄⋊Q₈

—

145

C₄⋊Q₈

—

146

C₄⋊Q₈

—

147

C₂×C₄⋊C₄

—

148

C₂×C₄⋊C₄

—

149

C₂×C₄⋊C₄

—

150

C₂×C₄⋊C₄

—

151

C₂×C₄⋊C₄

—

152

C₂×C₄²

C₂²×D₄₇

yes

characteristic

153

#107

—

154

#103

—

188

—

155

C₄×C₈

—

156

C₄×C₈

—

157

C₄×C₈

—

158

C₂×Q₈

—

159

C₂×Q₈

—

160

C₂×Q₈

—

188

—

161

C₂×Q₈

—

188

—

162

C₂×Q₈

—

188

—

163

C₂×Q₈

—

188

—

164

C₂×Q₈

—

188

—

165

C₂×Q₈

—

188

—

166

C₂²×C₄

—

188

—

167

C₂²×C₄

—

188

—

168

C₂²×C₄

—

188

—

169

C₂²×C₄

C₂×D₁₈₈

yes

characteristic

170

C₂²×C₄

C₂×D₁₈₈

yes

—

171

Q₁₆

—

188

—

172

C₂×C₈

—

173

C₂×C₈

—

174

C₂×C₈

—

175

C₂×C₈

—

176

C₂×C₈

D₃₇₆

yes

—

177

C₂×C₈

Dic₁₈₈

yes

complemented

178

C₄⋊C₄

—

179

C₄⋊C₄

—

180

C₄⋊C₄

—

181

C₄⋊C₄

—

188

—

182

C₄⋊C₄

—

188

—

183

C₄⋊C₄

—

188

—

184

C₄⋊C₄

—

188

—

185

C₄⋊C₄

—

188

—

186

C₄²

D₄×D₄₇

yes

—

187

C₄²

D₄×D₄₇

yes

characteristic

188

C₄²

#10625354

yes

characteristic

189

C₂³

#13094021

yes

characteristic

190

Q₈

—

376

—

191

Q₈

—

376

—

192

Q₈

—

188

—

193

Q₈

—

188

—

194

C₂×C₄

—

376

—

195

C₂×C₄

—

376

—

196

C₂×C₄

—

376

—

197

C₂×C₄

—

376

—

198

C₂×C₄

—

376

—

199

C₂×C₄

—

376

—

200

C₂×C₄

—

188

—

201

C₂×C₄

—

188

—

202

C₂×C₄

#13093994

yes

characteristic

203

C₂×C₄

#13093994

yes

—

204

C₂×C₄

#13093999

yes

characteristic

205

C₂×C₄

#13093999

yes

—

206

C₂×C₄

C₂×D₃₇₆

yes

—

207

C₂×C₄

C₂×Dic₁₈₈

yes

—

208

C₂×C₄

#13094052

yes

—

209

C₂×C₄

#13094057

yes

—

210

C₈

—

211

C₈

—

212

C₂²

#15689033

yes

characteristic

213

C₂²

#15689500

yes

characteristic

214

C₂²

#15689503

yes

—

215

C₂²

#15689512

yes

characteristic

216

C₂²

#15689517

yes

—

217

C₄

—

376

—

218

C₄

—

376

—

219

C₄

—

752

—

220

C₄

#15689259

yes

—

221

C₄

#15689281

yes

—

222

C₄

#15689319

yes

—

223

C₄

#15689334

yes

—

224

C₂

#20243006

yes

characteristic

225

C₂

#20243059

yes

characteristic

226

C₂

#20243065

yes

—

227

C₂

#20244461

yes

characteristic

228

C₂

#20244490

yes

—

	subgroup	G/H
center	C₂³	#13094021
commutator, cosocle	C₂×C₄²×C₄₇	C₂³
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1273	C₂
socle, prime radical	C₂³×C₄₇	#127
hypercenter	#1273	D₄₇
hypocenter	C₄₇	#10309
LCS	G ▷ C₂×C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1273

subgroup

G/H

center

C₂³

#13094021

commutator, cosocle

C₂×C₄²×C₄₇

C₂³

Frattini

C₂×C₄²

C₂²×D₄₇

Fitting

C₄₇×#1273

C₂

C₂³×C₄₇

D₄₇

C₄₇

G ▷ C₂×C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1273

N	H ≅ G/N	type
61. C₂×C₈×C₄₇	Q₁₆	semidirect
132. C₄₇	#10309	semidirect
177. C₂×C₈	Dic₁₈₈	semidirect

H ≅ G/N

type

61. C₂×C₈×C₄₇

Q₁₆

semidirect

132. C₄₇

#10309

semidirect

177. C₂×C₈

Dic₁₈₈

semidirect

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

subgroup	index
C₂×#15689528	2
C₂×#15689503	2
#20244349	2
#20244190	2
C₄₇×#1273	2
#10309	47

subgroup

index

C₄₇×#1273

Subgroups — #26119076

Conjugacy classes of subgroups

Complements

Maximal subgroups