Group #26120735 — subgroups

Subgroups — #26120735

Order 12032.

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

normal subgroups only

This group has 406 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	#20248630	C₂	1	1	yes	maximal, characteristic
2	#20244144	C₂	1	2	yes	maximal
3	#20241271	C₂	1	1	yes	maximal, characteristic
4	#20241228	C₂	1	2	yes	maximal
5	C₄₇×#1844	C₂	1	1	yes	maximal, characteristic
6	C₂×#13093957	—	2	2	no	—
7	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
8	#15689062	—	2	2	no	—
9	#15689022	—	2	2	no	—
10	#15688988	—	2	2	no	—
11	C₄×C₄⋊Dic₄₇	C₂²	1	2	yes	—
12	#15688885	—	2	2	no	—
13	C₂×#106×C₄₇	—	2	2	no	—
14	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
15	C₄₇×#384	C₂²	1	1	yes	characteristic
16	C₄₇×#384	C₂²	1	1	yes	characteristic
17	C₂²×C₄₇⋊C₈	—	2	2	no	—
18	C₂×C₄⋊Dic₄₇	—	2	2	no	—
19	C₂×C₄⋊Dic₄₇	D₄	1	2	yes	—
20	C₂×C₄⋊Dic₄₇	D₄	1	2	yes	—
21	C₂×C₄×Dic₄₇	—	2	2	no	—
22	#13093957	—	4	4	no	—
23	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
24	C₄⋊(C₄₇⋊C₈)	Q₈	1	4	yes	—
25	C₄₇⋊#95	—	2	2	no	—
26	C₄₇⋊#95	—	2	4	no	—
27	C₄₇⋊#95	—	2	2	no	—
28	C₂²×C₈×C₄₇	—	2	1	no	—
29	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
30	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
31	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
32	C₂×C₄⋊C₄×C₄₇	D₄	1	2	yes	—
33	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
34	#106×C₄₇	—	4	4	no	—
35	C₄×C₈×C₄₇	—	2	1	no	—
36	C₄×C₈×C₄₇	Q₈	1	2	yes	—
37	C₂²×C₄×C₄₇	—	2	2	no	—
38	C₂²×C₄×C₄₇	—	4	1	no	—
39	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
40	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
41	C₂²×Dic₄₇	—	2	2	no	—
42	C₂²×Dic₄₇	—	2	2	no	—
43	C₂×C₈×C₄₇	—	2	2	no	—
44	C₂×C₈×C₄₇	—	2	2	no	—
45	C₂×C₈×C₄₇	—	2	2	no	—
46	C₄⋊C₄×C₄₇	—	4	2	no	—
47	C₄⋊C₄×C₄₇	—	4	2	no	—
48	C₄⋊C₄×C₄₇	—	4	2	no	—
49	C₄⋊C₄×C₄₇	—	4	2	no	—
50	C₄⋊C₄×C₄₇	—	4	2	no	—
51	C₄⋊C₄×C₄₇	—	4	2	no	—
52	C₄²×C₄₇	C₂×Q₈	1	1	yes	characteristic
53	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
54	C₄²×C₄₇	#41	1	1	yes	characteristic
55	C₄⋊Dic₄₇	—	2	2	no	—
56	C₄⋊Dic₄₇	—	2	2	no	—
57	C₄⋊Dic₄₇	—	2	2	no	—
58	C₄⋊Dic₄₇	—	2	2	no	—
59	C₄⋊Dic₄₇	—	8	2	no	—
60	C₄×Dic₄₇	—	4	2	no	—
61	C₄×Dic₄₇	—	4	2	no	—
62	C₂×C₄₇⋊C₈	—	2	4	no	—
63	C₂×C₄₇⋊C₈	—	4	2	no	—
64	C₂×C₄₇⋊C₈	—	4	2	no	—
65	C₂³×C₄₇	#120	1	1	yes	characteristic
66	C₂×C₄×C₄₇	—	2	1	no	—
67	C₂×C₄×C₄₇	—	2	1	no	—
68	C₂×C₄×C₄₇	—	4	4	no	—
69	C₂×C₄×C₄₇	—	4	2	no	—
70	C₂×C₄×C₄₇	—	8	2	no	—
71	C₂×C₄×C₄₇	—	8	1	no	—
72	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
73	C₂×C₄×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
74	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
75	C₂×C₄×C₄₇	C₄⋊Q₈	1	2	yes	—
76	C₂×C₄×C₄₇	#135	1	2	yes	—
77	C₂×C₄×C₄₇	#135	1	2	yes	—
78	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
79	C₂×C₄×C₄₇	#137	1	2	yes	—
80	C₂×Dic₄₇	—	4	2	no	—
81	C₂×Dic₄₇	—	4	2	no	—
82	C₂×Dic₄₇	—	4	2	no	—
83	C₂×Dic₄₇	—	4	2	no	—
84	C₂×Dic₄₇	—	4	2	no	—
85	C₂×Dic₄₇	—	4	2	no	—
86	C₈×C₄₇	—	2	4	no	—
87	C₄₇⋊C₈	—	4	4	no	—
88	#10577	—	47	1	no	maximal, Sylow
89	C₂²×C₄₇	#395	1	1	yes	characteristic
90	C₂²×C₄₇	#449	1	2	yes	—
91	C₂²×C₄₇	#452	1	2	yes	—
92	C₂²×C₄₇	#453	1	1	yes	characteristic
93	C₂²×C₄₇	#456	1	1	yes	characteristic
94	C₄×C₄₇	—	2	1	no	—
95	C₄×C₄₇	—	2	1	no	—
96	C₄×C₄₇	—	8	2	no	—
97	C₄×C₄₇	—	16	1	no	—
98	C₄×C₄₇	#478	1	2	yes	—
99	C₄×C₄₇	#479	1	2	yes	—
100	C₄×C₄₇	#499	1	2	yes	—
101	C₄×C₄₇	#501	1	2	yes	—
102	Dic₄₇	—	8	2	no	—
103	Dic₄₇	—	8	2	no	—
104	#2571	—	47	1	no	—
105	#1947	—	47	2	no	—
106	#1844	D₄₇	1	1	yes	characteristic
107	#1296	—	47	1	no	—
108	#1279	—	47	2	no	—
109	C₂×C₄₇	#1624	1	1	yes	characteristic
110	C₂×C₄₇	#1630	1	1	yes	characteristic
111	C₂×C₄₇	#1652	1	2	yes	—
112	C₂×C₄₇	#2059	1	2	yes	—
113	C₂×C₄₇	#2060	1	1	yes	characteristic
114	C₂×#107	—	94	2	no	—
115	C₂×#106	—	2	2	no	—
116	C₂×C₄⋊C₈	—	47	2	no	—
117	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
118	#400	—	94	2	no	—
119	#389	—	94	2	no	—
120	#384	C₂×D₄₇	1	1	yes	characteristic
121	#384	C₂×D₄₇	1	1	yes	characteristic
122	#382	—	94	2	no	—
123	C₄×C₄⋊C₄	—	47	2	no	—
124	#340	—	94	2	no	—
125	C₄₇	#10577	1	1	yes	characteristic, complemented, Sylow
126	C₂²×C₈	—	2	1	no	—
127	C₂²×C₈	—	94	2	no	—
128	C₂×C₄⋊C₄	—	2	2	no	—
129	C₂×C₄⋊C₄	—	2	2	no	—
130	C₂×C₄⋊C₄	—	2	1	no	—
131	C₂×C₄⋊C₄	—	47	2	no	—
132	C₂×C₄⋊C₄	—	47	2	no	—
133	C₂×C₄⋊C₄	—	94	2	no	—
134	C₂×C₄⋊C₄	C₂²⋊D₄₇	1	2	yes	—
135	C₂×C₄²	—	94	2	no	—
136	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
137	#107	—	188	4	no	—
138	#106	—	4	4	no	—
139	C₄⋊C₈	—	47	4	no	—
140	C₄⋊C₈	—	94	2	no	—
141	C₄×C₈	—	2	1	no	—
142	C₄×C₈	Dic₉₄	1	2	yes	—
143	#95	—	94	2	no	—
144	#95	—	94	4	no	—
145	#95	—	94	2	no	—
146	C₂²×C₄	—	2	2	no	—
147	C₂²×C₄	—	4	1	no	—
148	C₂²×C₄	—	94	2	no	—
149	C₂²×C₄	—	94	2	no	—
150	C₂²×C₄	D₄×D₄₇	1	2	yes	—
151	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
152	C₂×C₈	—	2	2	no	—
153	C₂×C₈	—	2	2	no	—
154	C₂×C₈	—	2	2	no	—
155	C₂×C₈	—	94	4	no	—
156	C₂×C₈	—	188	2	no	—
157	C₂×C₈	—	188	2	no	—
158	C₄⋊C₄	—	4	2	no	—
159	C₄⋊C₄	—	4	2	no	—
160	C₄⋊C₄	—	4	2	no	—
161	C₄⋊C₄	—	4	2	no	—
162	C₄⋊C₄	—	4	2	no	—
163	C₄⋊C₄	—	4	2	no	—
164	C₄⋊C₄	—	94	2	no	—
165	C₄⋊C₄	—	94	2	no	—
166	C₄⋊C₄	—	94	2	no	—
167	C₄⋊C₄	—	94	2	no	—
168	C₄⋊C₄	—	376	2	no	—
169	C₄²	—	188	2	no	—
170	C₄²	—	188	2	no	—
171	C₄²	C₂×Dic₉₄	1	1	yes	characteristic
172	C₄²	#10625350	1	1	yes	characteristic
173	C₄²	Q₈×D₄₇	1	2	yes	—
174	C₂³	#13093992	1	1	yes	characteristic
175	C₂×C₄	—	2	1	no	—
176	C₂×C₄	—	2	1	no	—
177	C₂×C₄	—	4	4	no	—
178	C₂×C₄	—	4	2	no	—
179	C₂×C₄	—	8	2	no	—
180	C₂×C₄	—	8	1	no	—
181	C₂×C₄	—	188	2	no	—
182	C₂×C₄	—	188	2	no	—
183	C₂×C₄	—	188	2	no	—
184	C₂×C₄	—	188	2	no	—
185	C₂×C₄	—	188	2	no	—
186	C₂×C₄	—	188	2	no	—
187	C₂×C₄	#13094002	1	2	yes	—
188	C₂×C₄	#13094003	1	1	yes	characteristic
189	C₂×C₄	#13094022	1	1	yes	characteristic
190	C₂×C₄	#13094023	1	2	yes	—
191	C₂×C₄	#13094045	1	2	yes	—
192	C₂×C₄	#13094045	1	2	yes	—
193	C₂×C₄	#13094051	1	2	yes	—
194	C₂×C₄	#13094053	1	2	yes	—
195	C₈	—	2	4	no	—
196	C₈	—	188	4	no	—
197	C₂²	#15689043	1	1	yes	characteristic
198	C₂²	#15689267	1	2	yes	—
199	C₂²	#15689282	1	2	yes	—
200	C₂²	#15689290	1	1	yes	characteristic
201	C₂²	#15689305	1	1	yes	characteristic
202	C₄	—	2	1	no	—
203	C₄	—	2	1	no	—
204	C₄	—	8	2	no	—
205	C₄	—	16	1	no	—
206	C₄	—	376	2	no	—
207	C₄	—	376	2	no	—
208	C₄	#15689422	1	2	yes	—
209	C₄	#15689433	1	2	yes	—
210	C₄	#15689523	1	2	yes	—
211	C₄	#15689530	1	2	yes	—
212	C₂	#20242469	1	1	yes	characteristic
213	C₂	#20242507	1	1	yes	characteristic
214	C₂	#20242655	1	2	yes	—
215	C₂	#20244728	1	2	yes	—
216	C₂	#20244733	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20248630

C₂

yes

maximal, characteristic

#20244144

C₂

yes

maximal

#20241271

C₂

yes

maximal, characteristic

#20241228

C₂

yes

maximal

C₄₇×#1844

C₂

yes

maximal, characteristic

C₂×#13093957

—

C₂×C₄⋊(C₄₇⋊C₈)

C₂²

yes

—

#15689062

—

#15689022

—

#15688988

—

C₄×C₄⋊Dic₄₇

C₂²

yes

—

#15688885

—

C₂×#106×C₄₇

—

C₂×C₄×C₈×C₄₇

C₂²

yes

characteristic

C₄₇×#384

C₂²

yes

characteristic

C₄₇×#384

C₂²

yes

characteristic

C₂²×C₄₇⋊C₈

—

C₂×C₄⋊Dic₄₇

—

C₂×C₄⋊Dic₄₇

D₄

yes

—

C₂×C₄⋊Dic₄₇

D₄

yes

—

C₂×C₄×Dic₄₇

—

#13093957

—

C₄⋊(C₄₇⋊C₈)

—

C₄⋊(C₄₇⋊C₈)

Q₈

yes

—

C₄₇⋊#95

—

C₄₇⋊#95

—

C₄₇⋊#95

—

C₂²×C₈×C₄₇

—

C₂×C₄⋊C₄×C₄₇

—

C₂×C₄⋊C₄×C₄₇

—

C₂×C₄⋊C₄×C₄₇

—

C₂×C₄⋊C₄×C₄₇

D₄

yes

—

C₂×C₄²×C₄₇

C₂³

yes

characteristic

#106×C₄₇

—

C₄×C₈×C₄₇

—

C₄×C₈×C₄₇

Q₈

yes

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₄₇

C₂×D₄

yes

—

C₂²×Dic₄₇

—

C₂²×Dic₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄²×C₄₇

C₂×Q₈

yes

characteristic

C₄²×C₄₇

C₂×Q₈

yes

—

C₄²×C₄₇

#41

yes

characteristic

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄×Dic₄₇

—

C₄×Dic₄₇

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

#120

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂²⋊Q₈

yes

—

C₂×C₄×C₄₇

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₄₇

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₄₇

C₄⋊Q₈

yes

—

C₂×C₄×C₄₇

#135

yes

—

C₂×C₄×C₄₇

#135

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

—

C₈×C₄₇

—

C₄₇⋊C₈

—

—

C₂²×C₄₇

yes

C₂²×C₄₇

yes

—

C₂²×C₄₇

#452

yes

—

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#478

yes

—

C₄×C₄₇

#479

yes

—

100

C₄×C₄₇

#499

yes

—

101

C₄×C₄₇

#501

yes

—

102

Dic₄₇

—

103

Dic₄₇

—

104

#2571

—

105

#1947

—

106

#1844

D₄₇

yes

characteristic

107

#1296

—

108

#1279

—

109

C₂×C₄₇

#1624

yes

characteristic

110

C₂×C₄₇

#1630

yes

characteristic

111

C₂×C₄₇

#1652

yes

—

112

C₂×C₄₇

#2059

yes

—

113

C₂×C₄₇

#2060

yes

characteristic

114

C₂×#107

—

115

C₂×#106

—

116

C₂×C₄⋊C₈

—

117

C₂×C₄×C₈

C₂×D₄₇

yes

characteristic

118

#400

—

119

#389

—

120

#384

C₂×D₄₇

yes

characteristic

121

#384

C₂×D₄₇

yes

characteristic

122

#382

—

123

C₄×C₄⋊C₄

—

124

#340

—

125

C₄₇

#10577

yes

characteristic, complemented, Sylow

126

C₂²×C₈

—

127

C₂²×C₈

—

128

C₂×C₄⋊C₄

—

129

C₂×C₄⋊C₄

—

130

C₂×C₄⋊C₄

—

131

C₂×C₄⋊C₄

—

132

C₂×C₄⋊C₄

—

133

C₂×C₄⋊C₄

—

134

C₂×C₄⋊C₄

C₂²⋊D₄₇

yes

—

135

C₂×C₄²

—

136

C₂×C₄²

C₂²×D₄₇

yes

characteristic

137

#107

—

188

—

138

#106

—

139

C₄⋊C₈

—

140

C₄⋊C₈

—

141

C₄×C₈

—

142

C₄×C₈

Dic₉₄

yes

—

143

#95

—

144

#95

—

145

#95

—

146

C₂²×C₄

—

147

C₂²×C₄

—

148

C₂²×C₄

—

149

C₂²×C₄

—

150

C₂²×C₄

D₄×D₄₇

yes

—

151

C₂²×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

152

C₂×C₈

—

153

C₂×C₈

—

154

C₂×C₈

—

155

C₂×C₈

—

156

C₂×C₈

—

188

—

157

C₂×C₈

—

188

—

158

C₄⋊C₄

—

159

C₄⋊C₄

—

160

C₄⋊C₄

—

161

C₄⋊C₄

—

162

C₄⋊C₄

—

163

C₄⋊C₄

—

164

C₄⋊C₄

—

165

C₄⋊C₄

—

166

C₄⋊C₄

—

167

C₄⋊C₄

—

168

C₄⋊C₄

—

376

—

169

C₄²

—

188

—

170

C₄²

—

188

—

171

C₄²

C₂×Dic₉₄

yes

characteristic

172

C₄²

#10625350

yes

characteristic

173

C₄²

Q₈×D₄₇

yes

—

174

C₂³

#13093992

yes

characteristic

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

—

181

C₂×C₄

—

188

—

182

C₂×C₄

—

188

—

183

C₂×C₄

—

188

—

184

C₂×C₄

—

188

—

185

C₂×C₄

—

188

—

186

C₂×C₄

—

188

—

187

C₂×C₄

#13094002

yes

—

188

C₂×C₄

#13094003

yes

characteristic

189

C₂×C₄

#13094022

yes

characteristic

190

C₂×C₄

#13094023

yes

—

191

C₂×C₄

#13094045

yes

—

192

C₂×C₄

#13094045

yes

—

193

C₂×C₄

#13094051

yes

—

194

C₂×C₄

#13094053

yes

—

195

C₈

—

196

C₈

—

188

—

197

C₂²

#15689043

yes

characteristic

198

C₂²

#15689267

yes

—

199

C₂²

#15689282

yes

—

200

C₂²

#15689290

yes

characteristic

201

C₂²

#15689305

yes

characteristic

202

C₄

—

203

C₄

—

204

C₄

—

205

C₄

—

206

C₄

—

376

—

207

C₄

—

376

—

208

C₄

#15689422

yes

—

209

C₄

#15689433

yes

—

210

C₄

#15689523

yes

—

211

C₄

#15689530

yes

—

212

C₂

#20242469

yes

characteristic

213

C₂

#20242507

yes

characteristic

214

C₂

#20242655

yes

—

215

C₂

#20244728

yes

—

216

C₂

#20244733

yes

characteristic

	subgroup	G/H
center	C₂³	#13093992
commutator, cosocle	C₂×C₄²×C₄₇	C₂³
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1844	C₂
socle, prime radical	C₂³×C₄₇	#120
hypercenter	#1844	D₄₇
hypocenter	C₄₇	#10577
LCS	G ▷ C₂×C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1844

subgroup

G/H

center

C₂³

#13093992

commutator, cosocle

C₂×C₄²×C₄₇

C₂³

Frattini

C₂×C₄²

C₂²×D₄₇

Fitting

C₄₇×#1844

C₂

C₂³×C₄₇

D₄₇

C₄₇

G ▷ C₂×C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1844

N	H ≅ G/N	type
125. C₄₇	#10577	semidirect

H ≅ G/N

type

125. C₄₇

#10577

semidirect

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

subgroup	index
#20248630	2
#20244144	2
#20241271	2
#20241228	2
C₄₇×#1844	2
#10577	47

subgroup

index

C₄₇×#1844

Subgroups — #26120735

Conjugacy classes of subgroups

Complements

Maximal subgroups