Group #26121330 — subgroups

Subgroups — #26121330

Order 12032.

442 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 442 conjugacy classes of subgroups — too many to draw. Only its 155 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	#20244350	C₂	1	2	yes	maximal
2	#20243548	C₂	1	1	yes	maximal, characteristic
3	#20243547	C₂	1	2	yes	maximal
4	C₄₇⋊#1273	C₂	1	1	yes	maximal, characteristic
5	C₄₇×#1956	C₂	1	1	yes	maximal, characteristic
6	C₂×#13093958	—	2	2	no	—
7	C₂×#13093957	—	2	4	no	—
8	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
9	#15688993	C₂²	1	2	yes	—
10	#15688885	—	2	4	no	—
11	C₂×C₄⋊Q₈×C₄₇	C₂²	1	1	yes	characteristic
12	C₂×#103×C₄₇	—	2	2	no	—
13	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
14	C₄₇×#384	C₂²	1	1	yes	characteristic
15	C₂²×C₄₇⋊C₈	D₄	1	4	yes	—
16	C₂×C₄⋊Dic₄₇	—	2	2	no	—
17	C₂×C₄⋊Dic₄₇	—	2	2	no	—
18	C₂×C₄⋊Dic₄₇	—	2	2	no	—
19	#13093958	—	4	4	no	—
20	#13093957	—	4	8	no	—
21	C₄×C₄₇⋊C₈	—	2	2	no	—
22	C₄×C₄₇⋊C₈	—	2	2	no	—
23	C₂²×Q₈×C₄₇	—	2	1	no	—
24	C₂²×C₈×C₄₇	D₄	1	2	yes	—
25	C₄⋊Q₈×C₄₇	—	2	2	no	—
26	C₄⋊Q₈×C₄₇	—	2	1	no	—
27	C₄⋊Q₈×C₄₇	Q₈	1	2	yes	—
28	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
29	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
30	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
31	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
32	#103×C₄₇	—	4	4	no	—
33	C₄×C₈×C₄₇	—	2	2	no	—
34	C₂×Q₈×C₄₇	—	2	2	no	—
35	C₂×Q₈×C₄₇	—	4	1	no	—
36	C₂×Q₈×C₄₇	—	4	2	no	—
37	C₂×Q₈×C₄₇	—	4	2	no	—
38	C₂²×C₄×C₄₇	—	4	1	no	—
39	C₂²×C₄×C₄₇	—	4	1	no	—
40	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
41	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
42	C₂²×Dic₄₇	—	4	2	no	—
43	C₂×C₈×C₄₇	—	2	2	no	—
44	C₂×C₈×C₄₇	—	2	2	no	—
45	C₂×C₈×C₄₇	D₈	1	4	yes	—
46	C₄⋊C₄×C₄₇	—	2	4	no	—
47	C₄⋊C₄×C₄₇	—	4	2	no	—
48	C₄⋊C₄×C₄₇	—	4	2	no	—
49	C₄⋊C₄×C₄₇	—	4	4	no	—
50	C₄²×C₄₇	C₂×Q₈	1	1	yes	characteristic
51	C₄²×C₄₇	#41	1	1	yes	characteristic
52	C₄²×C₄₇	#41	1	2	yes	—
53	C₄⋊Dic₄₇	—	4	4	no	—
54	C₄⋊Dic₄₇	—	4	4	no	—
55	C₄⋊Dic₄₇	—	4	4	no	—
56	C₂×C₄₇⋊C₈	—	2	4	no	—
57	C₂×C₄₇⋊C₈	—	2	4	no	—
58	C₂×C₄₇⋊C₈	Q₁₆	1	8	yes	—
59	C₂³×C₄₇	#127	1	1	yes	characteristic
60	Q₈×C₄₇	—	4	4	no	—
61	Q₈×C₄₇	—	8	2	no	—
62	C₂×C₄×C₄₇	—	4	2	no	—
63	C₂×C₄×C₄₇	—	8	1	no	—
64	C₂×C₄×C₄₇	—	8	1	no	—
65	C₂×C₄×C₄₇	—	8	1	no	—
66	C₂×C₄×C₄₇	—	8	2	no	—
67	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
68	C₂×C₄×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
69	C₂×C₄×C₄₇	#123	1	1	yes	characteristic
70	C₂×C₄×C₄₇	#123	1	2	yes	—
71	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
72	C₂×C₄×C₄₇	C₂×Q₁₆	1	4	yes	—
73	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	4	yes	—
74	C₂×C₄×C₄₇	#137	1	2	yes	—
75	C₂×Dic₄₇	—	8	2	no	—
76	C₂×Dic₄₇	—	8	2	no	—
77	C₂×Dic₄₇	—	8	2	no	—
78	C₈×C₄₇	—	2	4	no	—
79	C₄₇⋊C₈	—	2	8	no	—
80	#10674	—	47	1	no	maximal, Sylow
81	C₂²×C₄₇	#399	1	1	yes	characteristic
82	C₂²×C₄₇	#493	1	1	yes	characteristic
83	C₂²×C₄₇	#494	1	2	yes	—
84	C₂²×C₄₇	#496	1	2	yes	—
85	C₂²×C₄₇	#497	1	1	yes	characteristic
86	C₄×C₄₇	—	8	2	no	—
87	C₄×C₄₇	—	16	1	no	—
88	C₄×C₄₇	#474	1	2	yes	—
89	C₄×C₄₇	#477	1	4	yes	—
90	C₄×C₄₇	#480	1	2	yes	—
91	C₄×C₄₇	#484	1	4	yes	—
92	Dic₄₇	—	16	2	no	—
93	#1990	—	47	2	no	—
94	#1956	D₄₇	1	1	yes	characteristic
95	#1845	—	47	1	no	—
96	#1845	—	47	2	no	—
97	#1273	—	47	1	no	—
98	C₂×C₄₇	#1713	1	1	yes	characteristic
99	C₂×C₄₇	#1721	1	1	yes	characteristic
100	C₂×C₄₇	#1723	1	2	yes	—
101	C₂×C₄₇	#2093	1	1	yes	characteristic
102	C₂×C₄₇	#2095	1	2	yes	—
103	C₂×C₄⋊Q₈	C₂×D₄₇	1	1	yes	characteristic
104	C₂×#107	—	94	4	no	—
105	C₂×#107	—	94	2	no	—
106	C₂×#103	—	2	2	no	—
107	C₂×C₄×C₈	—	47	2	no	—
108	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
109	#384	—	47	2	no	—
110	#384	C₂×D₄₇	1	1	yes	characteristic
111	#340	—	94	4	no	—
112	C₄₇	#10674	1	1	yes	characteristic, complemented, Sylow
113	C₂²×Q₈	—	2	1	no	—
114	C₂²×C₈	—	47	4	no	—
115	C₂²×C₈	C₂²⋊D₄₇	1	2	yes	—
116	C₄⋊Q₈	—	2	1	no	—
117	C₄⋊Q₈	—	2	2	no	—
118	C₄⋊Q₈	Dic₉₄	1	2	yes	—
119	C₂×C₄⋊C₄	—	2	2	no	—
120	C₂×C₄⋊C₄	—	2	2	no	—
121	C₂×C₄⋊C₄	—	2	1	no	—
122	C₂×C₄⋊C₄	—	94	2	no	—
123	C₂×C₄⋊C₄	—	94	2	no	—
124	C₂×C₄⋊C₄	—	94	2	no	—
125	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
126	#107	—	188	8	no	—
127	#107	—	188	4	no	—
128	#103	—	4	4	no	—
129	C₄×C₈	—	2	2	no	—
130	C₄×C₈	—	94	2	no	—
131	C₄×C₈	—	94	2	no	—
132	C₂×Q₈	—	2	2	no	—
133	C₂×Q₈	—	4	1	no	—
134	C₂×Q₈	—	4	2	no	—
135	C₂×Q₈	—	4	2	no	—
136	C₂²×C₄	—	4	1	no	—
137	C₂²×C₄	—	4	1	no	—
138	C₂²×C₄	—	188	2	no	—
139	C₂²×C₄	D₄×D₄₇	1	2	yes	—
140	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
141	C₂×C₈	—	2	2	no	—
142	C₂×C₈	—	2	2	no	—
143	C₂×C₈	—	47	8	no	—
144	C₂×C₈	—	94	4	no	—
145	C₂×C₈	—	94	4	no	—
146	C₂×C₈	#10625334	1	4	yes	—
147	C₄⋊C₄	—	2	4	no	—
148	C₄⋊C₄	—	4	2	no	—
149	C₄⋊C₄	—	4	2	no	—
150	C₄⋊C₄	—	4	4	no	—
151	C₄⋊C₄	—	188	4	no	—
152	C₄⋊C₄	—	188	4	no	—
153	C₄⋊C₄	—	188	4	no	—
154	C₄²	C₂×Dic₉₄	1	1	yes	characteristic
155	C₄²	#10625350	1	1	yes	characteristic
156	C₄²	#10625352	1	2	yes	—
157	C₂³	#13094020	1	1	yes	characteristic
158	Q₈	—	4	4	no	—
159	Q₈	—	8	2	no	—
160	C₂×C₄	—	4	2	no	—
161	C₂×C₄	—	8	1	no	—
162	C₂×C₄	—	8	1	no	—
163	C₂×C₄	—	8	1	no	—
164	C₂×C₄	—	8	2	no	—
165	C₂×C₄	—	376	2	no	—
166	C₂×C₄	—	376	2	no	—
167	C₂×C₄	—	376	2	no	—
168	C₂×C₄	#13094001	1	2	yes	—
169	C₂×C₄	#13094003	1	1	yes	characteristic
170	C₂×C₄	#13094005	1	1	yes	characteristic
171	C₂×C₄	#13094006	1	2	yes	—
172	C₂×C₄	C₂×#10625334	1	2	yes	—
173	C₂×C₄	Q₁₆×D₄₇	1	4	yes	—
174	C₂×C₄	#13094051	1	4	yes	—
175	C₂×C₄	#13094054	1	2	yes	—
176	C₈	—	2	4	no	—
177	C₈	—	94	8	no	—
178	C₂²	#15689059	1	1	yes	characteristic
179	C₂²	#15689499	1	1	yes	characteristic
180	C₂²	#15689501	1	2	yes	—
181	C₂²	#15689509	1	2	yes	—
182	C₂²	#15689518	1	1	yes	characteristic
183	C₄	—	8	2	no	—
184	C₄	—	16	1	no	—
185	C₄	—	752	2	no	—
186	C₄	#15689403	1	2	yes	—
187	C₄	#15689416	1	4	yes	—
188	C₄	#15689434	1	2	yes	—
189	C₄	#15689457	1	4	yes	—
190	C₂	#20243004	1	1	yes	characteristic
191	C₂	#20243052	1	1	yes	characteristic
192	C₂	#20243068	1	2	yes	—
193	C₂	#20244916	1	1	yes	characteristic
194	C₂	#20244924	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20244350

C₂

yes

maximal

#20243548

C₂

yes

maximal, characteristic

#20243547

C₂

yes

maximal

C₄₇⋊#1273

C₂

yes

maximal, characteristic

C₄₇×#1956

C₂

yes

maximal, characteristic

C₂×#13093958

—

C₂×#13093957

—

C₂×C₄×C₄₇⋊C₈

C₂²

yes

—

#15688993

C₂²

yes

—

#15688885

—

C₂×C₄⋊Q₈×C₄₇

C₂²

yes

characteristic

C₂×#103×C₄₇

—

C₂×C₄×C₈×C₄₇

C₂²

yes

characteristic

C₄₇×#384

C₂²

yes

characteristic

C₂²×C₄₇⋊C₈

D₄

yes

—

C₂×C₄⋊Dic₄₇

—

C₂×C₄⋊Dic₄₇

—

C₂×C₄⋊Dic₄₇

—

#13093958

—

#13093957

—

C₄×C₄₇⋊C₈

—

C₄×C₄₇⋊C₈

—

C₂²×Q₈×C₄₇

—

C₂²×C₈×C₄₇

D₄

yes

—

C₄⋊Q₈×C₄₇

—

C₄⋊Q₈×C₄₇

—

C₄⋊Q₈×C₄₇

Q₈

yes

—

C₂×C₄⋊C₄×C₄₇

—

C₂×C₄⋊C₄×C₄₇

—

C₂×C₄⋊C₄×C₄₇

—

C₂×C₄²×C₄₇

C₂³

yes

characteristic

#103×C₄₇

—

C₄×C₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₄₇

C₂×D₄

yes

—

C₂²×Dic₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

D₈

yes

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄²×C₄₇

C₂×Q₈

yes

C₄²×C₄₇

yes

C₄²×C₄₇

yes

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

Q₁₆

yes

—

C₂³×C₄₇

#127

yes

characteristic

Q₈×C₄₇

—

Q₈×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

C₂×C₄×C₄₇

yes

C₂×C₄×C₄₇

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

—

C₈×C₄₇

—

C₄₇⋊C₈

—

—

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

—

C₂²×C₄₇

#496

yes

—

C₂²×C₄₇

#497

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#474

yes

—

C₄×C₄₇

#477

yes

—

C₄×C₄₇

#480

yes

—

C₄×C₄₇

#484

yes

—

Dic₄₇

—

#1990

—

#1956

D₄₇

yes

—

—

—

C₂×C₄₇

yes

C₂×C₄₇

yes

100

C₂×C₄₇

#1723

yes

—

101

C₂×C₄₇

#2093

yes

characteristic

102

C₂×C₄₇

#2095

yes

—

103

C₂×C₄⋊Q₈

C₂×D₄₇

yes

characteristic

104

C₂×#107

—

105

C₂×#107

—

106

C₂×#103

—

107

C₂×C₄×C₈

—

108

C₂×C₄×C₈

C₂×D₄₇

yes

characteristic

109

#384

—

110

#384

C₂×D₄₇

yes

characteristic

111

#340

—

112

C₄₇

#10674

yes

characteristic, complemented, Sylow

113

C₂²×Q₈

—

114

C₂²×C₈

—

115

C₂²×C₈

C₂²⋊D₄₇

yes

—

116

C₄⋊Q₈

—

117

C₄⋊Q₈

—

118

C₄⋊Q₈

Dic₉₄

yes

—

119

C₂×C₄⋊C₄

—

120

C₂×C₄⋊C₄

—

121

C₂×C₄⋊C₄

—

122

C₂×C₄⋊C₄

—

123

C₂×C₄⋊C₄

—

124

C₂×C₄⋊C₄

—

125

C₂×C₄²

C₂²×D₄₇

yes

characteristic

126

#107

—

188

—

127

#107

—

188

—

128

#103

—

129

C₄×C₈

—

130

C₄×C₈

—

131

C₄×C₈

—

132

C₂×Q₈

—

133

C₂×Q₈

—

134

C₂×Q₈

—

135

C₂×Q₈

—

136

C₂²×C₄

—

137

C₂²×C₄

—

138

C₂²×C₄

—

188

—

139

C₂²×C₄

D₄×D₄₇

yes

—

140

C₂²×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

141

C₂×C₈

—

142

C₂×C₈

—

143

C₂×C₈

—

144

C₂×C₈

—

145

C₂×C₈

—

146

C₂×C₈

#10625334

yes

—

147

C₄⋊C₄

—

148

C₄⋊C₄

—

149

C₄⋊C₄

—

150

C₄⋊C₄

—

151

C₄⋊C₄

—

188

—

152

C₄⋊C₄

—

188

—

153

C₄⋊C₄

—

188

—

154

C₄²

C₂×Dic₉₄

yes

characteristic

155

C₄²

#10625350

yes

characteristic

156

C₄²

#10625352

yes

—

157

C₂³

#13094020

yes

characteristic

158

Q₈

—

159

Q₈

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

376

—

166

C₂×C₄

—

376

—

167

C₂×C₄

—

376

—

168

C₂×C₄

#13094001

yes

—

169

C₂×C₄

#13094003

yes

characteristic

170

C₂×C₄

#13094005

yes

characteristic

171

C₂×C₄

#13094006

yes

—

172

C₂×C₄

C₂×#10625334

yes

—

173

C₂×C₄

Q₁₆×D₄₇

yes

—

174

C₂×C₄

#13094051

yes

—

175

C₂×C₄

#13094054

yes

—

176

C₈

—

177

C₈

—

178

C₂²

#15689059

yes

characteristic

179

C₂²

#15689499

yes

characteristic

180

C₂²

#15689501

yes

—

181

C₂²

#15689509

yes

—

182

C₂²

#15689518

yes

characteristic

183

C₄

—

184

C₄

—

185

C₄

—

752

—

186

C₄

#15689403

yes

—

187

C₄

#15689416

yes

—

188

C₄

#15689434

yes

—

189

C₄

#15689457

yes

—

190

C₂

#20243004

yes

characteristic

191

C₂

#20243052

yes

characteristic

192

C₂

#20243068

yes

—

193

C₂

#20244916

yes

characteristic

194

C₂

#20244924

yes

—

	subgroup	G/H
center	C₂³	#13094020
commutator, cosocle	C₂×C₄²×C₄₇	C₂³
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1956	C₂
socle, prime radical	C₂³×C₄₇	#127
hypercenter	#1956	D₄₇
hypocenter	C₄₇	#10674
LCS	G ▷ C₂×C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1956

subgroup

G/H

center

C₂³

#13094020

commutator, cosocle

C₂×C₄²×C₄₇

C₂³

Frattini

C₂×C₄²

C₂²×D₄₇

Fitting

C₄₇×#1956

C₂

C₂³×C₄₇

D₄₇

C₄₇

G ▷ C₂×C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1956

N	H ≅ G/N	type
112. C₄₇	#10674	semidirect

H ≅ G/N

type

112. C₄₇

#10674

semidirect

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

subgroup	index
#20244350	2
#20243548	2
#20243547	2
C₄₇⋊#1273	2
C₄₇×#1956	2
#10674	47

subgroup

index

#20244350

#20243548

#20243547

C₄₇⋊#1273

C₄₇×#1956

#10674

Subgroups — #26121330

Conjugacy classes of subgroups

Complements

Maximal subgroups