Group #26121398 — subgroups

Subgroups — #26121398

Order 12032.

410 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 147 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	#20243545	C₂	1	1	yes	maximal, characteristic
3	#20243515	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×#13093956	—	2	2	no	—
7	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
8	#15688993	C₂²	1	2	yes	—
9	#15688885	—	2	4	no	—
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	#13093956	—	4	4	no	—
20	C₄×C₄₇⋊C₈	—	2	2	no	—
21	C₄×C₄₇⋊C₈	—	2	2	no	—
22	C₂²×Q₈×C₄₇	—	2	1	no	—
23	C₂²×C₈×C₄₇	D₄	1	2	yes	—
24	C₄⋊Q₈×C₄₇	—	2	2	no	—
25	C₄⋊Q₈×C₄₇	—	2	1	no	—
26	C₄⋊Q₈×C₄₇	Q₈	1	2	yes	—
27	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
28	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
29	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
30	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
31	#103×C₄₇	—	4	4	no	—
32	C₄×C₈×C₄₇	—	2	2	no	—
33	C₂×Q₈×C₄₇	—	2	2	no	—
34	C₂×Q₈×C₄₇	—	4	1	no	—
35	C₂×Q₈×C₄₇	—	4	2	no	—
36	C₂×Q₈×C₄₇	—	4	2	no	—
37	C₂²×C₄×C₄₇	—	4	1	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	—	4	2	no	—
42	C₂×C₈×C₄₇	—	2	2	no	—
43	C₂×C₈×C₄₇	—	2	2	no	—
44	C₂×C₈×C₄₇	SD₁₆	1	4	yes	—
45	C₄⋊C₄×C₄₇	—	2	4	no	—
46	C₄⋊C₄×C₄₇	—	4	2	no	—
47	C₄⋊C₄×C₄₇	—	4	2	no	—
48	C₄⋊C₄×C₄₇	—	4	4	no	—
49	C₄²×C₄₇	C₂×Q₈	1	1	yes	characteristic
50	C₄²×C₄₇	#41	1	1	yes	characteristic
51	C₄²×C₄₇	#41	1	2	yes	—
52	C₄⋊Dic₄₇	—	4	4	no	—
53	C₄⋊Dic₄₇	—	4	4	no	—
54	C₄⋊Dic₄₇	—	4	4	no	—
55	C₂×C₄₇⋊C₈	—	2	4	no	—
56	C₂×C₄₇⋊C₈	—	2	4	no	—
57	C₂×C₄₇⋊C₈	—	2	4	no	—
58	C₂³×C₄₇	#127	1	1	yes	characteristic
59	Q₈×C₄₇	—	4	4	no	—
60	Q₈×C₄₇	—	8	2	no	—
61	C₂×C₄×C₄₇	—	4	2	no	—
62	C₂×C₄×C₄₇	—	8	1	no	—
63	C₂×C₄×C₄₇	—	8	1	no	—
64	C₂×C₄×C₄₇	—	8	1	no	—
65	C₂×C₄×C₄₇	—	8	2	no	—
66	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
67	C₂×C₄×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
68	C₂×C₄×C₄₇	#123	1	1	yes	characteristic
69	C₂×C₄×C₄₇	#123	1	2	yes	—
70	C₂×C₄×C₄₇	C₂×SD₁₆	1	2	yes	—
71	C₂×C₄×C₄₇	#135	1	4	yes	—
72	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
73	C₂×C₄×C₄₇	#137	1	4	yes	—
74	C₂×Dic₄₇	—	8	2	no	—
75	C₂×Dic₄₇	—	8	2	no	—
76	C₂×Dic₄₇	—	8	2	no	—
77	C₈×C₄₇	—	2	4	no	—
78	C₄₇⋊C₈	—	4	4	no	—
79	#10686	—	47	1	no	maximal, Sylow
80	C₂²×C₄₇	#399	1	1	yes	characteristic
81	C₂²×C₄₇	#492	1	1	yes	characteristic
82	C₂²×C₄₇	#495	1	2	yes	—
83	C₂²×C₄₇	#496	1	1	yes	characteristic
84	C₂²×C₄₇	#497	1	2	yes	—
85	C₄×C₄₇	—	8	2	no	—
86	C₄×C₄₇	—	16	1	no	—
87	C₄×C₄₇	#476	1	2	yes	—
88	C₄×C₄₇	#479	1	4	yes	—
89	C₄×C₄₇	#481	1	2	yes	—
90	C₄×C₄₇	#485	1	4	yes	—
91	Dic₄₇	—	16	2	no	—
92	#1990	—	47	2	no	—
93	#1956	D₄₇	1	1	yes	characteristic
94	#1844	—	47	1	no	—
95	#1834	—	47	2	no	—
96	#1273	—	47	1	no	—
97	C₂×C₄₇	#1708	1	1	yes	characteristic
98	C₂×C₄₇	#1716	1	1	yes	characteristic
99	C₂×C₄₇	#1719	1	2	yes	—
100	C₂×C₄₇	#2094	1	1	yes	characteristic
101	C₂×C₄₇	#2097	1	2	yes	—
102	C₂×C₄⋊Q₈	C₂×D₄₇	1	1	yes	characteristic
103	C₂×#106	—	94	2	no	—
104	C₂×#103	—	2	2	no	—
105	C₂×C₄×C₈	—	47	2	no	—
106	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
107	#384	—	47	2	no	—
108	#384	C₂×D₄₇	1	1	yes	characteristic
109	#340	—	94	4	no	—
110	#340	—	94	4	no	—
111	C₄₇	#10686	1	1	yes	characteristic, complemented, Sylow
112	C₂²×Q₈	—	2	1	no	—
113	C₂²×C₈	—	47	4	no	—
114	C₂²×C₈	C₂²⋊D₄₇	1	2	yes	—
115	C₄⋊Q₈	—	2	1	no	—
116	C₄⋊Q₈	—	2	2	no	—
117	C₄⋊Q₈	Dic₉₄	1	2	yes	—
118	C₂×C₄⋊C₄	—	2	2	no	—
119	C₂×C₄⋊C₄	—	2	2	no	—
120	C₂×C₄⋊C₄	—	2	1	no	—
121	C₂×C₄⋊C₄	—	94	2	no	—
122	C₂×C₄⋊C₄	—	94	2	no	—
123	C₂×C₄⋊C₄	—	94	2	no	—
124	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
125	#106	—	188	4	no	—
126	#103	—	4	4	no	—
127	C₄×C₈	—	2	2	no	—
128	C₄×C₈	—	94	2	no	—
129	C₄×C₈	—	94	2	no	—
130	C₂×Q₈	—	2	2	no	—
131	C₂×Q₈	—	4	1	no	—
132	C₂×Q₈	—	4	2	no	—
133	C₂×Q₈	—	4	2	no	—
134	C₂²×C₄	—	4	1	no	—
135	C₂²×C₄	—	4	1	no	—
136	C₂²×C₄	—	188	2	no	—
137	C₂²×C₄	D₄×D₄₇	1	2	yes	—
138	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
139	C₂×C₈	—	2	2	no	—
140	C₂×C₈	—	2	2	no	—
141	C₂×C₈	—	94	4	no	—
142	C₂×C₈	—	94	4	no	—
143	C₂×C₈	—	94	4	no	—
144	C₂×C₈	#10625335	1	4	yes	—
145	C₄⋊C₄	—	2	4	no	—
146	C₄⋊C₄	—	4	2	no	—
147	C₄⋊C₄	—	4	2	no	—
148	C₄⋊C₄	—	4	4	no	—
149	C₄⋊C₄	—	188	4	no	—
150	C₄⋊C₄	—	188	4	no	—
151	C₄⋊C₄	—	188	4	no	—
152	C₄²	C₂×Dic₉₄	1	1	yes	characteristic
153	C₄²	#10625350	1	1	yes	characteristic
154	C₄²	#10625352	1	2	yes	—
155	C₂³	#13094020	1	1	yes	characteristic
156	Q₈	—	4	4	no	—
157	Q₈	—	8	2	no	—
158	C₂×C₄	—	4	2	no	—
159	C₂×C₄	—	8	1	no	—
160	C₂×C₄	—	8	1	no	—
161	C₂×C₄	—	8	1	no	—
162	C₂×C₄	—	8	2	no	—
163	C₂×C₄	—	376	2	no	—
164	C₂×C₄	—	376	2	no	—
165	C₂×C₄	—	376	2	no	—
166	C₂×C₄	#13094001	1	2	yes	—
167	C₂×C₄	#13094003	1	1	yes	characteristic
168	C₂×C₄	#13094005	1	1	yes	characteristic
169	C₂×C₄	#13094006	1	2	yes	—
170	C₂×C₄	C₂×#10625335	1	2	yes	—
171	C₂×C₄	#13094045	1	4	yes	—
172	C₂×C₄	#13094049	1	2	yes	—
173	C₂×C₄	#13094056	1	4	yes	—
174	C₈	—	2	4	no	—
175	C₈	—	188	4	no	—
176	C₂²	#15689059	1	1	yes	characteristic
177	C₂²	#15689495	1	1	yes	characteristic
178	C₂²	#15689504	1	2	yes	—
179	C₂²	#15689513	1	1	yes	characteristic
180	C₂²	#15689516	1	2	yes	—
181	C₄	—	8	2	no	—
182	C₄	—	16	1	no	—
183	C₄	—	752	2	no	—
184	C₄	#15689413	1	2	yes	—
185	C₄	#15689428	1	4	yes	—
186	C₄	#15689439	1	2	yes	—
187	C₄	#15689462	1	4	yes	—
188	C₂	#20242987	1	1	yes	characteristic
189	C₂	#20243019	1	1	yes	characteristic
190	C₂	#20243043	1	2	yes	—
191	C₂	#20244921	1	1	yes	characteristic
192	C₂	#20244938	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20244350

C₂

yes

maximal

#20243545

C₂

yes

maximal, characteristic

#20243515

C₂

yes

maximal

C₄₇⋊#1273

C₂

yes

maximal, characteristic

C₄₇×#1956

C₂

yes

maximal, characteristic

C₂×#13093956

—

C₂×C₄×C₄₇⋊C₈

C₂²

yes

—

#15688993

C₂²

yes

—

#15688885

—

#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

—

#13093956

—

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

SD₁₆

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

—

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×SD₁₆

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

—

C₈×C₄₇

—

C₄₇⋊C₈

—

—

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

—

C₂²×C₄₇

#496

yes

characteristic

C₂²×C₄₇

#497

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#476

yes

—

C₄×C₄₇

#479

yes

—

C₄×C₄₇

#481

yes

—

C₄×C₄₇

#485

yes

—

Dic₄₇

—

#1990

—

#1956

D₄₇

yes

—

—

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

—

100

C₂×C₄₇

#2094

yes

characteristic

101

C₂×C₄₇

#2097

yes

—

102

C₂×C₄⋊Q₈

C₂×D₄₇

yes

characteristic

103

C₂×#106

—

104

C₂×#103

—

105

C₂×C₄×C₈

—

106

C₂×C₄×C₈

C₂×D₄₇

yes

characteristic

107

#384

—

108

#384

C₂×D₄₇

yes

characteristic

109

#340

—

110

#340

—

111

C₄₇

#10686

yes

characteristic, complemented, Sylow

112

C₂²×Q₈

—

113

C₂²×C₈

—

114

C₂²×C₈

C₂²⋊D₄₇

yes

—

115

C₄⋊Q₈

—

116

C₄⋊Q₈

—

117

C₄⋊Q₈

Dic₉₄

yes

—

118

C₂×C₄⋊C₄

—

119

C₂×C₄⋊C₄

—

120

C₂×C₄⋊C₄

—

121

C₂×C₄⋊C₄

—

122

C₂×C₄⋊C₄

—

123

C₂×C₄⋊C₄

—

124

C₂×C₄²

C₂²×D₄₇

yes

characteristic

125

#106

—

188

—

126

#103

—

127

C₄×C₈

—

128

C₄×C₈

—

129

C₄×C₈

—

130

C₂×Q₈

—

131

C₂×Q₈

—

132

C₂×Q₈

—

133

C₂×Q₈

—

134

C₂²×C₄

—

135

C₂²×C₄

—

136

C₂²×C₄

—

188

—

137

C₂²×C₄

D₄×D₄₇

yes

—

138

C₂²×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

139

C₂×C₈

—

140

C₂×C₈

—

141

C₂×C₈

—

142

C₂×C₈

—

143

C₂×C₈

—

144

C₂×C₈

#10625335

yes

—

145

C₄⋊C₄

—

146

C₄⋊C₄

—

147

C₄⋊C₄

—

148

C₄⋊C₄

—

149

C₄⋊C₄

—

188

—

150

C₄⋊C₄

—

188

—

151

C₄⋊C₄

—

188

—

152

C₄²

C₂×Dic₉₄

yes

characteristic

153

C₄²

#10625350

yes

characteristic

154

C₄²

#10625352

yes

—

155

C₂³

#13094020

yes

characteristic

156

Q₈

—

157

Q₈

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

376

—

164

C₂×C₄

—

376

—

165

C₂×C₄

—

376

—

166

C₂×C₄

#13094001

yes

—

167

C₂×C₄

#13094003

yes

characteristic

168

C₂×C₄

#13094005

yes

characteristic

169

C₂×C₄

#13094006

yes

—

170

C₂×C₄

C₂×#10625335

yes

—

171

C₂×C₄

#13094045

yes

—

172

C₂×C₄

#13094049

yes

—

173

C₂×C₄

#13094056

yes

—

174

C₈

—

175

C₈

—

188

—

176

C₂²

#15689059

yes

characteristic

177

C₂²

#15689495

yes

characteristic

178

C₂²

#15689504

yes

—

179

C₂²

#15689513

yes

characteristic

180

C₂²

#15689516

yes

—

181

C₄

—

182

C₄

—

183

C₄

—

752

—

184

C₄

#15689413

yes

—

185

C₄

#15689428

yes

—

186

C₄

#15689439

yes

—

187

C₄

#15689462

yes

—

188

C₂

#20242987

yes

characteristic

189

C₂

#20243019

yes

characteristic

190

C₂

#20243043

yes

—

191

C₂

#20244921

yes

characteristic

192

C₂

#20244938

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	#10686
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
111. C₄₇	#10686	semidirect

H ≅ G/N

type

111. C₄₇

#10686

semidirect

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

subgroup	index
#20244350	2
#20243545	2
#20243515	2
C₄₇⋊#1273	2
C₄₇×#1956	2
#10686	47

subgroup

index

#20244350

#20243545

#20243515

C₄₇⋊#1273

C₄₇×#1956

#10686

Subgroups — #26121398

Conjugacy classes of subgroups

Complements

Maximal subgroups