Group #26115770 — subgroups

Subgroups — #26115770

Order 12032.

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

normal subgroups only

This group has 506 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	#20243526	C₂	1	1	yes	maximal, characteristic, complemented
2	#20242060	C₂	1	2	yes	maximal, complemented
3	#20242056	C₂	1	2	yes	maximal
4	#20241680	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₄₇×#492	C₂	1	1	yes	maximal, characteristic
6	#15689176	C₂²	1	2	yes	—
7	#15689147	C₂²	1	2	yes	—
8	#15688888	—	2	2	no	—
9	#15688869	—	2	4	no	—
10	#15688867	—	2	4	no	—
11	#15688845	—	2	2	no	—
12	C₂²×SD₁₆×C₄₇	—	2	2	no	—
13	C₂×C₄⋊Q₈×C₄₇	C₂²	1	1	yes	characteristic
14	C₂×#127×C₄₇	C₂²	1	1	yes	characteristic
15	C₄₇×#492	—	2	2	no	—
16	C₄₇×#492	C₄	1	4	yes	—
17	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
18	C₂×#10625329	—	2	2	no	—
19	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
20	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
21	C₄⋊(C₄₇⋊C₈)	D₄	1	4	yes	—
22	C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
23	C₄×C₄₇⋊C₈	—	2	2	no	—
24	C₂²×Q₈×C₄₇	—	2	1	no	—
25	C₂²×D₄×C₄₇	—	2	1	no	—
26	C₂²×D₄×C₄₇	—	2	2	no	—
27	C₂×SD₁₆×C₄₇	—	2	8	no	—
28	C₂×SD₁₆×C₄₇	—	4	8	no	—
29	C₂²×C₈×C₄₇	D₄	1	2	yes	—
30	C₄⋊Q₈×C₄₇	—	2	2	no	—
31	C₄⋊Q₈×C₄₇	—	2	1	no	—
32	C₄⋊Q₈×C₄₇	C₂×C₄	1	2	yes	—
33	#127×C₄₇	—	2	1	no	—
34	#127×C₄₇	—	2	2	no	—
35	#127×C₄₇	C₂×C₄	1	2	yes	—
36	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
37	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
38	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
39	C₄×C₈×C₄₇	D₄	1	2	yes	—
40	C₂⁴×C₄₇	—	4	1	no	—
41	C₂×Q₈×C₄₇	—	2	2	no	—
42	C₂×Q₈×C₄₇	—	4	1	no	—
43	C₂×Q₈×C₄₇	—	4	4	no	—
44	C₂×D₄×C₄₇	—	2	2	no	—
45	C₂×D₄×C₄₇	—	2	4	no	—
46	C₂×D₄×C₄₇	—	4	2	no	—
47	C₂×D₄×C₄₇	—	4	1	no	—
48	C₂×D₄×C₄₇	—	4	4	no	—
49	C₂×D₄×C₄₇	—	8	4	no	—
50	C₂²×C₄×C₄₇	—	4	1	no	—
51	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
52	C₂²×C₄×C₄₇	#41	1	2	yes	—
53	SD₁₆×C₄₇	—	4	16	no	—
54	C₂×C₈×C₄₇	—	2	4	no	—
55	C₂×C₈×C₄₇	C₂²⋊C₄	1	4	yes	—
56	C₄⋊C₄×C₄₇	—	2	4	no	—
57	C₄⋊C₄×C₄₇	—	4	2	no	—
58	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
59	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
60	C₄²×C₄₇	C₂×D₄	1	2	yes	—
61	#10625329	—	4	4	no	—
62	C₂×C₄₇⋊C₈	—	2	4	no	—
63	C₂×C₄₇⋊C₈	—	4	2	no	—
64	C₂³×C₄₇	—	4	2	no	—
65	C₂³×C₄₇	—	8	4	no	—
66	C₂³×C₄₇	—	8	2	no	—
67	C₂³×C₄₇	#124	1	1	yes	characteristic
68	Q₈×C₄₇	—	4	4	no	—
69	Q₈×C₄₇	—	8	2	no	—
70	D₄×C₄₇	—	4	4	no	—
71	D₄×C₄₇	—	8	4	no	—
72	D₄×C₄₇	—	8	4	no	—
73	D₄×C₄₇	—	8	2	no	—
74	C₂×C₄×C₄₇	—	2	4	no	—
75	C₂×C₄×C₄₇	—	4	2	no	—
76	C₂×C₄×C₄₇	—	8	2	no	—
77	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
79	C₂×C₄×C₄₇	#121	1	2	yes	—
80	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
81	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	4	yes	—
82	C₈×C₄₇	—	2	4	no	—
83	C₄₇⋊C₈	—	4	4	no	—
84	C₄₇⋊C₈	—	8	2	no	—
85	#9777	—	47	1	no	maximal, Sylow
86	C₂²×C₄₇	—	2	2	no	—
87	C₂²×C₄₇	—	8	2	no	—
88	C₂²×C₄₇	—	8	4	no	—
89	C₂²×C₄₇	—	16	4	no	—
90	C₂²×C₄₇	#390	1	1	yes	characteristic
91	C₂²×C₄₇	#490	1	2	yes	—
92	C₄×C₄₇	—	2	2	no	—
93	C₄×C₄₇	—	8	2	no	—
94	C₄×C₄₇	#418	1	2	yes	—
95	C₄×C₄₇	#471	1	4	yes	—
96	C₄×C₄₇	#496	1	2	yes	—
97	C₂×#492	D₄₇	1	1	yes	characteristic
98	#1837	—	47	1	no	—
99	#1546	—	47	2	no	—
100	#1545	—	47	2	no	—
101	#1452	—	47	1	no	—
102	C₂×C₄₇	—	2	2	no	—
103	C₂×C₄₇	—	16	2	no	—
104	C₂×C₄₇	#1961	1	1	yes	characteristic
105	C₂×C₄₇	#1977	1	2	yes	—
106	C₂²×SD₁₆	—	2	2	no	—
107	C₂×C₄⋊Q₈	C₂×D₄₇	1	1	yes	characteristic
108	C₂×#127	C₂×D₄₇	1	1	yes	characteristic
109	#492	—	2	2	no	—
110	#492	Dic₄₇	1	4	yes	—
111	#431	—	47	2	no	—
112	#423	—	47	2	no	—
113	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
114	#341	—	94	2	no	—
115	#332	—	94	4	no	—
116	#331	—	94	4	no	—
117	#322	—	94	2	no	—
118	C₄₇	#9777	1	1	yes	characteristic, complemented, Sylow
119	C₂²×Q₈	—	2	1	no	—
120	C₂²×D₄	—	2	2	no	—
121	C₂²×D₄	—	2	1	no	—
122	C₂×SD₁₆	—	2	8	no	—
123	C₂×SD₁₆	—	4	8	no	—
124	C₂×M₁₆	—	94	2	no	—
125	C₂²×C₈	C₂²⋊D₄₇	1	2	yes	—
126	C₄⋊Q₈	—	2	2	no	—
127	C₄⋊Q₈	—	2	1	no	—
128	C₄⋊Q₈	C₂×Dic₄₇	1	2	yes	—
129	#127	—	2	2	no	—
130	#127	—	2	1	no	—
131	#127	C₂×Dic₄₇	1	2	yes	—
132	C₂×C₄⋊C₄	—	2	2	no	—
133	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
134	C₄⋊C₈	—	47	4	no	—
135	C₄⋊C₈	—	94	2	no	—
136	C₄⋊C₈	—	94	2	no	—
137	C₂²⋊C₈	—	94	2	no	—
138	C₄×C₈	—	94	2	no	—
139	C₄×C₈	C₂×Dic₄₇	1	2	yes	—
140	C₄×C₈	C₂²⋊D₄₇	1	2	yes	—
141	C₂⁴	—	4	1	no	—
142	C₂×Q₈	—	2	2	no	—
143	C₂×Q₈	—	4	1	no	—
144	C₂×Q₈	—	4	4	no	—
145	C₂×D₄	—	2	2	no	—
146	C₂×D₄	—	2	4	no	—
147	C₂×D₄	—	4	2	no	—
148	C₂×D₄	—	4	1	no	—
149	C₂×D₄	—	4	4	no	—
150	C₂×D₄	—	8	4	no	—
151	C₂²×C₄	—	4	1	no	—
152	C₂²×C₄	#10625352	1	2	yes	—
153	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
154	SD₁₆	—	4	16	no	—
155	M₁₆	—	188	4	no	—
156	C₂×C₈	—	2	4	no	—
157	C₂×C₈	—	94	4	no	—
158	C₂×C₈	—	188	2	no	—
159	C₂×C₈	C₂²⋊Dic₄₇	1	4	yes	—
160	C₄⋊C₄	—	2	4	no	—
161	C₄⋊C₄	—	4	2	no	—
162	C₄²	D₄×D₄₇	1	2	yes	—
163	C₄²	C₂²×Dic₄₇	1	1	yes	characteristic
164	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
165	C₂³	—	4	2	no	—
166	C₂³	—	8	4	no	—
167	C₂³	—	8	2	no	—
168	C₂³	#13094011	1	1	yes	characteristic
169	Q₈	—	4	4	no	—
170	Q₈	—	8	2	no	—
171	D₄	—	4	4	no	—
172	D₄	—	8	4	no	—
173	D₄	—	8	2	no	—
174	D₄	—	8	4	no	—
175	C₂×C₄	—	2	4	no	—
176	C₂×C₄	—	4	2	no	—
177	C₂×C₄	—	8	2	no	—
178	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
179	C₂×C₄	D₄×Dic₄₇	1	2	yes	—
180	C₂×C₄	#13093995	1	2	yes	—
181	C₂×C₄	#13094020	1	1	yes	characteristic
182	C₂×C₄	#13094050	1	4	yes	—
183	C₈	—	2	4	no	—
184	C₈	—	376	2	no	—
185	C₈	—	188	4	no	—
186	C₂²	—	2	2	no	—
187	C₂²	—	8	4	no	—
188	C₂²	—	8	2	no	—
189	C₂²	—	16	4	no	—
190	C₂²	#15689026	1	1	yes	characteristic
191	C₂²	#15689487	1	2	yes	—
192	C₄	—	2	2	no	—
193	C₄	—	8	2	no	—
194	C₄	#15689128	1	2	yes	—
195	C₄	#15689383	1	4	yes	—
196	C₄	#15689510	1	2	yes	—
197	C₂	—	2	2	no	—
198	C₂	—	16	2	no	—
199	C₂	#20244213	1	1	yes	characteristic
200	C₂	#20244293	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20243526

C₂

yes

maximal, characteristic, complemented

#20242060

C₂

yes

maximal, complemented

#20242056

C₂

yes

maximal

#20241680

C₂

yes

maximal, characteristic, complemented

C₂×C₄₇×#492

C₂

yes

maximal, characteristic

#15689176

C₂²

yes

—

#15689147

C₂²

yes

—

—

—

—

—

C₂²×SD₁₆×C₄₇

—

C₂×C₄⋊Q₈×C₄₇

C₂²

yes

characteristic

C₂×#127×C₄₇

C₂²

yes

characteristic

C₄₇×#492

—

C₄₇×#492

C₄

yes

—

C₂×C₄×C₈×C₄₇

C₂²

yes

characteristic

C₂×#10625329

—

C₄⋊(C₄₇⋊C₈)

—

C₄⋊(C₄₇⋊C₈)

—

C₄⋊(C₄₇⋊C₈)

D₄

yes

—

C₂²⋊(C₄₇⋊C₈)

—

C₄×C₄₇⋊C₈

—

C₂²×Q₈×C₄₇

—

C₂²×D₄×C₄₇

—

C₂²×D₄×C₄₇

—

C₂×SD₁₆×C₄₇

—

C₂×SD₁₆×C₄₇

—

C₂²×C₈×C₄₇

D₄

yes

—

C₄⋊Q₈×C₄₇

—

C₄⋊Q₈×C₄₇

—

C₄⋊Q₈×C₄₇

C₂×C₄

yes

—

#127×C₄₇

—

#127×C₄₇

—

#127×C₄₇

C₂×C₄

yes

—

C₂×C₄⋊C₄×C₄₇

—

C₂×C₄²×C₄₇

C₂³

yes

characteristic

C₄×C₈×C₄₇

C₂×C₄

yes

—

C₄×C₈×C₄₇

D₄

yes

—

C₂⁴×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₄₇

#41

yes

—

SD₁₆×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

C₂²⋊C₄

yes

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄²×C₄₇

C₂²×C₄

yes

characteristic

C₄²×C₄₇

C₂×D₄

yes

characteristic

C₄²×C₄₇

C₂×D₄

yes

—

#10625329

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

#124

yes

characteristic

Q₈×C₄₇

—

Q₈×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₇

C₄×D₄

yes

—

C₂×C₄×C₄₇

#121

yes

—

C₂×C₄×C₄₇

#127

yes

characteristic

C₂×C₄×C₄₇

AGL(1,ℤ/8)

yes

—

C₈×C₄₇

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

#9777

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

#390

yes

characteristic

C₂²×C₄₇

#490

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#418

yes

—

C₄×C₄₇

#471

yes

—

C₄×C₄₇

#496

yes

—

C₂×#492

D₄₇

yes

characteristic

#1837

—

#1546

—

100

#1545

—

101

#1452

—

102

C₂×C₄₇

—

103

C₂×C₄₇

—

104

C₂×C₄₇

#1961

yes

characteristic

105

C₂×C₄₇

#1977

yes

—

106

C₂²×SD₁₆

—

107

C₂×C₄⋊Q₈

C₂×D₄₇

yes

characteristic

108

C₂×#127

C₂×D₄₇

yes

characteristic

109

#492

—

110

#492

Dic₄₇

yes

—

111

#431

—

112

#423

—

113

C₂×C₄×C₈

C₂×D₄₇

yes

characteristic

114

#341

—

115

#332

—

116

#331

—

117

#322

—

118

C₄₇

#9777

yes

characteristic, complemented, Sylow

119

C₂²×Q₈

—

120

C₂²×D₄

—

121

C₂²×D₄

—

122

C₂×SD₁₆

—

123

C₂×SD₁₆

—

124

C₂×M₁₆

—

125

C₂²×C₈

C₂²⋊D₄₇

yes

—

126

C₄⋊Q₈

—

127

C₄⋊Q₈

—

128

C₄⋊Q₈

C₂×Dic₄₇

yes

—

129

#127

—

130

#127

—

131

#127

C₂×Dic₄₇

yes

—

132

C₂×C₄⋊C₄

—

133

C₂×C₄²

C₂²×D₄₇

yes

characteristic

134

C₄⋊C₈

—

135

C₄⋊C₈

—

136

C₄⋊C₈

—

137

C₂²⋊C₈

—

138

C₄×C₈

—

139

C₄×C₈

C₂×Dic₄₇

yes

—

140

C₄×C₈

C₂²⋊D₄₇

yes

—

141

C₂⁴

—

142

C₂×Q₈

—

143

C₂×Q₈

—

144

C₂×Q₈

—

145

C₂×D₄

—

146

C₂×D₄

—

147

C₂×D₄

—

148

C₂×D₄

—

149

C₂×D₄

—

150

C₂×D₄

—

151

C₂²×C₄

—

152

C₂²×C₄

#10625352

yes

—

153

C₂²×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

154

SD₁₆

—

155

M₁₆

—

188

—

156

C₂×C₈

—

157

C₂×C₈

—

158

C₂×C₈

—

188

—

159

C₂×C₈

C₂²⋊Dic₄₇

yes

—

160

C₄⋊C₄

—

161

C₄⋊C₄

—

162

C₄²

D₄×D₄₇

yes

—

163

C₄²

C₂²×Dic₄₇

yes

characteristic

164

C₄²

C₂×C₂²⋊D₄₇

yes

characteristic

165

C₂³

—

166

C₂³

—

167

C₂³

—

168

C₂³

#13094011

yes

characteristic

169

Q₈

—

170

Q₈

—

171

D₄

—

172

D₄

—

173

D₄

—

174

D₄

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

C₂×C₂²⋊Dic₄₇

yes

characteristic

179

C₂×C₄

D₄×Dic₄₇

yes

—

180

C₂×C₄

#13093995

yes

—

181

C₂×C₄

#13094020

yes

characteristic

182

C₂×C₄

#13094050

yes

—

183

C₈

—

184

C₈

—

376

—

185

C₈

—

188

—

186

C₂²

—

187

C₂²

—

188

C₂²

—

189

C₂²

—

190

C₂²

#15689026

yes

characteristic

191

C₂²

#15689487

yes

—

192

C₄

—

193

C₄

—

194

C₄

#15689128

yes

—

195

C₄

#15689383

yes

—

196

C₄

#15689510

yes

—

197

C₂

—

198

C₂

—

199

C₂

#20244213

yes

characteristic

200

C₂

#20244293

yes

—

	subgroup	G/H
center	C₂²	#15689026
commutator	C₄²×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₂×C₄₇×#492	C₂
socle	C₂²×C₄₇	#390
prime radical	C₂×#127×C₄₇	C₂²
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	C₂×#492	D₄₇
hypocenter	C₄₇	#9777
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#492

subgroup

G/H

center

C₂²

#15689026

commutator

C₄²×C₄₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₄₇

Fitting

C₂×C₄₇×#492

C₂

socle

C₂²×C₄₇

#390

prime radical

C₂×#127×C₄₇

C₂²

cosocle

C₂×C₄²×C₄₇

C₂³

D₄₇

C₄₇

G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#492

N	H ≅ G/N	type
1. #20243526	C₂	semidirect
2. #20242060	C₂	semidirect
4. #20241680	C₂	semidirect
118. C₄₇	#9777	semidirect

H ≅ G/N

type

1. #20243526

C₂

semidirect

2. #20242060

C₂

semidirect

4. #20241680

C₂

semidirect

118. C₄₇

#9777

semidirect

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

subgroup	index
#20243526	2
#20242060	2
#20242056	2
#20241680	2
C₂×C₄₇×#492	2
#9777	47

subgroup

index

C₂×C₄₇×#492

Subgroups — #26115770

Conjugacy classes of subgroups

Complements

Maximal subgroups