Group #26134387 — subgroups

Subgroups — #26134387

Order 12032.

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

normal subgroups only

This group has 294 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	#20245174	C₂	1	2	yes	maximal
2	C₂×#15688944	C₂	1	1	yes	maximal, characteristic
3	#20243267	C₂	1	1	yes	maximal, characteristic
4	#20241496	C₂	1	2	yes	maximal
5	C₄₇×#1846	C₂	1	1	yes	maximal, characteristic
6	C₂×#13093964	C₂²	1	2	yes	—
7	C₂×#13093960	C₂²	1	1	yes	characteristic
8	#15689161	C₂²	1	1	yes	characteristic
9	#15689078	—	2	1	no	—
10	#15688950	—	2	2	no	—
11	#15688948	—	2	2	no	—
12	#15688944	C₄	1	4	yes	complemented
13	#15688883	—	2	1	no	—
14	C₂×#107×C₄₇	C₂²	1	2	yes	—
15	C₂×#106×C₄₇	—	2	1	no	—
16	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
17	C₄₇×#382	—	2	1	no	—
18	C₂×#10625329	—	2	1	no	—
19	#13093983	—	2	1	no	—
20	#13093964	—	2	2	no	—
21	#13093964	C₂×C₄	1	4	yes	—
22	C₂×C₄₇⋊C₁₆	—	2	2	no	—
23	#13093960	—	2	1	no	—
24	#13093960	Q₈	1	2	yes	—
25	#13093958	D₄	1	2	yes	—
26	#13093956	—	2	1	no	—
27	#13093934	—	4	1	no	—
28	C₈×Dic₄₇	—	4	1	no	—
29	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
30	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
31	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
32	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
33	#107×C₄₇	—	2	2	no	—
34	#107×C₄₇	—	2	2	no	—
35	#106×C₄₇	—	4	1	no	—
36	#106×C₄₇	—	4	1	no	—
37	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
38	C₄×C₈×C₄₇	Q₈	1	2	yes	—
39	#95×C₄₇	—	4	2	no	—
40	C₂²×C₄×C₄₇	—	2	1	no	—
41	C₂²×C₄×C₄₇	—	4	2	no	—
42	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
43	C₂×C₈×C₄₇	C₄⋊C₄	1	2	yes	—
44	C₂×C₈×C₄₇	D₈	1	1	yes	characteristic
45	C₂×C₈×C₄₇	D₈	1	1	yes	characteristic
46	C₂×C₈×C₄₇	Q₁₆	1	2	yes	—
47	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
49	C₂×C₈×C₄₇	C₂×Q₈	1	1	yes	characteristic
50	C₂×C₈×C₄₇	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
52	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
53	C₄⋊C₄×C₄₇	—	4	2	no	—
54	C₄⋊C₄×C₄₇	—	4	2	no	—
55	C₄²×C₄₇	—	2	1	no	—
56	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
57	C₂²⋊Dic₄₇	—	4	1	no	—
58	C₄⋊Dic₄₇	—	2	2	no	—
59	C₄×Dic₄₇	—	4	1	no	—
60	#10625329	—	2	2	no	—
61	#10625329	—	4	1	no	—
62	C₂×C₄₇⋊C₈	—	4	1	no	—
63	C₄₇⋊C₁₆	—	2	4	no	—
64	C₂³×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
65	C₂×C₄×C₄₇	—	2	2	no	—
66	C₂×C₄×C₄₇	—	4	2	no	—
67	C₂×C₄×C₄₇	—	8	4	no	—
68	C₂×C₄×C₄₇	—	8	2	no	—
69	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
70	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
71	C₂×C₄×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
72	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
73	C₂×C₄×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
74	C₂×C₄×C₄₇	#137	1	1	yes	characteristic
75	C₂×Dic₄₇	—	4	2	no	—
76	C₈×C₄₇	#107	1	2	yes	—
77	C₈×C₄₇	#107	1	2	yes	—
78	C₈×C₄₇	C₄×D₄	1	1	yes	characteristic
79	C₈×C₄₇	C₄×Q₈	1	1	yes	characteristic
80	C₈×C₄₇	#125	1	1	yes	characteristic
81	C₈×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
82	C₄₇⋊C₈	—	4	2	no	—
83	#13180	—	47	1	no	maximal, Sylow
84	C₂²×C₄₇	—	2	2	no	—
85	C₂²×C₄₇	#428	1	1	yes	characteristic
86	C₂²×C₄₇	#459	1	1	yes	characteristic
87	C₂²×C₄₇	#477	1	1	yes	characteristic
88	C₄×C₄₇	—	4	2	no	—
89	C₄×C₄₇	—	16	2	no	—
90	C₄×C₄₇	#389	1	1	yes	characteristic
91	C₄×C₄₇	C₂×#107	1	1	yes	characteristic
92	C₄×C₄₇	#462	1	1	yes	characteristic
93	C₄×C₄₇	#474	1	1	yes	characteristic
94	Dic₄₇	—	8	2	no	—
95	#2159	—	47	2	no	—
96	C₂×#364	—	47	1	no	—
97	#1846	D₄₇	1	1	yes	characteristic
98	#1772	—	47	1	no	—
99	#1387	—	47	2	no	—
100	C₂×C₄₇	—	2	1	no	—
101	C₂×C₄₇	—	2	1	no	—
102	C₂×C₄₇	#1947	1	1	yes	characteristic
103	C₂×C₄₇	#2212	1	2	yes	—
104	C₂×M₃₂	—	47	2	no	—
105	C₂×#108	—	47	1	no	—
106	#427	—	47	1	no	—
107	C₂×#107	C₂×D₄₇	1	2	yes	—
108	C₂×#106	—	2	1	no	—
109	#405	—	94	1	no	—
110	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
111	#382	—	2	1	no	—
112	#367	—	94	2	no	—
113	#366	—	94	2	no	—
114	#364	—	47	4	no	—
115	#339	—	94	1	no	—
116	C₄₇	#13180	1	1	yes	characteristic, complemented, Sylow
117	C₂×M₁₆	—	94	1	no	—
118	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
119	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
120	#117	—	94	1	no	—
121	C₂×C₄⋊C₄	—	2	2	no	—
122	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
123	M₃₂	—	47	4	no	—
124	M₃₂	—	94	2	no	—
125	C₂×C₁₆	—	94	2	no	—
126	#108	—	47	2	no	—
127	#108	—	94	1	no	—
128	#107	—	2	2	no	—
129	#107	—	2	2	no	—
130	#107	—	47	2	no	—
131	#106	—	4	1	no	—
132	#106	—	4	1	no	—
133	#106	—	94	1	no	—
134	#97	—	188	1	no	—
135	C₄×C₈	—	188	1	no	—
136	C₄×C₈	Dic₉₄	1	2	yes	—
137	C₄×C₈	C₄×D₄₇	1	2	yes	—
138	#95	—	4	2	no	—
139	C₂²×C₄	—	2	1	no	—
140	C₂²×C₄	—	4	2	no	—
141	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
142	M₁₆	—	94	2	no	—
143	M₁₆	—	188	1	no	—
144	C₂×C₈	—	188	1	no	—
145	C₂×C₈	Dic₄₇⋊C₄	1	2	yes	—
146	C₂×C₈	#10625334	1	1	yes	characteristic
147	C₂×C₈	#10625334	1	1	yes	characteristic
148	C₂×C₈	#10625337	1	2	yes	—
149	C₂×C₈	C₂×Dic₉₄	1	1	yes	characteristic
150	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
151	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
152	C₂×C₈	#10625352	1	1	yes	characteristic
153	C₂×C₈	Q₈×D₄₇	1	1	yes	characteristic
154	C₂×C₈	#10625354	1	1	yes	characteristic
155	C₄⋊C₄	—	4	2	no	—
156	C₄⋊C₄	—	4	2	no	—
157	C₄⋊C₄	—	94	2	no	—
158	C₂²⋊C₄	—	188	1	no	—
159	C₄²	—	2	1	no	—
160	C₄²	—	188	1	no	—
161	C₄²	Dic₄₇⋊C₄	1	2	yes	—
162	C₁₆	—	94	4	no	—
163	C₂³	#13094050	1	1	yes	characteristic
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	4	2	no	—
166	C₂×C₄	—	8	4	no	—
167	C₂×C₄	—	8	2	no	—
168	C₂×C₄	—	188	2	no	—
169	C₂×C₄	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
170	C₂×C₄	#13093995	1	1	yes	characteristic
171	C₂×C₄	#13094000	1	1	yes	characteristic
172	C₂×C₄	C₂×#10625334	1	1	yes	characteristic
173	C₂×C₄	C₂×#10625337	1	1	yes	characteristic
174	C₂×C₄	#13094055	1	1	yes	characteristic
175	C₈	—	188	2	no	—
176	C₈	#13093957	1	2	yes	—
177	C₈	#13093957	1	2	yes	—
178	C₈	#13093986	1	1	yes	characteristic
179	C₈	#13093989	1	1	yes	characteristic
180	C₈	#13094015	1	1	yes	characteristic
181	C₈	#13094023	1	1	yes	characteristic
182	C₂²	—	2	2	no	—
183	C₂²	#15689163	1	1	yes	characteristic
184	C₂²	#15689315	1	1	yes	characteristic
185	C₂²	#15689415	1	1	yes	characteristic
186	C₄	—	4	2	no	—
187	C₄	—	16	2	no	—
188	C₄	—	376	2	no	—
189	C₄	#15689019	1	1	yes	characteristic
190	C₄	C₂×#13093957	1	1	yes	characteristic
191	C₄	#15689330	1	1	yes	characteristic
192	C₄	#15689400	1	1	yes	characteristic
193	C₂	—	2	1	no	—
194	C₂	—	2	1	no	—
195	C₂	#20244144	1	1	yes	characteristic
196	C₂	#20245454	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20245174

C₂

yes

maximal

C₂×#15688944

C₂

yes

maximal, characteristic

#20243267

C₂

yes

maximal, characteristic

#20241496

C₂

yes

maximal

C₄₇×#1846

C₂

yes

maximal, characteristic

C₂×#13093964

C₂²

yes

—

C₂×#13093960

C₂²

yes

characteristic

#15689161

C₂²

yes

—

—

—

C₄

yes

complemented

#15688883

—

C₂×#107×C₄₇

C₂²

yes

—

C₂×#106×C₄₇

—

C₂×C₄×C₈×C₄₇

C₂²

yes

characteristic

C₄₇×#382

—

—

—

—

C₂×C₄

yes

—

C₂×C₄₇⋊C₁₆

—

#13093960

—

#13093960

Q₈

yes

—

#13093958

D₄

yes

—

#13093956

—

#13093934

—

C₈×Dic₄₇

—

C₂²×C₈×C₄₇

D₄

yes

characteristic

C₂²×C₈×C₄₇

C₂³

yes

characteristic

C₂×C₄⋊C₄×C₄₇

—

C₂×C₄²×C₄₇

D₄

yes

characteristic

#107×C₄₇

—

#107×C₄₇

—

#106×C₄₇

—

#106×C₄₇

—

C₄×C₈×C₄₇

C₂×C₄

yes

—

C₄×C₈×C₄₇

Q₈

yes

—

#95×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₈×C₄₇

C₄⋊C₄

yes

—

C₂×C₈×C₄₇

D₈

yes

characteristic

C₂×C₈×C₄₇

D₈

yes

characteristic

C₂×C₈×C₄₇

Q₁₆

yes

—

C₂×C₈×C₄₇

C₂²×C₄

yes

characteristic

C₂×C₈×C₄₇

C₂×D₄

yes

characteristic

C₂×C₈×C₄₇

C₂×Q₈

yes

characteristic

C₂×C₈×C₄₇

C₂×Q₈

yes

C₂×C₈×C₄₇

yes

C₂×C₈×C₄₇

yes

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

C₄⋊C₄

yes

—

C₂²⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄×Dic₄₇

—

#10625329

—

#10625329

—

C₂×C₄₇⋊C₈

—

C₄₇⋊C₁₆

—

C₂³×C₄₇

AGL(1,ℤ/8)

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₄₇

#121

yes

characteristic

C₂×C₄×C₄₇

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₄₇

C₂×D₈

yes

characteristic

C₂×C₄×C₄₇

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₄₇

#137

yes

characteristic

C₂×Dic₄₇

—

C₈×C₄₇

#107

yes

—

C₈×C₄₇

#107

yes

—

C₈×C₄₇

C₄×D₄

yes

characteristic

C₈×C₄₇

C₄×Q₈

yes

characteristic

C₈×C₄₇

#125

yes

characteristic

C₈×C₄₇

C₄⋊Q₈

yes

characteristic

C₄₇⋊C₈

—

#13180

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

Dic₄₇

—

#2159

—

C₂×#364

—

#1846

D₄₇

yes

characteristic

#1772

—

#1387

—

100

C₂×C₄₇

—

101

C₂×C₄₇

—

102

C₂×C₄₇

#1947

yes

characteristic

103

C₂×C₄₇

#2212

yes

—

104

C₂×M₃₂

—

105

C₂×#108

—

106

#427

—

107

C₂×#107

C₂×D₄₇

yes

—

108

C₂×#106

—

109

#405

—

110

C₂×C₄×C₈

C₂×D₄₇

yes

characteristic

111

#382

—

112

#367

—

113

#366

—

114

#364

—

115

#339

—

116

C₄₇

#13180

yes

characteristic, complemented, Sylow

117

C₂×M₁₆

—

118

C₂²×C₈

C₂²⋊D₄₇

yes

characteristic

119

C₂²×C₈

C₂²×D₄₇

yes

characteristic

120

#117

—

121

C₂×C₄⋊C₄

—

122

C₂×C₄²

C₂²⋊D₄₇

yes

characteristic

123

M₃₂

—

124

M₃₂

—

125

C₂×C₁₆

—

126

#108

—

127

#108

—

128

#107

—

129

#107

—

130

#107

—

131

#106

—

132

#106

—

133

#106

—

134

#97

—

188

—

135

C₄×C₈

—

188

—

136

C₄×C₈

Dic₉₄

yes

—

137

C₄×C₈

C₄×D₄₇

yes

—

138

#95

—

139

C₂²×C₄

—

140

C₂²×C₄

—

141

C₂²×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

142

M₁₆

—

143

M₁₆

—

188

—

144

C₂×C₈

—

188

—

145

C₂×C₈

Dic₄₇⋊C₄

yes

—

146

C₂×C₈

#10625334

yes

characteristic

147

C₂×C₈

#10625334

yes

characteristic

148

C₂×C₈

#10625337

yes

—

149

C₂×C₈

C₂×Dic₉₄

yes

characteristic

150

C₂×C₈

C₂×C₄×D₄₇

yes

characteristic

151

C₂×C₈

D₄×D₄₇

yes

characteristic

152

C₂×C₈

#10625352

yes

characteristic

153

C₂×C₈

Q₈×D₄₇

yes

characteristic

154

C₂×C₈

#10625354

yes

characteristic

155

C₄⋊C₄

—

156

C₄⋊C₄

—

157

C₄⋊C₄

—

158

C₂²⋊C₄

—

188

—

159

C₄²

—

160

C₄²

—

188

—

161

C₄²

Dic₄₇⋊C₄

yes

—

162

C₁₆

—

163

C₂³

#13094050

yes

characteristic

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

188

—

169

C₂×C₄

C₂×Dic₄₇⋊C₄

yes

characteristic

170

C₂×C₄

#13093995

yes

characteristic

171

C₂×C₄

#13094000

yes

characteristic

172

C₂×C₄

C₂×#10625334

yes

characteristic

173

C₂×C₄

C₂×#10625337

yes

characteristic

174

C₂×C₄

#13094055

yes

characteristic

175

C₈

—

188

—

176

C₈

#13093957

yes

—

177

C₈

#13093957

yes

—

178

C₈

#13093986

yes

characteristic

179

C₈

#13093989

yes

characteristic

180

C₈

#13094015

yes

characteristic

181

C₈

#13094023

yes

characteristic

182

C₂²

—

183

C₂²

#15689163

yes

characteristic

184

C₂²

#15689315

yes

characteristic

185

C₂²

#15689415

yes

characteristic

186

C₄

—

187

C₄

—

188

C₄

—

376

—

189

C₄

#15689019

yes

characteristic

190

C₄

C₂×#13093957

yes

characteristic

191

C₄

#15689330

yes

characteristic

192

C₄

#15689400

yes

characteristic

193

C₂

—

194

C₂

—

195

C₂

#20244144

yes

characteristic

196

C₂

#20245454

yes

—

	subgroup	G/H
center	C₂²	#15689315
commutator	C₂×C₈×C₄₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₄₇
Fitting	C₄₇×#1846	C₂
socle	C₂²×C₄₇	#459
prime radical	C₂³×C₄₇	AGL(1,ℤ/8)
cosocle	C₂²×C₈×C₄₇	C₂³
hypercenter	#1846	D₄₇
hypocenter	C₄₇	#13180
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1846

subgroup

G/H

center

C₂²

#15689315

commutator

C₂×C₈×C₄₇

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₄₇

Fitting

C₄₇×#1846

C₂

C₂²×C₄₇

C₂³×C₄₇

C₂²×C₈×C₄₇

C₂³

D₄₇

C₄₇

G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1846

N	H ≅ G/N	type
12. #15688944	C₄	semidirect
116. C₄₇	#13180	semidirect

H ≅ G/N

type

12. #15688944

C₄

semidirect

116. C₄₇

#13180

semidirect

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

subgroup	index
#20245174	2
C₂×#15688944	2
#20243267	2
#20241496	2
C₄₇×#1846	2
#13180	47

subgroup

index

C₄₇×#1846

Subgroups — #26134387

Conjugacy classes of subgroups

Complements

Maximal subgroups