Group #26372710 — subgroups

Subgroups — #26372710

Order 12032.

464 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 143 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	#20254047	C₂	1	2	yes	maximal
2	#20245553	C₂	1	4	yes	maximal, complemented
3	#20245496	C₂	1	2	yes	maximal
4	#20245479	C₂	1	2	yes	maximal
5	#20245439	C₂	1	4	yes	maximal, complemented
6	C₄₇×#2151	C₂	1	1	yes	maximal, characteristic
7	#15689735	—	2	2	no	—
8	C₂×Dic₃₇₆	C₂²	1	2	yes	—
9	C₂×Dic₃₇₆	C₂²	1	2	yes	—
10	#15689525	C₂²	1	2	yes	—
11	#15689503	C₂²	1	2	yes	—
12	#15689492	—	2	2	no	—
13	#15689381	—	2	2	no	—
14	#15689364	C₂²	1	4	yes	—
15	#15689231	—	2	2	no	—
16	C₄×Dic₁₈₈	C₂²	1	4	yes	—
17	#15688951	C₂²	1	2	yes	—
18	#15688949	C₂²	1	2	yes	—
19	#15688928	C₂²	1	4	yes	—
20	#15688928	C₂²	1	4	yes	—
21	C₂×M₃₂×C₄₇	C₂²	1	1	yes	characteristic
22	C₂²×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
23	C₄₇×#405	C₂²	1	1	yes	characteristic
24	C₄₇×C₄⋊C₁₆	C₂²	1	4	yes	—
25	C₂×Dic₁₈₈	—	2	2	no	—
26	C₂×Dic₁₈₈	C₂³	1	4	yes	—
27	#13094024	—	2	2	no	—
28	#13094017	—	4	4	no	—
29	#13094016	—	2	2	no	—
30	#13094014	—	4	4	no	—
31	#13094003	—	2	4	no	—
32	C₄×Dic₉₄	—	2	4	no	—
33	Dic₃₇₆	—	4	2	no	—
34	Dic₃₇₆	—	4	2	no	—
35	#13093958	C₂³	1	4	yes	—
36	#13093956	—	2	2	no	—
37	#13093949	—	2	4	no	—
38	#13093949	—	2	4	no	—
39	C₂×M₁₆×C₄₇	D₄	1	1	yes	characteristic
40	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
41	#117×C₄₇	D₄	1	1	yes	characteristic
42	M₃₂×C₄₇	—	4	1	no	—
43	C₂×C₁₆×C₄₇	—	4	1	no	—
44	C₂×C₁₆×C₄₇	C₂³	1	2	yes	—
45	C₂×C₁₆×C₄₇	C₂³	1	2	yes	—
46	#97×C₄₇	D₄	1	2	yes	—
47	C₄×C₈×C₄₇	C₂³	1	2	yes	—
48	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
49	C₂×Dic₉₄	—	2	4	no	—
50	M₁₆×C₄₇	—	4	1	no	—
51	C₂×C₈×C₄₇	—	2	1	no	—
52	C₂×C₈×C₄₇	—	2	1	no	—
53	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
54	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
55	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
56	C₄⋊C₄×C₄₇	D₈	1	2	yes	—
57	C₂²⋊C₄×C₄₇	D₈	1	2	yes	—
58	C₄²×C₄₇	C₂×D₄	1	2	yes	—
59	C₂²⋊Dic₄₇	—	4	4	no	—
60	C₄⋊Dic₄₇	—	2	4	no	—
61	C₄⋊Dic₄₇	—	4	4	no	—
62	Dic₄₇⋊C₄	—	4	4	no	—
63	Dic₄₇⋊C₄	—	4	4	no	—
64	Dic₄₇⋊C₄	—	4	4	no	—
65	C₄×Dic₄₇	—	4	4	no	—
66	Dic₁₈₈	—	2	4	no	—
67	Dic₁₈₈	—	2	4	no	—
68	Dic₁₈₈	—	4	4	no	—
69	C₁₆×C₄₇	—	2	2	no	—
70	C₁₆×C₄₇	—	2	2	no	—
71	C₂³×C₄₇	C₂×D₈	1	1	yes	characteristic
72	C₂×C₄×C₄₇	—	2	1	no	—
73	C₂×C₄×C₄₇	—	2	1	no	—
74	C₂×C₄×C₄₇	C₂×D₈	1	4	yes	—
75	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
76	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
77	C₂×Dic₄₇	—	4	4	no	—
78	C₂×Dic₄₇	—	4	4	no	—
79	Dic₉₄	—	4	4	no	—
80	Dic₉₄	—	4	4	no	—
81	C₈×C₄₇	—	2	1	no	—
82	C₈×C₄₇	—	2	2	no	—
83	C₈×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
84	C₈×C₄₇	2₋¹⁺⁴	1	1	yes	characteristic
85	#33827	—	47	1	no	maximal, Sylow
86	C₂²×C₄₇	—	2	2	no	—
87	C₂²×C₄₇	—	2	1	no	—
88	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
89	C₄×C₄₇	—	2	4	no	—
90	C₄×C₄₇	—	2	1	no	—
91	C₄×C₄₇	#537	1	1	yes	characteristic
92	C₄×C₄₇	#575	1	1	yes	characteristic
93	Dic₄₇	—	8	4	no	—
94	Dic₄₇	—	8	4	no	—
95	#3177	—	47	2	no	—
96	#2229	—	47	4	no	—
97	#2219	—	47	2	no	—
98	#2216	—	47	2	no	—
99	#2209	—	47	4	no	—
100	#2151	D₄₇	1	1	yes	characteristic
101	C₂×C₄₇	—	4	1	no	—
102	C₂×C₄₇	#3245	1	1	yes	characteristic
103	C₂×C₄₇	#3417	1	2	yes	—
104	#541	—	94	2	no	—
105	C₂×Q₃₂	—	47	2	no	—
106	C₂×Q₃₂	—	47	2	no	—
107	C₂×M₃₂	C₂×D₄₇	1	1	yes	characteristic
108	C₂²×C₁₆	C₂×D₄₇	1	1	yes	characteristic
109	#499	—	47	2	no	—
110	#494	—	47	2	no	—
111	#491	—	94	2	no	—
112	#470	—	94	2	no	—
113	#467	—	47	4	no	—
114	#441	—	94	2	no	—
115	C₄×Q₁₆	—	47	4	no	—
116	#405	C₂×D₄₇	1	1	yes	characteristic
117	#367	—	47	2	no	—
118	#366	—	47	2	no	—
119	C₄⋊C₁₆	C₂×D₄₇	1	4	yes	—
120	#358	—	47	4	no	—
121	#358	—	47	4	no	—
122	C₄₇	#33827	1	1	yes	characteristic, complemented, Sylow
123	C₂×Q₁₆	—	47	4	no	—
124	C₂×Q₁₆	—	94	2	no	—
125	C₂×M₁₆	D₁₈₈	1	1	yes	characteristic
126	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
127	C₄⋊Q₈	—	94	2	no	—
128	#126	—	188	4	no	—
129	#125	—	94	2	no	—
130	#125	—	188	4	no	—
131	C₂²⋊Q₈	—	94	4	no	—
132	C₄×Q₈	—	94	4	no	—
133	#117	D₁₈₈	1	1	yes	characteristic
134	Q₃₂	—	188	2	no	—
135	Q₃₂	—	188	2	no	—
136	M₃₂	—	4	1	no	—
137	C₂×C₁₆	—	4	1	no	—
138	C₂×C₁₆	C₂²×D₄₇	1	2	yes	—
139	C₂×C₁₆	C₂²×D₄₇	1	2	yes	—
140	#107	—	47	4	no	—
141	#106	—	94	2	no	—
142	#103	—	94	4	no	—
143	#103	—	94	4	no	—
144	#97	D₁₈₈	1	2	yes	—
145	C₄×C₈	C₂²×D₄₇	1	2	yes	—
146	C₂×Q₈	—	94	4	no	—
147	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
148	Q₁₆	—	94	4	no	—
149	Q₁₆	—	94	4	no	—
150	Q₁₆	—	188	4	no	—
151	M₁₆	—	4	1	no	—
152	C₂×C₈	—	2	1	no	—
153	C₂×C₈	—	2	1	no	—
154	C₂×C₈	C₂×D₁₈₈	1	2	yes	—
155	C₂×C₈	C₂×D₁₈₈	1	1	yes	characteristic
156	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
157	C₄⋊C₄	—	94	4	no	—
158	C₄⋊C₄	—	188	4	no	—
159	C₄⋊C₄	—	188	4	no	—
160	C₄⋊C₄	—	188	4	no	—
161	C₄⋊C₄	—	188	4	no	—
162	C₄⋊C₄	D₃₇₆	1	2	yes	—
163	C₂²⋊C₄	—	188	4	no	—
164	C₂²⋊C₄	D₃₇₆	1	2	yes	—
165	C₄²	—	188	4	no	—
166	C₄²	C₂×D₁₈₈	1	2	yes	—
167	C₁₆	—	2	2	no	—
168	C₁₆	—	2	2	no	—
169	C₂³	C₂×D₃₇₆	1	1	yes	characteristic
170	Q₈	—	188	4	no	—
171	Q₈	—	188	4	no	—
172	C₂×C₄	—	2	1	no	—
173	C₂×C₄	—	2	1	no	—
174	C₂×C₄	—	188	4	no	—
175	C₂×C₄	—	188	4	no	—
176	C₂×C₄	C₂×D₃₇₆	1	4	yes	—
177	C₂×C₄	C₂×D₃₇₆	1	1	yes	characteristic
178	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
179	C₈	—	2	2	no	—
180	C₈	—	2	1	no	—
181	C₈	#13094069	1	1	yes	characteristic
182	C₈	#13094072	1	1	yes	characteristic
183	C₂²	—	2	2	no	—
184	C₂²	—	2	1	no	—
185	C₂²	C₂²×D₃₇₆	1	1	yes	characteristic
186	C₄	—	2	4	no	—
187	C₄	—	2	1	no	—
188	C₄	—	376	4	no	—
189	C₄	—	376	4	no	—
190	C₄	#15689704	1	1	yes	characteristic
191	C₄	#15689923	1	1	yes	characteristic
192	C₂	—	4	1	no	—
193	C₂	#20254698	1	1	yes	characteristic
194	C₂	#20256541	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20254047

C₂

yes

maximal

#20245553

C₂

yes

maximal, complemented

#20245496

C₂

yes

maximal

#20245479

C₂

yes

maximal

#20245439

C₂

yes

maximal, complemented

C₄₇×#2151

C₂

yes

maximal, characteristic

#15689735

—

C₂×Dic₃₇₆

C₂²

yes

—

C₂×Dic₃₇₆

C₂²

yes

—

#15689525

C₂²

yes

—

#15689503

C₂²

yes

—

#15689492

—

#15689381

—

#15689364

C₂²

yes

—

#15689231

—

C₄×Dic₁₈₈

C₂²

yes

—

#15688951

C₂²

yes

—

#15688949

C₂²

yes

—

#15688928

C₂²

yes

—

#15688928

C₂²

yes

—

C₂×M₃₂×C₄₇

C₂²

yes

characteristic

C₂²×C₁₆×C₄₇

C₂²

yes

characteristic

C₄₇×#405

C₂²

yes

characteristic

C₄₇×C₄⋊C₁₆

C₂²

yes

—

C₂×Dic₁₈₈

—

C₂×Dic₁₈₈

C₂³

yes

—

—

—

—

—

—

C₄×Dic₉₄

—

Dic₃₇₆

—

Dic₃₇₆

—

#13093958

C₂³

yes

—

#13093956

—

#13093949

—

#13093949

—

C₂×M₁₆×C₄₇

D₄

yes

characteristic

C₂²×C₈×C₄₇

C₂³

yes

characteristic

#117×C₄₇

D₄

yes

characteristic

M₃₂×C₄₇

—

C₂×C₁₆×C₄₇

—

C₂×C₁₆×C₄₇

C₂³

yes

—

C₂×C₁₆×C₄₇

C₂³

yes

—

#97×C₄₇

D₄

yes

—

C₄×C₈×C₄₇

C₂³

yes

—

C₂²×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×Dic₉₄

—

M₁₆×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

C₂×D₄

yes

characteristic

C₂×C₈×C₄₇

C₂×D₄

yes

—

C₂×C₈×C₄₇

C₂⁴

yes

characteristic

C₄⋊C₄×C₄₇

D₈

yes

—

C₂²⋊C₄×C₄₇

D₈

yes

—

C₄²×C₄₇

C₂×D₄

yes

—

C₂²⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

Dic₄₇⋊C₄

—

Dic₄₇⋊C₄

—

Dic₄₇⋊C₄

—

C₄×Dic₄₇

—

Dic₁₈₈

—

Dic₁₈₈

—

Dic₁₈₈

—

C₁₆×C₄₇

—

C₁₆×C₄₇

—

C₂³×C₄₇

C₂×D₈

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×D₈

yes

—

C₂×C₄×C₄₇

C₂×D₈

yes

characteristic

C₂×C₄×C₄₇

C₂²×D₄

yes

characteristic

C₂×Dic₄₇

—

C₂×Dic₄₇

—

Dic₉₄

—

Dic₉₄

—

C₈×C₄₇

—

C₈×C₄₇

—

C₈×C₄₇

H(5,𝔽₂)

yes

characteristic

C₈×C₄₇

2₋¹⁺⁴

yes

characteristic

#33827

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

C₂²×D₈

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

yes

C₄×C₄₇

yes

Dic₄₇

—

Dic₄₇

—

—

—

—

—

—

100

#2151

D₄₇

yes

characteristic

101

C₂×C₄₇

—

102

C₂×C₄₇

#3245

yes

characteristic

103

C₂×C₄₇

#3417

yes

—

104

#541

—

105

C₂×Q₃₂

—

106

C₂×Q₃₂

—

107

C₂×M₃₂

C₂×D₄₇

yes

characteristic

108

C₂²×C₁₆

C₂×D₄₇

yes

characteristic

109

#499

—

110

#494

—

111

#491

—

112

#470

—

113

#467

—

114

#441

—

115

C₄×Q₁₆

—

116

#405

C₂×D₄₇

yes

characteristic

117

#367

—

118

#366

—

119

C₄⋊C₁₆

C₂×D₄₇

yes

—

120

#358

—

121

#358

—

122

C₄₇

#33827

yes

characteristic, complemented, Sylow

123

C₂×Q₁₆

—

124

C₂×Q₁₆

—

125

C₂×M₁₆

D₁₈₈

yes

characteristic

126

C₂²×C₈

C₂²×D₄₇

yes

characteristic

127

C₄⋊Q₈

—

128

#126

—

188

—

129

#125

—

130

#125

—

188

—

131

C₂²⋊Q₈

—

132

C₄×Q₈

—

133

#117

D₁₈₈

yes

characteristic

134

Q₃₂

—

188

—

135

Q₃₂

—

188

—

136

M₃₂

—

137

C₂×C₁₆

—

138

C₂×C₁₆

C₂²×D₄₇

yes

—

139

C₂×C₁₆

C₂²×D₄₇

yes

—

140

#107

—

141

#106

—

142

#103

—

143

#103

—

144

#97

D₁₈₈

yes

—

145

C₄×C₈

C₂²×D₄₇

yes

—

146

C₂×Q₈

—

147

C₂²×C₄

C₂×D₁₈₈

yes

characteristic

148

Q₁₆

—

149

Q₁₆

—

150

Q₁₆

—

188

—

151

M₁₆

—

152

C₂×C₈

—

153

C₂×C₈

—

154

C₂×C₈

C₂×D₁₈₈

yes

—

155

C₂×C₈

C₂×D₁₈₈

yes

characteristic

156

C₂×C₈

C₂³×D₄₇

yes

characteristic

157

C₄⋊C₄

—

158

C₄⋊C₄

—

188

—

159

C₄⋊C₄

—

188

—

160

C₄⋊C₄

—

188

—

161

C₄⋊C₄

—

188

—

162

C₄⋊C₄

D₃₇₆

yes

—

163

C₂²⋊C₄

—

188

—

164

C₂²⋊C₄

D₃₇₆

yes

—

165

C₄²

—

188

—

166

C₄²

C₂×D₁₈₈

yes

—

167

C₁₆

—

168

C₁₆

—

169

C₂³

C₂×D₃₇₆

yes

characteristic

170

Q₈

—

188

—

171

Q₈

—

188

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

188

—

175

C₂×C₄

—

188

—

176

C₂×C₄

C₂×D₃₇₆

yes

—

177

C₂×C₄

C₂×D₃₇₆

yes

characteristic

178

C₂×C₄

C₂²×D₁₈₈

yes

characteristic

179

C₈

—

180

C₈

—

181

C₈

#13094069

yes

characteristic

182

C₈

#13094072

yes

characteristic

183

C₂²

—

184

C₂²

—

185

C₂²

C₂²×D₃₇₆

yes

characteristic

186

C₄

—

187

C₄

—

188

C₄

—

376

—

189

C₄

—

376

—

190

C₄

#15689704

yes

characteristic

191

C₄

#15689923

yes

characteristic

192

C₂

—

193

C₂

#20254698

yes

characteristic

194

C₂

#20256541

yes

—

	subgroup	G/H
center	C₂²	C₂²×D₃₇₆
commutator, cosocle	C₂×C₈×C₄₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₇
Fitting	C₄₇×#2151	C₂
socle	C₂²×C₄₇	C₂²×D₈
prime radical	C₂³×C₄₇	C₂×D₈
hypercenter	#2151	D₄₇
hypocenter	C₄₇	#33827
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ #2151

subgroup

G/H

center

C₂²

C₂²×D₃₇₆

commutator, cosocle

C₂×C₈×C₄₇

C₂⁴

Frattini

C₂×C₈

C₂³×D₄₇

Fitting

C₄₇×#2151

C₂

socle

C₂²×C₄₇

C₂²×D₈

prime radical

C₂³×C₄₇

C₂×D₈

D₄₇

C₄₇

G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ #2151

N	H ≅ G/N	type
2. #20245553	C₂	semidirect
5. #20245439	C₂	semidirect
122. C₄₇	#33827	semidirect

H ≅ G/N

type

2. #20245553

C₂

semidirect

5. #20245439

C₂

semidirect

122. C₄₇

#33827

semidirect

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

subgroup	index
#20254047	2
#20245553	2
#20245496	2
#20245479	2
#20245439	2
C₄₇×#2151	2
#33827	47

subgroup

index

C₄₇×#2151

Subgroups — #26372710

Conjugacy classes of subgroups

Complements

Maximal subgroups