Group #26203985 — subgroups

Subgroups — #26203985

Order 12032.

706 conjugacy classes of subgroups; only the 187 normal subgroups are shown.

normal subgroups only

This group has 706 conjugacy classes of subgroups — too many to draw. Only its 187 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	#20253973	C₂	1	4	yes	maximal
2	#20253899	C₂	1	2	yes	maximal
3	#20243005	C₂	1	2	yes	maximal, complemented
4	#20242998	C₂	1	1	yes	maximal, characteristic, complemented
5	#20242984	C₂	1	4	yes	maximal, complemented
6	C₈²×D₄₇	C₂	1	1	yes	maximal, characteristic
7	C₄₇×#1710	C₂	1	1	yes	maximal, characteristic, complemented
8	#15689678	C₂²	1	4	yes	—
9	#15689528	C₂²	1	2	yes	—
10	#15689526	C₂²	1	4	yes	—
11	#15689524	—	2	4	no	—
12	#15689519	C₂²	1	4	yes	—
13	#15689502	C₂²	1	4	yes	—
14	#15689474	C₂²	1	4	yes	—
15	#15689363	—	2	8	no	—
16	#15689215	—	2	8	no	—
17	#15689160	—	2	4	no	—
18	C₄×C₈×D₄₇	C₂²	1	2	yes	—
19	C₄×C₈×D₄₇	C₂²	1	1	yes	characteristic
20	C₈×C₄₇⋊C₈	C₂²	1	2	yes	—
21	C₈×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
22	C₄₇×#500	C₂²	1	2	yes	—
23	C₄₇×#487	C₂²	1	4	yes	—
24	C₈²×C₄₇	C₂²	1	1	yes	characteristic
25	C₂×#10625337	—	2	8	no	—
26	C₂×C₈×D₄₇	D₄	1	4	yes	—
27	C₂×C₈×D₄₇	D₄	1	2	yes	—
28	#13094024	C₂³	1	4	yes	—
29	#13094015	—	2	4	no	—
30	#13094004	—	2	4	no	—
31	#13094000	—	2	4	no	—
32	Q₈×Dic₄₇	—	2	4	no	—
33	#13093989	—	2	4	no	—
34	#13093982	—	2	4	no	—
35	C₄²×D₄₇	C₂³	1	1	yes	characteristic
36	#13093958	—	2	4	no	—
37	#13093957	—	2	8	no	—
38	#13093955	—	2	8	no	—
39	#13093949	—	2	8	no	—
40	C₄×C₄₇⋊C₈	C₂³	1	2	yes	—
41	C₄×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
42	C₈×Dic₄₇	D₄	1	2	yes	—
43	C₈×Dic₄₇	D₄	1	4	yes	—
44	C₄⋊Q₈×C₄₇	C₂³	1	4	yes	—
45	#107×C₄₇	—	2	4	no	—
46	#103×C₄₇	—	2	8	no	—
47	C₄×C₈×C₄₇	C₂³	1	2	yes	—
48	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
49	C₂×Q₈×C₄₇	—	2	4	no	—
50	C₂×C₄×D₄₇	C₂×D₄	1	2	yes	—
51	C₂×C₄×D₄₇	C₂×D₄	1	1	yes	characteristic
52	C₂×Dic₉₄	—	2	4	no	—
53	C₂×C₈×C₄₇	C₂×D₄	1	4	yes	—
54	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
55	C₄⋊C₄×C₄₇	—	2	4	no	—
56	C₄⋊C₄×C₄₇	—	2	4	no	—
57	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
58	#10625337	—	4	16	no	—
59	(C₂×D₄₇)⋊C₄	—	4	4	no	—
60	C₄⋊Dic₄₇	—	2	4	no	—
61	C₄⋊Dic₄₇	—	2	4	no	—
62	C₄⋊Dic₄₇	—	4	4	no	—
63	Dic₄₇⋊C₄	—	4	4	no	—
64	C₄×Dic₄₇	—	4	4	no	—
65	C₄×Dic₄₇	C₂×D₄	1	2	yes	—
66	C₄×Dic₄₇	C₂×D₄	1	1	yes	characteristic
67	C₂×C₄₇⋊C₈	C₂×D₄	1	2	yes	—
68	C₂×C₄₇⋊C₈	C₂×D₄	1	4	yes	—
69	C₈×D₄₇	—	2	8	no	—
70	C₈×D₄₇	Q₁₆	1	8	yes	complemented
71	C₂²×D₄₇	#127	1	1	yes	characteristic
72	Q₈×C₄₇	—	4	8	no	—
73	C₂×C₄×C₄₇	—	4	4	no	—
74	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
75	C₂×C₄×C₄₇	C₂²×D₄	1	2	yes	—
76	C₂×Dic₄₇	—	4	4	no	—
77	C₂×Dic₄₇	#127	1	2	yes	—
78	C₂×Dic₄₇	#127	1	1	yes	characteristic
79	C₄×D₄₇	—	2	4	no	—
80	C₄×D₄₇	C₂×Q₁₆	1	4	yes	—
81	Dic₉₄	—	4	8	no	—
82	C₈×C₄₇	—	2	4	no	—
83	C₈×C₄₇	C₂×Q₁₆	1	4	yes	—
84	C₄₇⋊C₈	C₂×Q₁₆	1	4	yes	—
85	C₄₇⋊C₈	#135	1	8	yes	—
86	#19604	—	47	1	no	maximal, Sylow
87	C₂²×C₄₇	C₂×#127	1	1	yes	characteristic
88	C₂×D₄₇	—	2	2	no	—
89	C₂×D₄₇	#494	1	2	yes	—
90	C₄×C₄₇	—	8	4	no	—
91	C₄×C₄₇	C₂²×Q₁₆	1	2	yes	—
92	C₄×C₄₇	C₂×#135	1	4	yes	—
93	Dic₄₇	—	2	2	no	—
94	Dic₄₇	—	8	4	no	—
95	Dic₄₇	#494	1	2	yes	—
96	#3170	—	47	4	no	—
97	#3160	—	47	2	no	—
98	#1713	—	47	2	no	—
99	#1710	—	47	1	no	—
100	#1710	D₄₇	1	1	yes	characteristic, complemented
101	#1707	—	47	4	no	—
102	C₂×C₈²	—	47	1	no	—
103	C₂×C₄₇	C₂×#494	1	1	yes	characteristic
104	C₂×C₄₇	#3147	1	2	yes	—
105	D₄₇	—	2	2	no	—
106	#533	—	47	4	no	—
107	#500	—	47	4	no	—
108	#500	—	47	2	no	—
109	#500	C₂×D₄₇	1	2	yes	—
110	#499	—	94	4	no	—
111	#498	—	47	4	no	—
112	#494	—	47	4	no	—
113	#487	—	47	4	no	—
114	#487	C₂×D₄₇	1	4	yes	—
115	#467	—	94	8	no	—
116	C₄×Q₁₆	—	94	8	no	—
117	#427	—	94	4	no	—
118	C₂×C₄×C₈	—	47	1	no	—
119	C₂×C₄×C₈	—	47	2	no	—
120	C₈²	—	47	2	no	—
121	C₈²	—	47	1	no	—
122	C₈²	C₂×D₄₇	1	1	yes	characteristic
123	C₄₇	#19604	1	1	yes	characteristic, complemented, Sylow
124	C₂×Q₁₆	—	94	8	no	—
125	C₂²×C₈	—	47	2	no	—
126	C₂²×C₈	—	47	4	no	—
127	C₄⋊Q₈	—	47	4	no	—
128	C₄⋊Q₈	C₂²×D₄₇	1	4	yes	—
129	#125	—	94	4	no	—
130	C₂²⋊Q₈	—	94	4	no	—
131	C₂²⋊Q₈	—	94	4	no	—
132	C₄×Q₈	—	94	4	no	—
133	C₄×Q₈	—	94	4	no	—
134	#117	—	94	4	no	—
135	C₂×C₄²	—	47	1	no	—
136	#107	—	2	4	no	—
137	#107	—	94	8	no	—
138	#107	—	94	4	no	—
139	#106	—	94	8	no	—
140	#103	—	2	8	no	—
141	#103	—	94	8	no	—
142	C₄×C₈	—	47	4	no	—
143	C₄×C₈	—	47	2	no	—
144	C₄×C₈	—	47	2	no	—
145	C₄×C₈	—	47	1	no	—
146	C₄×C₈	C₂²×D₄₇	1	2	yes	—
147	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
148	C₂×Q₈	—	2	4	no	—
149	C₂×Q₈	—	94	4	no	—
150	C₂²×C₄	—	47	1	no	—
151	C₂²×C₄	—	47	2	no	—
152	Q₁₆	—	188	16	no	—
153	C₂×C₈	—	47	2	no	—
154	C₂×C₈	—	47	4	no	—
155	C₂×C₈	—	47	8	no	—
156	C₂×C₈	—	94	8	no	—
157	C₂×C₈	D₄×D₄₇	1	4	yes	—
158	C₂×C₈	D₄×D₄₇	1	2	yes	—
159	C₄⋊C₄	—	2	4	no	—
160	C₄⋊C₄	—	2	4	no	—
161	C₄⋊C₄	—	94	4	no	—
162	C₄⋊C₄	—	94	4	no	—
163	C₄⋊C₄	—	188	4	no	—
164	C₄⋊C₄	—	188	4	no	—
165	C₂²⋊C₄	—	188	4	no	—
166	C₄²	—	47	2	no	—
167	C₄²	—	47	1	no	—
168	C₄²	—	188	4	no	—
169	C₄²	C₂³×D₄₇	1	1	yes	characteristic
170	C₂³	—	47	1	no	—
171	Q₈	—	4	8	no	—
172	Q₈	—	188	8	no	—
173	C₂×C₄	—	4	4	no	—
174	C₂×C₄	—	47	1	no	—
175	C₂×C₄	—	47	4	no	—
176	C₂×C₄	—	47	2	no	—
177	C₂×C₄	—	94	4	no	—
178	C₂×C₄	—	188	4	no	—
179	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
180	C₂×C₄	C₂×D₄×D₄₇	1	2	yes	—
181	C₈	—	2	4	no	—
182	C₈	—	47	4	no	—
183	C₈	—	47	8	no	—
184	C₈	Q₁₆×D₄₇	1	4	yes	—
185	C₂²	—	47	2	no	—
186	C₂²	—	94	2	no	—
187	C₂²	#127×D₄₇	1	1	yes	characteristic
188	C₄	—	8	4	no	—
189	C₄	—	47	2	no	—
190	C₄	—	94	2	no	—
191	C₄	—	376	4	no	—
192	C₄	C₂×Q₁₆×D₄₇	1	2	yes	—
193	C₄	#135×D₄₇	1	4	yes	—
194	C₂	—	94	2	no	—
195	C₂	#494×D₄₇	1	1	yes	characteristic
196	C₂	#20253813	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20253973

C₂

yes

maximal

#20253899

C₂

yes

maximal

#20243005

C₂

yes

maximal, complemented

#20242998

C₂

yes

maximal, characteristic, complemented

#20242984

C₂

yes

maximal, complemented

C₈²×D₄₇

C₂

yes

maximal, characteristic

C₄₇×#1710

C₂

yes

maximal, characteristic, complemented

#15689678

C₂²

yes

—

#15689528

C₂²

yes

—

#15689526

C₂²

yes

—

#15689524

—

#15689519

C₂²

yes

—

#15689502

C₂²

yes

—

#15689474

C₂²

yes

—

#15689363

—

#15689215

—

#15689160

—

C₄×C₈×D₄₇

C₂²

yes

—

C₄×C₈×D₄₇

C₂²

yes

characteristic

C₈×C₄₇⋊C₈

C₂²

yes

—

C₈×C₄₇⋊C₈

C₂²

yes

characteristic

C₄₇×#500

C₂²

yes

—

C₄₇×#487

C₂²

yes

—

C₈²×C₄₇

C₂²

yes

characteristic

C₂×#10625337

—

C₂×C₈×D₄₇

D₄

yes

—

C₂×C₈×D₄₇

D₄

yes

—

#13094024

C₂³

yes

—

#13094015

—

#13094004

—

#13094000

—

Q₈×Dic₄₇

—

#13093989

—

#13093982

—

C₄²×D₄₇

C₂³

yes

—

—

—

—

C₄×C₄₇⋊C₈

C₂³

yes

—

C₄×C₄₇⋊C₈

C₂³

yes

characteristic

C₈×Dic₄₇

D₄

yes

—

C₈×Dic₄₇

D₄

yes

—

C₄⋊Q₈×C₄₇

C₂³

yes

—

#107×C₄₇

—

#103×C₄₇

—

C₄×C₈×C₄₇

C₂³

yes

—

C₄×C₈×C₄₇

C₂³

yes

characteristic

C₂×Q₈×C₄₇

—

C₂×C₄×D₄₇

C₂×D₄

yes

—

C₂×C₄×D₄₇

C₂×D₄

yes

characteristic

C₂×Dic₉₄

—

C₂×C₈×C₄₇

C₂×D₄

yes

—

C₂×C₈×C₄₇

C₂×D₄

yes

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄²×C₄₇

C₂⁴

yes

characteristic

#10625337

—

(C₂×D₄₇)⋊C₄

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

Dic₄₇⋊C₄

—

C₄×Dic₄₇

—

C₄×Dic₄₇

C₂×D₄

yes

—

C₄×Dic₄₇

C₂×D₄

yes

characteristic

C₂×C₄₇⋊C₈

C₂×D₄

yes

—

C₂×C₄₇⋊C₈

C₂×D₄

yes

—

C₈×D₄₇

—

C₈×D₄₇

Q₁₆

yes

complemented

C₂²×D₄₇

#127

yes

characteristic

Q₈×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂²×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂²×D₄

yes

—

C₂×Dic₄₇

—

C₂×Dic₄₇

#127

yes

—

C₂×Dic₄₇

#127

yes

characteristic

C₄×D₄₇

—

C₄×D₄₇

C₂×Q₁₆

yes

—

Dic₉₄

—

C₈×C₄₇

—

C₈×C₄₇

C₂×Q₁₆

yes

—

C₄₇⋊C₈

C₂×Q₁₆

yes

—

C₄₇⋊C₈

#135

yes

—

—

C₂²×C₄₇

yes

C₂×D₄₇

—

C₂×D₄₇

#494

yes

—

C₄×C₄₇

—

C₄×C₄₇

C₂²×Q₁₆

yes

—

C₄×C₄₇

C₂×#135

yes

—

Dic₄₇

—

Dic₄₇

—

Dic₄₇

#494

yes

—

—

—

—

—

100

#1710

D₄₇

yes

characteristic, complemented

101

#1707

—

102

C₂×C₈²

—

103

C₂×C₄₇

C₂×#494

yes

characteristic

104

C₂×C₄₇

#3147

yes

—

105

D₄₇

—

106

#533

—

107

#500

—

108

#500

—

109

#500

C₂×D₄₇

yes

—

110

#499

—

111

#498

—

112

#494

—

113

#487

—

114

#487

C₂×D₄₇

yes

—

115

#467

—

116

C₄×Q₁₆

—

117

#427

—

118

C₂×C₄×C₈

—

119

C₂×C₄×C₈

—

120

C₈²

—

121

C₈²

—

122

C₈²

C₂×D₄₇

yes

characteristic

123

C₄₇

#19604

yes

characteristic, complemented, Sylow

124

C₂×Q₁₆

—

125

C₂²×C₈

—

126

C₂²×C₈

—

127

C₄⋊Q₈

—

128

C₄⋊Q₈

C₂²×D₄₇

yes

—

129

#125

—

130

C₂²⋊Q₈

—

131

C₂²⋊Q₈

—

132

C₄×Q₈

—

133

C₄×Q₈

—

134

#117

—

135

C₂×C₄²

—

136

#107

—

137

#107

—

138

#107

—

139

#106

—

140

#103

—

141

#103

—

142

C₄×C₈

—

143

C₄×C₈

—

144

C₄×C₈

—

145

C₄×C₈

—

146

C₄×C₈

C₂²×D₄₇

yes

—

147

C₄×C₈

C₂²×D₄₇

yes

characteristic

148

C₂×Q₈

—

149

C₂×Q₈

—

150

C₂²×C₄

—

151

C₂²×C₄

—

152

Q₁₆

—

188

—

153

C₂×C₈

—

154

C₂×C₈

—

155

C₂×C₈

—

156

C₂×C₈

—

157

C₂×C₈

D₄×D₄₇

yes

—

158

C₂×C₈

D₄×D₄₇

yes

—

159

C₄⋊C₄

—

160

C₄⋊C₄

—

161

C₄⋊C₄

—

162

C₄⋊C₄

—

163

C₄⋊C₄

—

188

—

164

C₄⋊C₄

—

188

—

165

C₂²⋊C₄

—

188

—

166

C₄²

—

167

C₄²

—

168

C₄²

—

188

—

169

C₄²

C₂³×D₄₇

yes

characteristic

170

C₂³

—

171

Q₈

—

172

Q₈

—

188

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

188

—

179

C₂×C₄

C₂×D₄×D₄₇

yes

characteristic

180

C₂×C₄

C₂×D₄×D₄₇

yes

—

181

C₈

—

182

C₈

—

183

C₈

—

184

C₈

Q₁₆×D₄₇

yes

—

185

C₂²

—

186

C₂²

—

187

C₂²

#127×D₄₇

yes

characteristic

188

C₄

—

189

C₄

—

190

C₄

—

191

C₄

—

376

—

192

C₄

C₂×Q₁₆×D₄₇

yes

—

193

C₄

#135×D₄₇

yes

—

194

C₂

—

195

C₂

#494×D₄₇

yes

characteristic

196

C₂

#20253813

yes

—

	subgroup	G/H
center	C₂²	#127×D₄₇
commutator, cosocle	C₄²×C₄₇	C₂⁴
Frattini	C₄²	C₂³×D₄₇
Fitting	C₄₇×#1710	C₂
socle	C₂²×C₄₇	C₂×#127
prime radical	C₂²×D₄₇	#127
hypercenter	#1710	D₄₇
hypocenter	C₄₇	#19604
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #1710

subgroup

G/H

center

C₂²

#127×D₄₇

commutator, cosocle

C₄²×C₄₇

C₂⁴

Frattini

C₄²

C₂³×D₄₇

Fitting

C₄₇×#1710

C₂

C₂²×C₄₇

C₂²×D₄₇

D₄₇

C₄₇

G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #1710

N	H ≅ G/N	type
3. #20243005	C₂	semidirect
4. #20242998	C₂	semidirect
5. #20242984	C₂	semidirect
7. C₄₇×#1710	C₂	semidirect
70. C₈×D₄₇	Q₁₆	semidirect
100. #1710	D₄₇	semidirect
123. C₄₇	#19604	semidirect

H ≅ G/N

type

3. #20243005

C₂

semidirect

4. #20242998

C₂

semidirect

5. #20242984

C₂

semidirect

7. C₄₇×#1710

C₂

semidirect

70. C₈×D₄₇

Q₁₆

semidirect

100. #1710

D₄₇

semidirect

123. C₄₇

#19604

semidirect

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

subgroup	index
#20253973	2
#20253899	2
#20243005	2
#20242998	2
#20242984	2
C₈²×D₄₇	2
C₄₇×#1710	2
#19604	47

subgroup

index

C₈²×D₄₇

C₄₇×#1710

#19604

Subgroups — #26203985

Conjugacy classes of subgroups

Complements

Maximal subgroups