Group #20253883 — subgroups

Subgroups — #20253883

Order 6016.

532 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 532 conjugacy classes of subgroups — too many to draw. Only its 149 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	C₂×#13094054	C₂	1	2	yes	maximal, complemented
2	C₂²×Dic₁₈₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#15689694	C₂	1	2	yes	maximal, complemented
4	#15689514	C₂	1	4	yes	maximal, complemented
5	#15689506	C₂	1	2	yes	maximal, complemented
6	#15689501	C₂	1	2	yes	maximal, complemented
7	#15689078	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#135×C₄₇	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₉₄	C₂²	1	2	yes	—
10	#13094054	—	2	8	no	—
11	C₂×Dic₁₈₈	—	2	4	no	—
12	C₂×Dic₁₈₈	C₂²	1	2	yes	—
13	C₂×Dic₁₈₈	C₂²	1	2	yes	—
14	C₂×#10625337	C₂²	1	4	yes	—
15	C₂×#10625335	C₂²	1	4	yes	—
16	C₂×#10625329	C₂²	1	1	yes	characteristic
17	#13094022	C₂²	1	4	yes	—
18	#13094011	C₂²	1	4	yes	—
19	#13094005	—	2	4	no	—
20	#13094003	—	2	4	no	—
21	#13093983	C₂²	1	1	yes	characteristic
22	#13093934	C₂²	1	2	yes	—
23	C₈×Dic₄₇	C₂²	1	2	yes	—
24	C₂×#41×C₄₇	C₂²	1	2	yes	—
25	#135×C₄₇	—	2	4	no	—
26	C₂×Q₁₆×C₄₇	C₂²	1	1	yes	characteristic
27	C₂×SD₁₆×C₄₇	C₂²	1	2	yes	—
28	C₂×D₈×C₄₇	C₂²	1	1	yes	characteristic
29	C₂²×C₈×C₄₇	C₂²	1	1	yes	characteristic
30	#41×C₄₇	—	2	4	no	—
31	#41×C₄₇	—	2	4	no	—
32	C₂×Q₈×C₄₇	C₂³	1	2	yes	—
33	C₂×D₄×C₄₇	—	2	2	no	—
34	C₂×D₄×C₄₇	C₂³	1	2	yes	—
35	C₂²×C₄×C₄₇	—	2	2	no	—
36	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
37	C₂²×Dic₄₇	—	2	2	no	—
38	C₂×Dic₉₄	—	2	4	no	—
39	C₂×Dic₉₄	—	2	4	no	—
40	C₂×Dic₉₄	—	2	2	no	—
41	C₂×Dic₉₄	C₂³	1	4	yes	—
42	Q₁₆×C₄₇	—	2	2	no	—
43	SD₁₆×C₄₇	—	2	4	no	—
44	D₈×C₄₇	—	2	2	no	—
45	C₂×C₈×C₄₇	D₄	1	4	yes	—
46	C₂×C₈×C₄₇	C₂³	1	1	yes	characteristic
47	C₂×C₈×C₄₇	C₂³	1	1	yes	characteristic
48	C₂²⋊Dic₄₇	—	2	8	no	—
49	C₂²⋊Dic₄₇	D₄	1	2	yes	—
50	#10625337	—	2	8	no	—
51	#10625335	—	2	8	no	—
52	C₄⋊Dic₄₇	D₄	1	2	yes	—
53	Dic₄₇⋊C₄	—	2	8	no	—
54	C₄×Dic₄₇	C₂³	1	2	yes	—
55	#10625329	D₄	1	4	yes	—
56	C₂×C₄₇⋊C₈	C₂³	1	2	yes	—
57	Dic₁₈₈	—	2	4	no	—
58	Dic₁₈₈	—	2	4	no	—
59	C₂³×C₄₇	—	2	2	no	—
60	C₂³×C₄₇	C₂×D₄	1	1	yes	characteristic
61	Q₈×C₄₇	—	2	2	no	—
62	Q₈×C₄₇	—	2	2	no	—
63	D₄×C₄₇	—	2	2	no	—
64	D₄×C₄₇	—	2	2	no	—
65	D₄×C₄₇	—	4	4	no	—
66	C₂×C₄×C₄₇	—	2	2	no	—
67	C₂×C₄×C₄₇	—	4	4	no	—
68	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
69	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
70	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
71	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
72	C₂×Dic₄₇	—	2	4	no	—
73	C₂×Dic₄₇	—	4	4	no	—
74	C₂×Dic₄₇	C₂×D₄	1	4	yes	—
75	Dic₉₄	—	2	4	no	—
76	Dic₉₄	—	2	4	no	—
77	Dic₉₄	—	4	4	no	—
78	C₈×C₄₇	C₂×D₄	1	2	yes	—
79	C₈×C₄₇	C₂×D₄	1	2	yes	—
80	C₄₇⋊C₈	C₂×D₄	1	4	yes	—
81	C₂²×C₄₇	—	2	2	no	—
82	C₂²×C₄₇	—	4	2	no	—
83	C₂²×C₄₇	—	4	4	no	—
84	C₂²×C₄₇	#127	1	2	yes	—
85	C₂²×C₄₇	C₂²×D₄	1	1	yes	characteristic
86	C₄×C₄₇	—	4	2	no	—
87	C₄×C₄₇	#127	1	2	yes	—
88	C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
89	C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
90	Dic₄₇	—	2	4	no	—
91	Dic₄₇	—	4	4	no	—
92	#3156	—	47	1	no	maximal, Sylow
93	C₂×C₄₇	—	2	2	no	—
94	C₂×C₄₇	—	8	2	no	—
95	C₂×C₄₇	C₂×#127	1	1	yes	characteristic
96	C₂×C₄₇	#578	1	2	yes	—
97	C₂×#137	—	47	2	no	—
98	C₂×#135	D₄₇	1	1	yes	characteristic
99	C₂²×Q₁₆	—	47	1	no	—
100	#536	—	47	2	no	—
101	#497	—	47	4	no	—
102	#495	—	47	2	no	—
103	#494	—	47	2	no	—
104	#405	—	47	1	no	—
105	C₄₇	#3156	1	1	yes	characteristic, complemented, Sylow
106	C₂×#41	C₂×D₄₇	1	2	yes	—
107	C₂²×Q₈	—	47	2	no	—
108	#137	—	94	8	no	—
109	#135	—	2	4	no	—
110	C₂×Q₁₆	—	47	4	no	—
111	C₂×Q₁₆	—	47	2	no	—
112	C₂×Q₁₆	—	47	2	no	—
113	C₂×Q₁₆	—	94	4	no	—
114	C₂×Q₁₆	C₂×D₄₇	1	1	yes	characteristic
115	C₂×SD₁₆	—	47	4	no	—
116	C₂×SD₁₆	C₂×D₄₇	1	2	yes	—
117	C₂×D₈	C₂×D₄₇	1	1	yes	characteristic
118	C₂×M₁₆	—	47	1	no	—
119	C₂²×C₈	C₂×D₄₇	1	1	yes	characteristic
120	C₄⋊Q₈	—	47	4	no	—
121	#124	—	47	4	no	—
122	#123	—	94	4	no	—
123	C₂²⋊Q₈	—	94	4	no	—
124	#117	—	47	1	no	—
125	#97	—	47	2	no	—
126	C₄×C₈	—	47	2	no	—
127	#41	—	2	4	no	—
128	#41	—	2	4	no	—
129	C₂×Q₈	—	47	4	no	—
130	C₂×Q₈	—	94	2	no	—
131	C₂×Q₈	—	94	4	no	—
132	C₂×Q₈	—	94	4	no	—
133	C₂×Q₈	C₂²×D₄₇	1	2	yes	—
134	C₂×D₄	—	2	2	no	—
135	C₂×D₄	C₂²×D₄₇	1	2	yes	—
136	C₂²×C₄	—	2	2	no	—
137	C₂²×C₄	—	94	2	no	—
138	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
139	Q₁₆	—	2	2	no	—
140	Q₁₆	—	94	8	no	—
141	Q₁₆	—	94	4	no	—
142	Q₁₆	—	94	4	no	—
143	SD₁₆	—	2	4	no	—
144	SD₁₆	—	94	8	no	—
145	D₈	—	2	2	no	—
146	M₁₆	—	47	4	no	—
147	C₂×C₈	—	47	2	no	—
148	C₂×C₈	C₂²⋊D₄₇	1	4	yes	—
149	C₂×C₈	C₂²×D₄₇	1	1	yes	characteristic
150	C₂×C₈	C₂²×D₄₇	1	1	yes	characteristic
151	C₄⋊C₄	—	47	2	no	—
152	C₄⋊C₄	—	94	8	no	—
153	C₂²⋊C₄	—	47	2	no	—
154	C₂²⋊C₄	—	94	8	no	—
155	C₄²	—	47	2	no	—
156	C₂³	—	2	2	no	—
157	C₂³	D₄×D₄₇	1	1	yes	characteristic
158	Q₈	—	2	2	no	—
159	Q₈	—	2	2	no	—
160	Q₈	—	94	4	no	—
161	Q₈	—	94	4	no	—
162	Q₈	—	188	4	no	—
163	D₄	—	2	2	no	—
164	D₄	—	2	2	no	—
165	D₄	—	4	4	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	—	4	4	no	—
168	C₂×C₄	—	47	4	no	—
169	C₂×C₄	—	94	4	no	—
170	C₂×C₄	—	188	4	no	—
171	C₂×C₄	D₄×D₄₇	1	1	yes	characteristic
172	C₂×C₄	D₄×D₄₇	1	2	yes	—
173	C₂×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
174	C₂×C₄	C₂³×D₄₇	1	1	yes	characteristic
175	C₈	—	47	4	no	—
176	C₈	C₂×C₂²⋊D₄₇	1	2	yes	—
177	C₈	C₂×C₂²⋊D₄₇	1	2	yes	—
178	C₂²	—	2	2	no	—
179	C₂²	—	4	2	no	—
180	C₂²	—	4	4	no	—
181	C₂²	#13094020	1	2	yes	—
182	C₂²	C₂×D₄×D₄₇	1	1	yes	characteristic
183	C₄	—	4	2	no	—
184	C₄	—	94	4	no	—
185	C₄	—	188	4	no	—
186	C₄	#13094020	1	2	yes	—
187	C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
188	C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
189	C₂	—	2	2	no	—
190	C₂	—	8	2	no	—
191	C₂	C₂×#13094020	1	1	yes	characteristic
192	C₂	#15689937	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13094054

C₂

yes

maximal, complemented

C₂²×Dic₁₈₈

C₂

yes

maximal, characteristic, complemented

#15689694

C₂

yes

maximal, complemented

#15689514

C₂

yes

maximal, complemented

#15689506

C₂

yes

maximal, complemented

#15689501

C₂

yes

maximal, complemented

#15689078

C₂

yes

maximal, characteristic, complemented

C₂×#135×C₄₇

C₂

yes

maximal, characteristic

C₂²×Dic₉₄

C₂²

yes

—

#13094054

—

C₂×Dic₁₈₈

—

C₂×Dic₁₈₈

C₂²

yes

—

C₂×Dic₁₈₈

C₂²

yes

—

C₂×#10625337

C₂²

yes

—

C₂×#10625335

C₂²

yes

—

C₂×#10625329

C₂²

yes

characteristic

#13094022

C₂²

yes

—

#13094011

C₂²

yes

—

#13094005

—

#13094003

—

#13093983

C₂²

yes

characteristic

#13093934

C₂²

yes

—

C₈×Dic₄₇

C₂²

yes

—

C₂×#41×C₄₇

C₂²

yes

—

#135×C₄₇

—

C₂×Q₁₆×C₄₇

C₂²

yes

characteristic

C₂×SD₁₆×C₄₇

C₂²

yes

—

C₂×D₈×C₄₇

C₂²

yes

characteristic

C₂²×C₈×C₄₇

C₂²

yes

characteristic

#41×C₄₇

—

#41×C₄₇

—

C₂×Q₈×C₄₇

C₂³

yes

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

C₂³

yes

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×Dic₄₇

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₂×Dic₉₄

C₂³

yes

—

Q₁₆×C₄₇

—

SD₁₆×C₄₇

—

D₈×C₄₇

—

C₂×C₈×C₄₇

D₄

yes

—

C₂×C₈×C₄₇

C₂³

yes

characteristic

C₂×C₈×C₄₇

C₂³

yes

characteristic

C₂²⋊Dic₄₇

—

C₂²⋊Dic₄₇

D₄

yes

—

#10625337

—

#10625335

—

C₄⋊Dic₄₇

D₄

yes

—

Dic₄₇⋊C₄

—

C₄×Dic₄₇

C₂³

yes

—

#10625329

D₄

yes

—

C₂×C₄₇⋊C₈

C₂³

yes

—

Dic₁₈₈

—

Dic₁₈₈

—

C₂³×C₄₇

—

C₂³×C₄₇

C₂×D₄

yes

characteristic

Q₈×C₄₇

—

Q₈×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

C₂⁴

yes

characteristic

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

C₂×D₄

yes

—

Dic₉₄

—

Dic₉₄

—

Dic₉₄

—

C₈×C₄₇

C₂×D₄

yes

—

C₈×C₄₇

C₂×D₄

yes

—

C₄₇⋊C₈

C₂×D₄

yes

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

#127

yes

—

C₂²×C₄₇

C₂²×D₄

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

#127

yes

—

C₄×C₄₇

C₂²×D₄

yes

characteristic

C₄×C₄₇

C₂²×D₄

yes

characteristic

Dic₄₇

—

Dic₄₇

—

#3156

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

C₂×#127

yes

characteristic

C₂×C₄₇

#578

yes

—

C₂×#137

—

C₂×#135

D₄₇

yes

characteristic

C₂²×Q₁₆

—

100

#536

—

101

#497

—

102

#495

—

103

#494

—

104

#405

—

105

C₄₇

#3156

yes

characteristic, complemented, Sylow

106

C₂×#41

C₂×D₄₇

yes

—

107

C₂²×Q₈

—

108

#137

—

109

#135

—

110

C₂×Q₁₆

—

111

C₂×Q₁₆

—

112

C₂×Q₁₆

—

113

C₂×Q₁₆

—

114

C₂×Q₁₆

C₂×D₄₇

yes

characteristic

115

C₂×SD₁₆

—

116

C₂×SD₁₆

C₂×D₄₇

yes

—

117

C₂×D₈

C₂×D₄₇

yes

characteristic

118

C₂×M₁₆

—

119

C₂²×C₈

C₂×D₄₇

yes

characteristic

120

C₄⋊Q₈

—

121

#124

—

122

#123

—

123

C₂²⋊Q₈

—

124

#117

—

125

#97

—

126

C₄×C₈

—

127

#41

—

128

#41

—

129

C₂×Q₈

—

130

C₂×Q₈

—

131

C₂×Q₈

—

132

C₂×Q₈

—

133

C₂×Q₈

C₂²×D₄₇

yes

—

134

C₂×D₄

—

135

C₂×D₄

C₂²×D₄₇

yes

—

136

C₂²×C₄

—

137

C₂²×C₄

—

138

C₂²×C₄

C₂²×D₄₇

yes

characteristic

139

Q₁₆

—

140

Q₁₆

—

141

Q₁₆

—

142

Q₁₆

—

143

SD₁₆

—

144

SD₁₆

—

145

D₈

—

146

M₁₆

—

147

C₂×C₈

—

148

C₂×C₈

C₂²⋊D₄₇

yes

—

149

C₂×C₈

C₂²×D₄₇

yes

characteristic

150

C₂×C₈

C₂²×D₄₇

yes

characteristic

151

C₄⋊C₄

—

152

C₄⋊C₄

—

153

C₂²⋊C₄

—

154

C₂²⋊C₄

—

155

C₄²

—

156

C₂³

—

157

C₂³

D₄×D₄₇

yes

characteristic

158

Q₈

—

159

Q₈

—

160

Q₈

—

161

Q₈

—

162

Q₈

—

188

—

163

D₄

—

164

D₄

—

165

D₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

188

—

171

C₂×C₄

D₄×D₄₇

yes

characteristic

172

C₂×C₄

D₄×D₄₇

yes

—

173

C₂×C₄

C₂×C₂²⋊D₄₇

yes

—

174

C₂×C₄

C₂³×D₄₇

yes

characteristic

175

C₈

—

176

C₈

C₂×C₂²⋊D₄₇

yes

—

177

C₈

C₂×C₂²⋊D₄₇

yes

—

178

C₂²

—

179

C₂²

—

180

C₂²

—

181

C₂²

#13094020

yes

—

182

C₂²

C₂×D₄×D₄₇

yes

characteristic

183

C₄

—

184

C₄

—

185

C₄

—

188

—

186

C₄

#13094020

yes

—

187

C₄

C₂×D₄×D₄₇

yes

characteristic

188

C₄

C₂²×C₂²⋊D₄₇

yes

characteristic

189

C₂

—

190

C₂

—

191

C₂

C₂×#13094020

yes

characteristic

192

C₂

#15689937

yes

—

	subgroup	G/H
center	C₂²	C₂×D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₄₇
Fitting	C₂×#135×C₄₇	C₂
socle	C₂²×C₄₇	C₂²×D₄
hypercenter	C₂×#135	D₄₇
hypocenter	C₄₇	#3156
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#135

subgroup

G/H

center

C₂²

C₂×D₄×D₄₇

commutator

C₂×C₄×C₄₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₄₇

Fitting

C₂×#135×C₄₇

C₂

socle

C₂²×C₄₇

C₂²×D₄

D₄₇

C₄₇

G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#135

N	H ≅ G/N	type
1. C₂×#13094054	C₂	semidirect
2. C₂²×Dic₁₈₈	C₂	semidirect
3. #15689694	C₂	semidirect
4. #15689514	C₂	semidirect
5. #15689506	C₂	semidirect
6. #15689501	C₂	semidirect
7. #15689078	C₂	semidirect
105. C₄₇	#3156	semidirect

H ≅ G/N

type

1. C₂×#13094054

C₂

semidirect

2. C₂²×Dic₁₈₈

C₂

semidirect

3. #15689694

C₂

semidirect

4. #15689514

C₂

semidirect

5. #15689506

C₂

semidirect

6. #15689501

C₂

semidirect

7. #15689078

C₂

semidirect

105. C₄₇

#3156

semidirect

Subgroups — #20253883

Conjugacy classes of subgroups

Complements