Group #20253846 — subgroups

Subgroups — #20253846

Order 6016.

552 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 145 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×#13094056	C₂	1	4	yes	maximal, complemented
2	#15689674	C₂	1	1	yes	maximal, characteristic
3	C₄⋊Q₈×D₄₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#15689516	C₂	1	4	yes	maximal, complemented
5	#15689502	C₂	1	1	yes	maximal, characteristic, complemented
6	#15689497	C₂	1	1	yes	maximal, characteristic, complemented
7	#15689496	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#10625324	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₇×#494	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#10625354	C₂²	1	2	yes	—
11	C₂×Q₈×D₄₇	C₂²	1	2	yes	—
12	#13094056	—	2	16	no	—
13	C₂×#10625337	C₂²	1	4	yes	—
14	C₂×#10625325	C₂²	1	4	yes	—
15	C₂×#10625336	C₂²	1	4	yes	—
16	C₂×#10625324	C₂²	1	2	yes	—
17	#13094024	C₂²	1	1	yes	characteristic
18	#13094023	—	2	2	no	—
19	#13094022	C₂²	1	2	yes	—
20	#13094021	C₂²	1	1	yes	characteristic
21	#13094013	C₂²	1	2	yes	—
22	#13093999	—	2	2	no	—
23	#13093986	—	2	2	no	—
24	C₄⋊C₄×D₄₇	—	2	2	no	—
25	C₄²×D₄₇	C₂²	1	1	yes	characteristic
26	#13093934	C₂²	1	2	yes	—
27	C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
28	C₂×Q₁₆×C₄₇	C₂²	1	4	yes	—
29	C₄⋊Q₈×C₄₇	C₂²	1	1	yes	characteristic
30	C₄⋊Q₈×C₄₇	C₂²	1	1	yes	characteristic
31	C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
32	C₂×Q₈×C₄₇	C₂³	1	2	yes	—
33	C₂×Q₈×C₄₇	C₂³	1	2	yes	—
34	#10625354	—	2	8	no	—
35	Q₈×D₄₇	—	2	8	no	—
36	C₂×D₁₈₈	—	2	2	no	—
37	C₂×D₁₈₈	—	2	2	no	—
38	C₂×D₁₈₈	C₂³	1	2	yes	—
39	C₂×C₄×D₄₇	—	2	2	no	—
40	C₂×C₄×D₄₇	—	2	2	no	—
41	C₂×C₄×D₄₇	D₄	1	2	yes	—
42	C₂×C₄×D₄₇	C₂³	1	1	yes	characteristic
43	C₂×Dic₉₄	—	2	2	no	—
44	C₂×Dic₉₄	C₂³	1	2	yes	—
45	Q₁₆×C₄₇	—	2	8	no	—
46	C₂×C₈×C₄₇	C₂³	1	2	yes	—
47	C₄⋊C₄×C₄₇	—	2	2	no	—
48	C₄⋊C₄×C₄₇	—	2	2	no	—
49	C₄²×C₄₇	C₂³	1	1	yes	characteristic
50	#10625337	—	2	8	no	—
51	#10625336	—	2	8	no	—
52	(C₂×D₄₇)⋊C₄	—	2	4	no	—
53	C₄⋊Dic₄₇	—	2	2	no	—
54	Dic₄₇⋊C₄	—	2	4	no	—
55	C₄×Dic₄₇	D₄	1	2	yes	—
56	C₄×Dic₄₇	C₂³	1	1	yes	characteristic
57	C₂×C₄₇⋊C₈	C₂³	1	2	yes	—
58	#10625325	—	2	8	no	—
59	#10625324	D₄	1	8	yes	complemented
60	C₂²×D₄₇	—	2	2	no	—
61	C₂²×D₄₇	C₂×D₄	1	1	yes	characteristic
62	Q₈×C₄₇	—	2	4	no	—
63	Q₈×C₄₇	—	2	4	no	—
64	C₂×C₄×C₄₇	—	2	2	no	—
65	C₂×C₄×C₄₇	—	2	2	no	—
66	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
67	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
68	C₂×Dic₄₇	—	2	2	no	—
69	C₂×Dic₄₇	C₂×D₄	1	2	yes	—
70	C₂×Dic₄₇	C₂×D₄	1	1	yes	characteristic
71	D₁₈₈	—	2	4	no	—
72	D₁₈₈	—	4	4	no	—
73	D₁₈₈	—	4	4	no	—
74	C₄×D₄₇	—	2	4	no	—
75	C₄×D₄₇	—	4	4	no	—
76	C₄×D₄₇	—	4	4	no	—
77	C₄×D₄₇	C₂×D₄	1	4	yes	—
78	Dic₉₄	—	2	4	no	—
79	Dic₉₄	—	4	4	no	—
80	C₈×C₄₇	C₂×D₄	1	4	yes	—
81	C₄₇⋊C₈	C₂×D₄	1	4	yes	—
82	C₂²×C₄₇	C₂²×D₄	1	1	yes	characteristic
83	C₂×D₄₇	—	2	2	no	—
84	C₂×D₄₇	—	4	2	no	—
85	C₂×D₄₇	—	4	4	no	—
86	C₂×D₄₇	#127	1	2	yes	—
87	C₄×C₄₇	—	4	2	no	—
88	C₄×C₄₇	—	4	2	no	—
89	C₄×C₄₇	#137	1	4	yes	complemented
90	C₄×C₄₇	C₂²×D₄	1	2	yes	—
91	Dic₄₇	—	2	2	no	—
92	Dic₄₇	—	4	2	no	—
93	Dic₄₇	#127	1	2	yes	—
94	#3152	—	47	1	no	maximal, Sylow
95	C₂×C₄₇	C₂×#127	1	1	yes	characteristic
96	C₂×C₄₇	C₂×#137	1	2	yes	—
97	D₄₇	—	2	2	no	—
98	D₄₇	—	8	2	no	—
99	C₂×#137	—	47	4	no	—
100	#532	—	47	1	no	—
101	C₂×C₄⋊Q₈	—	47	1	no	—
102	#497	—	47	4	no	—
103	#494	—	47	1	no	—
104	#494	D₄₇	1	1	yes	characteristic, complemented
105	#492	—	47	1	no	—
106	#492	—	47	1	no	—
107	C₄×M₁₆	—	47	1	no	—
108	C₄₇	#3152	1	1	yes	characteristic, complemented, Sylow
109	C₂×#41	—	47	2	no	—
110	C₂²×Q₈	—	47	2	no	—
111	#137	—	94	16	no	—
112	C₂×Q₁₆	—	47	4	no	—
113	C₂×Q₁₆	C₂×D₄₇	1	4	yes	—
114	C₂×SD₁₆	—	47	4	no	—
115	C₂×SD₁₆	—	47	4	no	—
116	C₂×M₁₆	—	47	2	no	—
117	C₄⋊Q₈	—	47	2	no	—
118	C₄⋊Q₈	—	47	1	no	—
119	C₄⋊Q₈	—	94	2	no	—
120	C₄⋊Q₈	C₂×D₄₇	1	1	yes	characteristic
121	C₄⋊Q₈	C₂×D₄₇	1	1	yes	characteristic
122	#127	—	47	1	no	—
123	#124	—	47	2	no	—
124	#121	—	94	2	no	—
125	C₄×D₄	—	94	2	no	—
126	C₂×C₄⋊C₄	—	94	2	no	—
127	C₂×C₄²	—	47	1	no	—
128	#97	—	47	2	no	—
129	C₄×C₈	—	47	1	no	—
130	C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
131	#41	—	94	8	no	—
132	C₂×Q₈	—	47	2	no	—
133	C₂×Q₈	—	94	2	no	—
134	C₂×Q₈	—	94	8	no	—
135	C₂×Q₈	C₂²×D₄₇	1	2	yes	—
136	C₂×Q₈	C₂²×D₄₇	1	2	yes	—
137	C₂×D₄	—	47	2	no	—
138	C₂×D₄	—	94	2	no	—
139	C₂×D₄	—	94	2	no	—
140	C₂²×C₄	—	47	1	no	—
141	C₂²×C₄	—	47	2	no	—
142	C₂²×C₄	—	94	2	no	—
143	C₂²×C₄	—	94	2	no	—
144	Q₁₆	—	2	8	no	—
145	Q₁₆	—	94	8	no	—
146	SD₁₆	—	94	8	no	—
147	SD₁₆	—	94	8	no	—
148	M₁₆	—	47	8	no	—
149	C₂×C₈	—	47	2	no	—
150	C₂×C₈	C₂²×D₄₇	1	2	yes	—
151	C₄⋊C₄	—	2	2	no	—
152	C₄⋊C₄	—	2	2	no	—
153	C₄⋊C₄	—	94	4	no	—
154	C₄⋊C₄	—	94	2	no	—
155	C₂²⋊C₄	—	94	4	no	—
156	C₄²	—	47	2	no	—
157	C₄²	—	47	1	no	—
158	C₄²	C₂²×D₄₇	1	1	yes	characteristic
159	C₂³	—	47	1	no	—
160	C₂³	—	94	2	no	—
161	Q₈	—	2	4	no	—
162	Q₈	—	2	4	no	—
163	Q₈	—	94	4	no	—
164	Q₈	—	188	4	no	—
165	D₄	—	94	4	no	—
166	D₄	—	188	4	no	—
167	D₄	—	188	4	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	47	1	no	—
171	C₂×C₄	—	47	4	no	—
172	C₂×C₄	—	47	2	no	—
173	C₂×C₄	—	94	4	no	—
174	C₂×C₄	—	94	2	no	—
175	C₂×C₄	—	188	4	no	—
176	C₂×C₄	—	188	4	no	—
177	C₂×C₄	D₄×D₄₇	1	2	yes	—
178	C₂×C₄	C₂³×D₄₇	1	1	yes	characteristic
179	C₈	—	47	4	no	—
180	C₈	D₄×D₄₇	1	4	yes	—
181	C₂²	—	47	2	no	—
182	C₂²	—	94	2	no	—
183	C₂²	—	188	2	no	—
184	C₂²	—	188	4	no	—
185	C₂²	C₂×D₄×D₄₇	1	1	yes	characteristic
186	C₄	—	4	2	no	—
187	C₄	—	4	2	no	—
188	C₄	—	47	2	no	—
189	C₄	—	94	2	no	—
190	C₄	—	188	2	no	—
191	C₄	#13094056	1	4	yes	complemented
192	C₄	C₂×D₄×D₄₇	1	2	yes	—
193	C₂	—	94	2	no	—
194	C₂	—	376	2	no	—
195	C₂	#127×D₄₇	1	1	yes	characteristic
196	C₂	C₂×#13094056	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13094056

C₂

yes

maximal, complemented

#15689674

C₂

yes

maximal, characteristic

C₄⋊Q₈×D₄₇

C₂

yes

maximal, characteristic, complemented

#15689516

C₂

yes

maximal, complemented

#15689502

C₂

yes

maximal, characteristic, complemented

#15689497

C₂

yes

maximal, characteristic, complemented

#15689496

C₂

yes

maximal, characteristic, complemented

C₄×#10625324

C₂

yes

maximal, characteristic, complemented

C₄₇×#494

C₂

yes

maximal, characteristic, complemented

C₂×#10625354

C₂²

yes

—

C₂×Q₈×D₄₇

C₂²

yes

—

#13094056

—

C₂×#10625337

C₂²

yes

—

C₂×#10625325

C₂²

yes

—

C₂×#10625336

C₂²

yes

—

C₂×#10625324

C₂²

yes

—

#13094024

C₂²

yes

characteristic

#13094023

—

#13094022

C₂²

yes

—

#13094021

C₂²

yes

characteristic

#13094013

C₂²

yes

—

#13093999

—

#13093986

—

C₄⋊C₄×D₄₇

—

C₄²×D₄₇

C₂²

yes

characteristic

#13093934

C₂²

yes

—

C₄×C₄₇⋊C₈

C₂²

yes

characteristic

C₂×Q₁₆×C₄₇

C₂²

yes

—

C₄⋊Q₈×C₄₇

C₂²

yes

characteristic

C₄⋊Q₈×C₄₇

C₂²

yes

characteristic

C₄×C₈×C₄₇

C₂²

yes

characteristic

C₂×Q₈×C₄₇

C₂³

yes

—

C₂×Q₈×C₄₇

C₂³

yes

—

#10625354

—

Q₈×D₄₇

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

C₂³

yes

—

C₂×C₄×D₄₇

—

C₂×C₄×D₄₇

—

C₂×C₄×D₄₇

D₄

yes

—

C₂×C₄×D₄₇

C₂³

yes

characteristic

C₂×Dic₉₄

—

C₂×Dic₉₄

C₂³

yes

—

Q₁₆×C₄₇

—

C₂×C₈×C₄₇

C₂³

yes

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄²×C₄₇

C₂³

yes

characteristic

#10625337

—

#10625336

—

(C₂×D₄₇)⋊C₄

—

C₄⋊Dic₄₇

—

Dic₄₇⋊C₄

—

C₄×Dic₄₇

D₄

yes

—

C₄×Dic₄₇

C₂³

yes

characteristic

C₂×C₄₇⋊C₈

C₂³

yes

—

#10625325

—

#10625324

D₄

yes

complemented

C₂²×D₄₇

—

C₂²×D₄₇

C₂×D₄

yes

characteristic

Q₈×C₄₇

—

Q₈×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

C₂⁴

yes

characteristic

C₂×Dic₄₇

—

C₂×Dic₄₇

C₂×D₄

yes

—

C₂×Dic₄₇

C₂×D₄

yes

characteristic

D₁₈₈

—

D₁₈₈

—

D₁₈₈

—

C₄×D₄₇

—

C₄×D₄₇

—

C₄×D₄₇

—

C₄×D₄₇

C₂×D₄

yes

—

Dic₉₄

—

Dic₉₄

—

C₈×C₄₇

C₂×D₄

yes

—

C₄₇⋊C₈

C₂×D₄

yes

—

C₂²×C₄₇

C₂²×D₄

yes

characteristic

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

#127

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#137

yes

complemented

C₄×C₄₇

C₂²×D₄

yes

—

Dic₄₇

—

Dic₄₇

—

Dic₄₇

#127

yes

—

—

C₂×C₄₇

yes

C₂×C₄₇

yes

—

D₄₇

—

D₄₇

—

C₂×#137

—

100

#532

—

101

C₂×C₄⋊Q₈

—

102

#497

—

103

#494

—

104

#494

D₄₇

yes

characteristic, complemented

105

#492

—

106

#492

—

107

C₄×M₁₆

—

108

C₄₇

#3152

yes

characteristic, complemented, Sylow

109

C₂×#41

—

110

C₂²×Q₈

—

111

#137

—

112

C₂×Q₁₆

—

113

C₂×Q₁₆

C₂×D₄₇

yes

—

114

C₂×SD₁₆

—

115

C₂×SD₁₆

—

116

C₂×M₁₆

—

117

C₄⋊Q₈

—

118

C₄⋊Q₈

—

119

C₄⋊Q₈

—

120

C₄⋊Q₈

C₂×D₄₇

yes

characteristic

121

C₄⋊Q₈

C₂×D₄₇

yes

characteristic

122

#127

—

123

#124

—

124

#121

—

125

C₄×D₄

—

126

C₂×C₄⋊C₄

—

127

C₂×C₄²

—

128

#97

—

129

C₄×C₈

—

130

C₄×C₈

C₂×D₄₇

yes

characteristic

131

#41

—

132

C₂×Q₈

—

133

C₂×Q₈

—

134

C₂×Q₈

—

135

C₂×Q₈

C₂²×D₄₇

yes

—

136

C₂×Q₈

C₂²×D₄₇

yes

—

137

C₂×D₄

—

138

C₂×D₄

—

139

C₂×D₄

—

140

C₂²×C₄

—

141

C₂²×C₄

—

142

C₂²×C₄

—

143

C₂²×C₄

—

144

Q₁₆

—

145

Q₁₆

—

146

SD₁₆

—

147

SD₁₆

—

148

M₁₆

—

149

C₂×C₈

—

150

C₂×C₈

C₂²×D₄₇

yes

—

151

C₄⋊C₄

—

152

C₄⋊C₄

—

153

C₄⋊C₄

—

154

C₄⋊C₄

—

155

C₂²⋊C₄

—

156

C₄²

—

157

C₄²

—

158

C₄²

C₂²×D₄₇

yes

characteristic

159

C₂³

—

160

C₂³

—

161

Q₈

—

162

Q₈

—

163

Q₈

—

164

Q₈

—

188

—

165

D₄

—

166

D₄

—

188

—

167

D₄

—

188

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

—

188

—

176

C₂×C₄

—

188

—

177

C₂×C₄

D₄×D₄₇

yes

—

178

C₂×C₄

C₂³×D₄₇

yes

characteristic

179

C₈

—

180

C₈

D₄×D₄₇

yes

—

181

C₂²

—

182

C₂²

—

183

C₂²

—

188

—

184

C₂²

—

188

—

185

C₂²

C₂×D₄×D₄₇

yes

characteristic

186

C₄

—

187

C₄

—

188

C₄

—

189

C₄

—

190

C₄

—

188

—

191

C₄

#13094056

yes

complemented

192

C₄

C₂×D₄×D₄₇

yes

—

193

C₂

—

194

C₂

—

376

—

195

C₂

#127×D₄₇

yes

characteristic

196

C₂

C₂×#13094056

yes

—

	subgroup	G/H
center	C₂²	C₂×D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₄₇
Fitting	C₄₇×#494	C₂
socle	C₂²×C₄₇	C₂²×D₄
hypercenter	#494	D₄₇
hypocenter	C₄₇	#3152
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #494

subgroup

G/H

center

C₂²

C₂×D₄×D₄₇

commutator

C₂×C₄×C₄₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₄₇

Fitting

C₄₇×#494

C₂

socle

C₂²×C₄₇

C₂²×D₄

D₄₇

C₄₇

G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #494

N	H ≅ G/N	type
1. C₂×#13094056	C₂	semidirect
3. C₄⋊Q₈×D₄₇	C₂	semidirect
4. #15689516	C₂	semidirect
5. #15689502	C₂	semidirect
6. #15689497	C₂	semidirect
7. #15689496	C₂	semidirect
8. C₄×#10625324	C₂	semidirect
9. C₄₇×#494	C₂	semidirect
59. #10625324	D₄	semidirect
89. C₄×C₄₇	#137	semidirect
104. #494	D₄₇	semidirect
108. C₄₇	#3152	semidirect
191. C₄	#13094056	semidirect

H ≅ G/N

type

1. C₂×#13094056

C₂

semidirect

3. C₄⋊Q₈×D₄₇

C₂

semidirect

4. #15689516

C₂

semidirect

5. #15689502

C₂

semidirect

6. #15689497

C₂

semidirect

7. #15689496

C₂

semidirect

8. C₄×#10625324

C₂

semidirect

9. C₄₇×#494

C₂

semidirect

59. #10625324

D₄

semidirect

89. C₄×C₄₇

#137

semidirect

104. #494

D₄₇

semidirect

108. C₄₇

#3152

semidirect

191. C₄

#13094056

semidirect

Subgroups — #20253846

Conjugacy classes of subgroups

Complements