Group #20253680 — subgroups

Subgroups — #20253680

Order 6016.

382 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 121 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	#15689728	C₂	1	1	yes	maximal, characteristic
2	#15689600	C₂	1	1	yes	maximal, characteristic, complemented
3	#15689448	C₂	1	2	yes	maximal, complemented
4	#15689431	C₂	1	2	yes	maximal, complemented
5	#15689324	C₂	1	2	yes	maximal, complemented
6	#15689264	C₂	1	2	yes	maximal, complemented
7	#15689225	C₂	1	2	yes	maximal, complemented
8	C₄×#10625325	C₂	1	2	yes	maximal, complemented
9	C₄₇×#433	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×Dic₉₄	C₂²	1	1	yes	characteristic
11	C₂×#10625325	C₂²	1	2	yes	—
12	C₂×#10625325	C₂²	1	2	yes	—
13	#13094021	C₂²	1	1	yes	characteristic
14	#13094016	C₂²	1	1	yes	characteristic
15	#13094009	—	2	2	no	—
16	#13093998	C₂²	1	2	yes	—
17	#13093997	—	2	2	no	—
18	C₄×Dic₉₄	C₂²	1	2	yes	—
19	C₄×Dic₉₄	C₂²	1	2	yes	—
20	#13093989	—	2	2	no	—
21	C₄×D₁₈₈	C₂²	1	2	yes	—
22	#13093983	C₂²	1	1	yes	characteristic
23	#13093980	—	2	2	no	—
24	#13093958	C₂²	1	2	yes	—
25	#13093956	C₂²	1	2	yes	—
26	#13093949	C₂²	1	2	yes	—
27	#13093949	C₂²	1	2	yes	—
28	#13093946	C₂²	1	2	yes	—
29	#13093946	C₂²	1	2	yes	—
30	#117×C₄₇	C₂²	1	1	yes	characteristic
31	C₄⋊C₈×C₄₇	C₂²	1	2	yes	—
32	C₂²⋊C₈×C₄₇	C₂²	1	2	yes	—
33	#97×C₄₇	C₂²	1	1	yes	characteristic
34	C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
35	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
36	C₂×C₂²⋊D₄₇	—	2	2	no	—
37	C₂²×Dic₄₇	—	2	1	no	—
38	C₂×D₁₈₈	—	2	2	no	—
39	C₂×D₁₈₈	C₂³	1	2	yes	—
40	C₂×C₄×D₄₇	—	2	2	no	—
41	C₂×Dic₉₄	—	2	1	no	—
42	C₂×Dic₉₄	—	2	4	no	—
43	C₂×Dic₉₄	C₂³	1	2	yes	—
44	C₂×C₈×C₄₇	C₂³	1	2	yes	—
45	C₂×C₈×C₄₇	C₂³	1	2	yes	—
46	C₄⋊C₄×C₄₇	D₄	1	2	yes	—
47	C₂²⋊C₄×C₄₇	D₄	1	2	yes	—
48	C₄²×C₄₇	C₂³	1	1	yes	characteristic
49	C₄²×C₄₇	C₂³	1	1	yes	characteristic
50	C₂²⋊Dic₄₇	—	2	1	no	—
51	(C₂×D₄₇)⋊C₄	—	2	2	no	—
52	(C₂×D₄₇)⋊C₄	—	2	2	no	—
53	C₄⋊Dic₄₇	C₂³	1	2	yes	—
54	C₄⋊Dic₄₇	C₂³	1	2	yes	—
55	Dic₄₇⋊C₄	—	2	2	no	—
56	Dic₄₇⋊C₄	—	2	2	no	—
57	Dic₄₇⋊C₄	—	2	2	no	—
58	C₄×Dic₄₇	—	2	2	no	—
59	C₄×Dic₄₇	—	2	2	no	—
60	C₄×Dic₄₇	—	2	1	no	—
61	#10625325	—	2	4	no	—
62	#10625325	—	2	4	no	—
63	C₂³×C₄₇	C₂×D₄	1	1	yes	characteristic
64	C₂²×D₄₇	—	2	2	no	—
65	C₂×C₄×C₄₇	—	2	2	no	—
66	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
67	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
68	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
69	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
70	C₂²⋊D₄₇	—	8	2	no	—
71	C₂×Dic₄₇	—	2	4	no	—
72	C₂×Dic₄₇	—	2	2	no	—
73	C₂×Dic₄₇	—	2	2	no	—
74	C₂×Dic₄₇	—	2	2	no	—
75	D₁₈₈	—	2	2	no	—
76	D₁₈₈	—	2	2	no	—
77	D₁₈₈	—	8	2	no	—
78	C₄×D₄₇	—	8	2	no	—
79	Dic₉₄	—	2	4	no	—
80	Dic₉₄	—	2	2	no	—
81	Dic₉₄	#41	1	4	yes	—
82	C₈×C₄₇	—	2	2	no	—
83	C₈×C₄₇	—	2	2	no	—
84	C₂²×C₄₇	—	2	1	no	—
85	C₂²×C₄₇	—	2	2	no	—
86	C₂²×C₄₇	C₂²×D₄	1	1	yes	characteristic
87	C₂×D₄₇	—	4	2	no	—
88	C₂×D₄₇	—	4	2	no	—
89	C₂×D₄₇	—	4	2	no	—
90	C₄×C₄₇	—	2	1	no	—
91	C₄×C₄₇	—	2	2	no	—
92	C₄×C₄₇	—	2	2	no	—
93	C₄×C₄₇	C₂×#41	1	2	yes	—
94	Dic₄₇	—	2	4	no	—
95	Dic₄₇	—	4	2	no	—
96	Dic₄₇	—	4	2	no	—
97	#3126	—	47	1	no	maximal, Sylow
98	C₂×C₄₇	—	4	1	no	—
99	C₂×C₄₇	#525	1	1	yes	characteristic
100	C₂×C₄₇	#577	1	2	yes	—
101	D₄₇	—	8	2	no	—
102	#540	—	47	1	no	—
103	#519	—	47	1	no	—
104	#482	—	47	2	no	—
105	#479	—	47	2	no	—
106	#460	—	47	2	no	—
107	#448	—	47	2	no	—
108	#440	—	47	2	no	—
109	C₄×SD₁₆	—	47	2	no	—
110	#433	D₄₇	1	1	yes	characteristic, complemented
111	C₄₇	#3126	1	1	yes	characteristic, complemented, Sylow
112	C₂²×Q₈	—	47	1	no	—
113	C₂×SD₁₆	—	47	2	no	—
114	C₂×SD₁₆	—	47	2	no	—
115	#127	—	47	1	no	—
116	#125	—	47	1	no	—
117	#123	—	94	2	no	—
118	#121	—	47	2	no	—
119	#121	—	94	2	no	—
120	C₄×Q₈	—	47	2	no	—
121	C₄×Q₈	—	47	2	no	—
122	C₄×Q₈	—	94	2	no	—
123	C₄×D₄	—	47	2	no	—
124	#117	—	47	1	no	—
125	#117	—	94	2	no	—
126	#117	C₂×D₄₇	1	1	yes	characteristic
127	#107	—	47	2	no	—
128	#106	—	47	2	no	—
129	C₄⋊C₈	C₂×D₄₇	1	2	yes	—
130	#103	—	47	2	no	—
131	#103	—	47	2	no	—
132	#102	—	47	2	no	—
133	#102	—	47	2	no	—
134	C₂²⋊C₈	C₂×D₄₇	1	2	yes	—
135	#97	C₂×D₄₇	1	1	yes	characteristic
136	C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
137	C₂×Q₈	—	47	2	no	—
138	C₂×Q₈	—	94	4	no	—
139	C₂×Q₈	—	94	1	no	—
140	C₂×D₄	—	47	2	no	—
141	C₂×D₄	—	94	2	no	—
142	C₂×D₄	—	94	2	no	—
143	C₂²×C₄	—	94	1	no	—
144	C₂²×C₄	—	94	2	no	—
145	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
146	SD₁₆	—	94	4	no	—
147	SD₁₆	—	94	4	no	—
148	C₂×C₈	C₂²×D₄₇	1	2	yes	—
149	C₂×C₈	C₂²×D₄₇	1	2	yes	—
150	C₄⋊C₄	—	47	2	no	—
151	C₄⋊C₄	—	47	2	no	—
152	C₄⋊C₄	—	94	2	no	—
153	C₄⋊C₄	—	94	2	no	—
154	C₄⋊C₄	—	94	2	no	—
155	C₄⋊C₄	D₁₈₈	1	2	yes	—
156	C₂²⋊C₄	—	94	1	no	—
157	C₂²⋊C₄	—	94	2	no	—
158	C₂²⋊C₄	—	94	2	no	—
159	C₂²⋊C₄	D₁₈₈	1	2	yes	—
160	C₄²	—	94	1	no	—
161	C₄²	—	94	2	no	—
162	C₄²	—	94	2	no	—
163	C₄²	C₂²×D₄₇	1	1	yes	characteristic
164	C₄²	C₂²×D₄₇	1	1	yes	characteristic
165	C₂³	—	94	2	no	—
166	C₂³	C₂×D₁₈₈	1	1	yes	characteristic
167	Q₈	—	47	4	no	—
168	Q₈	—	94	4	no	—
169	Q₈	—	94	2	no	—
170	D₄	—	94	2	no	—
171	D₄	—	94	2	no	—
172	D₄	—	376	2	no	—
173	D₄	—	376	2	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	94	4	no	—
176	C₂×C₄	—	94	2	no	—
177	C₂×C₄	—	94	2	no	—
178	C₂×C₄	—	94	2	no	—
179	C₂×C₄	—	376	2	no	—
180	C₂×C₄	C₂×D₁₈₈	1	2	yes	—
181	C₂×C₄	C₂×D₁₈₈	1	2	yes	—
182	C₂×C₄	C₂×D₁₈₈	1	1	yes	characteristic
183	C₂×C₄	C₂³×D₄₇	1	1	yes	characteristic
184	C₈	—	2	2	no	—
185	C₈	—	2	2	no	—
186	C₂²	—	2	1	no	—
187	C₂²	—	2	2	no	—
188	C₂²	—	188	2	no	—
189	C₂²	—	188	2	no	—
190	C₂²	—	188	2	no	—
191	C₂²	C₂²×D₁₈₈	1	1	yes	characteristic
192	C₄	—	2	1	no	—
193	C₄	—	2	2	no	—
194	C₄	—	2	2	no	—
195	C₄	—	94	4	no	—
196	C₄	—	188	2	no	—
197	C₄	—	188	2	no	—
198	C₄	#41×D₄₇	1	2	yes	—
199	C₂	—	4	1	no	—
200	C₂	—	376	2	no	—
201	C₂	#15689635	1	1	yes	characteristic
202	C₂	#15689935	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#15689728

C₂

yes

maximal, characteristic

#15689600

C₂

yes

maximal, characteristic, complemented

#15689448

C₂

yes

maximal, complemented

#15689431

C₂

yes

maximal, complemented

#15689324

C₂

yes

maximal, complemented

#15689264

C₂

yes

maximal, complemented

#15689225

C₂

yes

maximal, complemented

C₄×#10625325

C₂

yes

maximal, complemented

C₄₇×#433

C₂

yes

maximal, characteristic, complemented

C₂²×Dic₉₄

C₂²

yes

characteristic

C₂×#10625325

C₂²

yes

—

C₂×#10625325

C₂²

yes

—

#13094021

C₂²

yes

characteristic

#13094016

C₂²

yes

characteristic

#13094009

—

#13093998

C₂²

yes

—

#13093997

—

C₄×Dic₉₄

C₂²

yes

—

C₄×Dic₉₄

C₂²

yes

—

#13093989

—

C₄×D₁₈₈

C₂²

yes

—

#13093983

C₂²

yes

characteristic

#13093980

—

#13093958

C₂²

yes

—

#13093956

C₂²

yes

—

#13093949

C₂²

yes

—

#13093949

C₂²

yes

—

#13093946

C₂²

yes

—

#13093946

C₂²

yes

—

#117×C₄₇

C₂²

yes

characteristic

C₄⋊C₈×C₄₇

C₂²

yes

—

C₂²⋊C₈×C₄₇

C₂²

yes

—

#97×C₄₇

C₂²

yes

characteristic

C₄×C₈×C₄₇

C₂²

yes

characteristic

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂×C₂²⋊D₄₇

—

C₂²×Dic₄₇

—

C₂×D₁₈₈

—

C₂×D₁₈₈

C₂³

yes

—

C₂×C₄×D₄₇

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₂×Dic₉₄

C₂³

yes

—

C₂×C₈×C₄₇

C₂³

yes

—

C₂×C₈×C₄₇

C₂³

yes

—

C₄⋊C₄×C₄₇

D₄

yes

—

C₂²⋊C₄×C₄₇

D₄

yes

—

C₄²×C₄₇

C₂³

yes

characteristic

C₄²×C₄₇

C₂³

yes

characteristic

C₂²⋊Dic₄₇

—

(C₂×D₄₇)⋊C₄

—

(C₂×D₄₇)⋊C₄

—

C₄⋊Dic₄₇

C₂³

yes

—

C₄⋊Dic₄₇

C₂³

yes

—

Dic₄₇⋊C₄

—

Dic₄₇⋊C₄

—

Dic₄₇⋊C₄

—

C₄×Dic₄₇

—

C₄×Dic₄₇

—

C₄×Dic₄₇

—

#10625325

—

#10625325

—

C₂³×C₄₇

C₂×D₄

yes

characteristic

C₂²×D₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂⁴

yes

characteristic

C₂²⋊D₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

D₁₈₈

—

D₁₈₈

—

D₁₈₈

—

C₄×D₄₇

—

Dic₉₄

—

Dic₉₄

—

Dic₉₄

#41

yes

—

C₈×C₄₇

—

C₈×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

C₂²×D₄

yes

characteristic

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₂×#41

yes

—

Dic₄₇

—

Dic₄₇

—

Dic₄₇

—

#3126

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

#525

yes

characteristic

100

C₂×C₄₇

#577

yes

—

101

D₄₇

—

102

#540

—

103

#519

—

104

#482

—

105

#479

—

106

#460

—

107

#448

—

108

#440

—

109

C₄×SD₁₆

—

110

#433

D₄₇

yes

characteristic, complemented

111

C₄₇

#3126

yes

characteristic, complemented, Sylow

112

C₂²×Q₈

—

113

C₂×SD₁₆

—

114

C₂×SD₁₆

—

115

#127

—

116

#125

—

117

#123

—

118

#121

—

119

#121

—

120

C₄×Q₈

—

121

C₄×Q₈

—

122

C₄×Q₈

—

123

C₄×D₄

—

124

#117

—

125

#117

—

126

#117

C₂×D₄₇

yes

characteristic

127

#107

—

128

#106

—

129

C₄⋊C₈

C₂×D₄₇

yes

—

130

#103

—

131

#103

—

132

#102

—

133

#102

—

134

C₂²⋊C₈

C₂×D₄₇

yes

—

135

#97

C₂×D₄₇

yes

characteristic

136

C₄×C₈

C₂×D₄₇

yes

characteristic

137

C₂×Q₈

—

138

C₂×Q₈

—

139

C₂×Q₈

—

140

C₂×D₄

—

141

C₂×D₄

—

142

C₂×D₄

—

143

C₂²×C₄

—

144

C₂²×C₄

—

145

C₂²×C₄

C₂²×D₄₇

yes

characteristic

146

SD₁₆

—

147

SD₁₆

—

148

C₂×C₈

C₂²×D₄₇

yes

—

149

C₂×C₈

C₂²×D₄₇

yes

—

150

C₄⋊C₄

—

151

C₄⋊C₄

—

152

C₄⋊C₄

—

153

C₄⋊C₄

—

154

C₄⋊C₄

—

155

C₄⋊C₄

D₁₈₈

yes

—

156

C₂²⋊C₄

—

157

C₂²⋊C₄

—

158

C₂²⋊C₄

—

159

C₂²⋊C₄

D₁₈₈

yes

—

160

C₄²

—

161

C₄²

—

162

C₄²

—

163

C₄²

C₂²×D₄₇

yes

characteristic

164

C₄²

C₂²×D₄₇

yes

characteristic

165

C₂³

—

166

C₂³

C₂×D₁₈₈

yes

characteristic

167

Q₈

—

168

Q₈

—

169

Q₈

—

170

D₄

—

171

D₄

—

172

D₄

—

376

—

173

D₄

—

376

—

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

—

376

—

180

C₂×C₄

C₂×D₁₈₈

yes

—

181

C₂×C₄

C₂×D₁₈₈

yes

—

182

C₂×C₄

C₂×D₁₈₈

yes

characteristic

183

C₂×C₄

C₂³×D₄₇

yes

characteristic

184

C₈

—

185

C₈

—

186

C₂²

—

187

C₂²

—

188

C₂²

—

188

—

189

C₂²

—

188

—

190

C₂²

—

188

—

191

C₂²

C₂²×D₁₈₈

yes

characteristic

192

C₄

—

193

C₄

—

194

C₄

—

195

C₄

—

196

C₄

—

188

—

197

C₄

—

188

—

198

C₄

#41×D₄₇

yes

—

199

C₂

—

200

C₂

—

376

—

201

C₂

#15689635

yes

characteristic

202

C₂

#15689935

yes

—

	subgroup	G/H
center	C₂²	C₂²×D₁₈₈
commutator	C₂×C₄×C₄₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₄₇
Fitting	C₄₇×#433	C₂
socle	C₂²×C₄₇	C₂²×D₄
hypercenter	#433	D₄₇
hypocenter	C₄₇	#3126
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ #117 ◁ #433

subgroup

G/H

center

C₂²

C₂²×D₁₈₈

commutator

C₂×C₄×C₄₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₄₇

Fitting

C₄₇×#433

C₂

socle

C₂²×C₄₇

C₂²×D₄

D₄₇

C₄₇

G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ #117 ◁ #433

N	H ≅ G/N	type
2. #15689600	C₂	semidirect
3. #15689448	C₂	semidirect
4. #15689431	C₂	semidirect
5. #15689324	C₂	semidirect
6. #15689264	C₂	semidirect
7. #15689225	C₂	semidirect
8. C₄×#10625325	C₂	semidirect
9. C₄₇×#433	C₂	semidirect
110. #433	D₄₇	semidirect
111. C₄₇	#3126	semidirect

H ≅ G/N

type

2. #15689600

C₂

semidirect

3. #15689448

C₂

semidirect

4. #15689431

C₂

semidirect

5. #15689324

C₂

semidirect

6. #15689264

C₂

semidirect

7. #15689225

C₂

semidirect

8. C₄×#10625325

C₂

semidirect

9. C₄₇×#433

C₂

semidirect

110. #433

D₄₇

semidirect

111. C₄₇

#3126

semidirect

Subgroups — #20253680

Conjugacy classes of subgroups

Complements