Group #14293731 — subgroups

Subgroups — #14293731

Order 1920.

440 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 116 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	#11750781	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×#64984	C₂	1	1	yes	maximal, characteristic
3	#11747071	C₂	1	2	yes	maximal, complemented
4	#11746890	C₂	1	2	yes	maximal, complemented
5	C₅×#4455	C₂	1	1	yes	maximal, characteristic, complemented
6	#10598884	—	3	1	no	maximal, Hall
7	#91758	C₂²	1	2	yes	—
8	C₂×C₃×D₄₀	—	2	2	no	—
9	C₃×#4020	C₂²	1	1	yes	characteristic
10	C₃×#4020	C₂²	1	1	yes	characteristic
11	C₂×#6657	—	2	4	no	—
12	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
13	#91101	—	2	2	no	—
14	C₄×C₅×C₃⋊C₈	C₂²	1	2	yes	—
15	#90965	—	2	4	no	—
16	#71277	—	5	1	no	maximal, Hall
17	#64984	D₃	1	1	yes	characteristic
18	#64983	—	3	2	no	—
19	#64982	—	3	2	no	—
20	#64969	—	3	1	no	—
21	C₅×#322	—	3	1	no	—
22	C₂×Dic₃₀	—	2	2	no	—
23	C₂×C₃×D₂₀	—	2	2	no	—
24	C₂×C₃×D₂₀	—	2	2	no	—
25	C₂×C₃×D₂₀	—	2	1	no	—
26	C₂×C₃×D₂₀	—	2	1	no	—
27	C₂×C₃×C₅×C₈	D₄	1	2	yes	—
28	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
29	C₄⋊Dic₁₅	—	2	2	no	—
30	C₄⋊Dic₁₅	—	2	2	no	—
31	C₂×C₅×C₃⋊C₈	D₄	1	4	yes	—
32	C₃×D₄₀	—	4	4	no	—
33	#6657	—	4	8	no	—
34	C₃×#493	—	5	1	no	—
35	#5076	—	5	2	no	—
36	#5070	—	5	2	no	—
37	#4680	—	5	1	no	—
38	#4455	D₅	1	1	yes	characteristic, complemented
39	C₂×D₄₀	—	2	2	no	—
40	C₂×#677	—	6	4	no	—
41	#4020	C₂×D₃	1	1	yes	characteristic
42	#4020	C₂×D₃	1	1	yes	characteristic
43	#4015	—	3	2	no	—
44	C₄×C₅×C₈	—	3	2	no	—
45	C₄×C₅×C₈	C₂×D₃	1	1	yes	characteristic
46	#3953	—	6	4	no	—
47	#3949	—	6	2	no	—
48	#1715	—	15	1	no	Sylow
49	C₂²×C₃×D₅	—	4	1	no	—
50	C₂²×C₃×D₅	—	4	1	no	—
51	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
52	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
53	C₂×Dic₁₅	—	4	2	no	—
54	Dic₃₀	—	4	4	no	—
55	C₃×D₂₀	—	4	2	no	—
56	C₃×D₂₀	—	4	2	no	—
57	C₃×D₂₀	—	4	4	no	—
58	C₃×D₂₀	—	4	4	no	—
59	C₃×C₅×C₈	SD₁₆	1	4	yes	complemented
60	C₅×C₃⋊C₈	—	2	4	no	—
61	C₂×C₃×D₈	—	10	2	no	—
62	C₃×#127	—	5	1	no	—
63	C₃×#127	—	5	1	no	—
64	C₂×#214	—	10	4	no	—
65	#869	—	5	2	no	—
66	C₃×C₄×C₈	C₂×D₅	1	1	yes	characteristic
67	#832	—	10	4	no	—
68	#817	—	10	2	no	—
69	C₄×C₃⋊C₈	C₂×D₅	1	2	yes	—
70	C₂×D₂₀	—	2	2	no	—
71	C₂×D₂₀	—	2	2	no	—
72	C₂×D₂₀	—	2	1	no	—
73	C₂×D₂₀	—	2	1	no	—
74	C₂×Dic₁₀	—	6	2	no	—
75	C₂×C₅×C₈	—	3	4	no	—
76	C₂×C₅×C₈	#67	1	2	yes	—
77	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
78	C₄⋊Dic₅	—	6	2	no	—
79	C₄⋊Dic₅	—	6	2	no	—
80	D₄₀	—	4	4	no	—
81	#677	—	12	8	no	—
82	#498	—	15	1	no	—
83	#493	—	5	1	no	—
84	#492	—	15	2	no	—
85	#486	—	15	2	no	—
86	#322	—	3	1	no	—
87	C₂²×C₃×C₅	#127	1	1	yes	characteristic
88	C₂×C₃×D₅	—	8	1	no	—
89	C₂×C₃×D₅	—	8	1	no	—
90	C₂×C₃×D₅	—	8	2	no	—
91	C₂×C₃×D₅	—	8	2	no	—
92	C₃×C₄×C₅	C₂×SD₁₆	1	2	yes	—
93	C₃×C₄×C₅	AGL(1,ℤ/8)	1	4	yes	—
94	Dic₁₅	—	8	2	no	—
95	C₂×C₃×D₄	—	10	2	no	—
96	C₂×C₃×D₄	—	10	2	no	—
97	C₂×C₃×D₄	—	10	1	no	—
98	C₂×C₃×D₄	—	10	1	no	—
99	C₂×Dic₆	—	10	2	no	—
100	C₃×D₈	—	20	4	no	—
101	C₂×C₃×C₈	D₂₀	1	2	yes	—
102	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
103	#214	—	20	8	no	—
104	C₄⋊Dic₃	—	10	2	no	—
105	C₄⋊Dic₃	—	10	2	no	—
106	C₂×C₃⋊C₈	D₂₀	1	4	yes	—
107	C₂²×D₅	—	4	1	no	—
108	C₂²×D₅	—	4	1	no	—
109	C₂×C₄×C₅	D₃×D₄	1	2	yes	—
110	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
111	C₂×Dic₅	—	12	2	no	—
112	D₂₀	—	4	4	no	—
113	D₂₀	—	4	4	no	—
114	D₂₀	—	4	2	no	—
115	D₂₀	—	4	2	no	—
116	Dic₁₀	—	12	4	no	—
117	C₅×C₈	—	6	4	no	—
118	C₅×C₈	#214	1	4	yes	complemented
119	C₂×SD₁₆	—	30	4	no	—
120	C₂×D₈	—	10	2	no	—
121	C₄⋊Q₈	—	15	2	no	—
122	#127	—	5	1	no	—
123	#127	—	5	1	no	—
124	#106	—	30	2	no	—
125	#102	—	30	4	no	—
126	C₄×C₈	—	3	2	no	—
127	C₄×C₈	D₃×D₅	1	1	yes	characteristic
128	C₂×C₃×C₅	#492	1	1	yes	characteristic
129	C₂×C₃×C₅	#496	1	2	yes	—
130	C₃×D₅	—	16	1	no	—
131	C₃×D₅	—	16	1	no	—
132	C₂³×C₃	—	20	1	no	—
133	C₂³×C₃	—	20	1	no	—
134	C₃×D₄	—	20	2	no	—
135	C₃×D₄	—	20	2	no	—
136	C₃×D₄	—	20	4	no	—
137	C₃×D₄	—	20	4	no	—
138	C₂×C₃×C₄	C₂×D₂₀	1	2	yes	—
139	C₂×C₃×C₄	C₂×D₂₀	1	1	yes	characteristic
140	C₂×Dic₃	—	20	2	no	—
141	Dic₆	—	20	4	no	—
142	C₃×C₈	#677	1	4	yes	complemented
143	C₃⋊C₈	—	2	4	no	—
144	C₂²×C₅	#940	1	1	yes	characteristic
145	C₂×D₅	—	8	2	no	—
146	C₂×D₅	—	8	2	no	—
147	C₂×D₅	—	8	1	no	—
148	C₂×D₅	—	8	1	no	—
149	C₄×C₅	#911	1	4	yes	—
150	C₄×C₅	C₂×#214	1	2	yes	—
151	Dic₅	—	24	2	no	—
152	C₂×Q₈	—	30	2	no	—
153	C₂×D₄	—	10	2	no	—
154	C₂×D₄	—	10	2	no	—
155	C₂×D₄	—	10	1	no	—
156	C₂×D₄	—	10	1	no	—
157	SD₁₆	—	60	8	no	—
158	D₈	—	20	4	no	—
159	C₂×C₈	—	3	4	no	—
160	C₂×C₈	#1202	1	2	yes	—
161	C₄⋊C₄	—	30	2	no	—
162	C₄⋊C₄	—	30	2	no	—
163	C₄²	C₂×D₃×D₅	1	1	yes	characteristic
164	C₃×C₅	#1715	1	1	yes	characteristic, complemented, Hall
165	C₂²×C₃	—	40	1	no	—
166	C₂²×C₃	—	40	1	no	—
167	C₂²×C₃	—	40	2	no	—
168	C₂²×C₃	—	40	2	no	—
169	C₂²×C₃	#4020	1	1	yes	characteristic
170	C₃×C₄	C₂×#677	1	2	yes	—
171	C₃×C₄	#4054	1	4	yes	—
172	Dic₃	—	40	2	no	—
173	C₂×C₅	#5076	1	1	yes	characteristic
174	C₂×C₅	#5154	1	2	yes	—
175	D₅	—	16	1	no	—
176	D₅	—	16	1	no	—
177	C₂³	—	20	1	no	—
178	C₂³	—	20	1	no	—
179	Q₈	—	60	4	no	—
180	D₄	—	20	4	no	—
181	D₄	—	20	4	no	—
182	D₄	—	20	2	no	—
183	D₄	—	20	2	no	—
184	C₂×C₄	—	60	2	no	—
185	C₂×C₄	D₃×D₂₀	1	2	yes	—
186	C₂×C₄	C₂×#1202	1	1	yes	characteristic
187	C₈	—	6	4	no	—
188	C₈	#6657	1	4	yes	complemented
189	C₂×C₃	—	80	1	no	—
190	C₂×C₃	—	80	1	no	—
191	C₂×C₃	#64982	1	1	yes	characteristic
192	C₂×C₃	#65001	1	2	yes	—
193	C₅	#71277	1	1	yes	characteristic, complemented, Sylow
194	C₂²	—	40	2	no	—
195	C₂²	—	40	2	no	—
196	C₂²	—	40	1	no	—
197	C₂²	—	40	1	no	—
198	C₂²	#91458	1	1	yes	characteristic
199	C₄	—	120	2	no	—
200	C₄	#91254	1	4	yes	—
201	C₄	C₂×#6657	1	2	yes	—
202	C₃	#10598884	1	1	yes	characteristic, complemented, Sylow
203	C₂	—	80	1	no	—
204	C₂	—	80	1	no	—
205	C₂	#11747071	1	1	yes	characteristic
206	C₂	#11748009	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#11750781

C₂

yes

maximal, characteristic, complemented

C₃×#64984

C₂

yes

maximal, characteristic

#11747071

C₂

yes

maximal, complemented

#11746890

C₂

yes

maximal, complemented

C₅×#4455

C₂

yes

maximal, characteristic, complemented

#10598884

—

maximal, Hall

#91758

C₂²

yes

—

C₂×C₃×D₄₀

—

C₃×#4020

C₂²

yes

characteristic

C₃×#4020

C₂²

yes

characteristic

C₂×#6657

—

C₃×C₄×C₅×C₈

C₂²

yes

characteristic

#91101

—

C₄×C₅×C₃⋊C₈

C₂²

yes

—

—

—

D₃

yes

—

—

—

—

C₂×Dic₃₀

—

C₂×C₃×D₂₀

—

C₂×C₃×D₂₀

—

C₂×C₃×D₂₀

—

C₂×C₃×D₂₀

—

C₂×C₃×C₅×C₈

D₄

yes

—

C₃×C₄²×C₅

C₂³

yes

characteristic

C₄⋊Dic₁₅

—

C₄⋊Dic₁₅

—

C₂×C₅×C₃⋊C₈

D₄

yes

—

C₃×D₄₀

—

—

—

—

—

—

D₅

yes

characteristic, complemented

C₂×D₄₀

—

C₂×#677

—

#4020

C₂×D₃

yes

characteristic

#4020

C₂×D₃

yes

characteristic

#4015

—

C₄×C₅×C₈

—

C₄×C₅×C₈

C₂×D₃

yes

—

—

—

C₂²×C₃×D₅

—

C₂²×C₃×D₅

—

C₂×C₃×C₄×C₅

C₂×D₄

yes

characteristic

C₂×C₃×C₄×C₅

C₂×D₄

yes

—

C₂×Dic₁₅

—

Dic₃₀

—

C₃×D₂₀

—

C₃×D₂₀

—

C₃×D₂₀

—

C₃×D₂₀

—

C₃×C₅×C₈

SD₁₆

yes

complemented

C₅×C₃⋊C₈

—

C₂×C₃×D₈

—

—

—

—

—

C₃×C₄×C₈

C₂×D₅

yes

characteristic

#832

—

#817

—

C₄×C₃⋊C₈

C₂×D₅

yes

—

C₂×D₂₀

—

C₂×D₂₀

—

C₂×D₂₀

—

C₂×D₂₀

—

C₂×Dic₁₀

—

C₂×C₅×C₈

—

C₂×C₅×C₈

#67

yes

—

C₄²×C₅

C₂²×D₃

yes

characteristic

C₄⋊Dic₅

—

C₄⋊Dic₅

—

D₄₀

—

—

—

—

—

—

—

C₂²×C₃×C₅

#127

yes

characteristic

C₂×C₃×D₅

—

C₂×C₃×D₅

—

C₂×C₃×D₅

—

C₂×C₃×D₅

—

C₃×C₄×C₅

C₂×SD₁₆

yes

—

C₃×C₄×C₅

AGL(1,ℤ/8)

yes

—

Dic₁₅

—

C₂×C₃×D₄

—

C₂×C₃×D₄

—

C₂×C₃×D₄

—

C₂×C₃×D₄

—

C₂×Dic₆

—

100

C₃×D₈

—

101

C₂×C₃×C₈

D₂₀

yes

—

102

C₃×C₄²

C₂²×D₅

yes

characteristic

103

#214

—

104

C₄⋊Dic₃

—

105

C₄⋊Dic₃

—

106

C₂×C₃⋊C₈

D₂₀

yes

—

107

C₂²×D₅

—

108

C₂²×D₅

—

109

C₂×C₄×C₅

D₃×D₄

yes

—

110

C₂×C₄×C₅

C₂×#67

yes

characteristic

111

C₂×Dic₅

—

112

D₂₀

—

113

D₂₀

—

114

D₂₀

—

115

D₂₀

—

116

Dic₁₀

—

117

C₅×C₈

—

118

C₅×C₈

#214

yes

complemented

119

C₂×SD₁₆

—

120

C₂×D₈

—

121

C₄⋊Q₈

—

122

#127

—

123

#127

—

124

#106

—

125

#102

—

126

C₄×C₈

—

127

C₄×C₈

D₃×D₅

yes

characteristic

128

C₂×C₃×C₅

#492

yes

characteristic

129

C₂×C₃×C₅

#496

yes

—

130

C₃×D₅

—

131

C₃×D₅

—

132

C₂³×C₃

—

133

C₂³×C₃

—

134

C₃×D₄

—

135

C₃×D₄

—

136

C₃×D₄

—

137

C₃×D₄

—

138

C₂×C₃×C₄

C₂×D₂₀

yes

—

139

C₂×C₃×C₄

C₂×D₂₀

yes

characteristic

140

C₂×Dic₃

—

141

Dic₆

—

142

C₃×C₈

#677

yes

complemented

143

C₃⋊C₈

—

144

C₂²×C₅

#940

yes

characteristic

145

C₂×D₅

—

146

C₂×D₅

—

147

C₂×D₅

—

148

C₂×D₅

—

149

C₄×C₅

#911

yes

—

150

C₄×C₅

C₂×#214

yes

—

151

Dic₅

—

152

C₂×Q₈

—

153

C₂×D₄

—

154

C₂×D₄

—

155

C₂×D₄

—

156

C₂×D₄

—

157

SD₁₆

—

158

D₈

—

159

C₂×C₈

—

160

C₂×C₈

#1202

yes

—

161

C₄⋊C₄

—

162

C₄⋊C₄

—

163

C₄²

C₂×D₃×D₅

yes

characteristic

164

C₃×C₅

#1715

yes

characteristic, complemented, Hall

165

C₂²×C₃

—

166

C₂²×C₃

—

167

C₂²×C₃

—

168

C₂²×C₃

—

169

C₂²×C₃

#4020

yes

characteristic

170

C₃×C₄

C₂×#677

yes

—

171

C₃×C₄

#4054

yes

—

172

Dic₃

—

173

C₂×C₅

#5076

yes

characteristic

174

C₂×C₅

#5154

yes

—

175

D₅

—

176

D₅

—

177

C₂³

—

178

C₂³

—

179

Q₈

—

180

D₄

—

181

D₄

—

182

D₄

—

183

D₄

—

184

C₂×C₄

—

185

C₂×C₄

D₃×D₂₀

yes

—

186

C₂×C₄

C₂×#1202

yes

characteristic

187

C₈

—

188

C₈

#6657

yes

complemented

189

C₂×C₃

—

190

C₂×C₃

—

191

C₂×C₃

#64982

yes

characteristic

192

C₂×C₃

#65001

yes

—

193

C₅

#71277

yes

characteristic, complemented, Sylow

194

C₂²

—

195

C₂²

—

196

C₂²

—

197

C₂²

—

198

C₂²

#91458

yes

characteristic

199

C₄

—

120

—

200

C₄

#91254

yes

—

201

C₄

C₂×#6657

yes

—

202

C₃

#10598884

yes

characteristic, complemented, Sylow

203

C₂

—

204

C₂

—

205

C₂

#11747071

yes

characteristic

206

C₂

#11748009

yes

—

	subgroup	G/H
center	C₂²	#91458
commutator, cosocle	C₃×C₄²×C₅	C₂³
Frattini	C₄²	C₂×D₃×D₅
Fitting	C₃×C₄×C₅×C₈	C₂²
socle	C₂²×C₃×C₅	#127
prime radical	C₃×#64984	C₂
hypercenter	C₄×C₈	D₃×D₅
hypocenter	C₃×C₅	#1715
LCS	G ▷ C₃×C₄²×C₅ ▷ C₂²×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈

subgroup

G/H

center

C₂²

#91458

commutator, cosocle

C₃×C₄²×C₅

C₂³

Frattini

C₄²

C₂×D₃×D₅

Fitting

C₃×C₄×C₅×C₈

C₂²

C₂²×C₃×C₅

C₂

C₄×C₈

D₃×D₅

hypocenter

C₃×C₅

#1715

LCS

G ▷ C₃×C₄²×C₅ ▷ C₂²×C₃×C₅ ▷ C₃×C₅

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈

N	H ≅ G/N	type
1. #11750781	C₂	semidirect
3. #11747071	C₂	semidirect
4. #11746890	C₂	semidirect
5. C₅×#4455	C₂	semidirect
38. #4455	D₅	semidirect
59. C₃×C₅×C₈	SD₁₆	semidirect
118. C₅×C₈	#214	semidirect
142. C₃×C₈	#677	semidirect
164. C₃×C₅	#1715	semidirect
188. C₈	#6657	semidirect
193. C₅	#71277	semidirect
202. C₃	#10598884	semidirect

H ≅ G/N

type

1. #11750781

C₂

semidirect

3. #11747071

C₂

semidirect

4. #11746890

C₂

semidirect

5. C₅×#4455

C₂

semidirect

38. #4455

D₅

semidirect

59. C₃×C₅×C₈

SD₁₆

semidirect

118. C₅×C₈

#214

semidirect

142. C₃×C₈

#677

semidirect

164. C₃×C₅

#1715

semidirect

188. C₈

#6657

semidirect

193. C₅

#71277

semidirect

202. C₃

#10598884

semidirect

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

subgroup	index
#11750781	2
C₃×#64984	2
#11747071	2
#11746890	2
C₅×#4455	2
#10598884	3
#71277	5

subgroup

index

Subgroups — #14293731

Conjugacy classes of subgroups

Complements

Maximal subgroups