Group #14292815 — subgroups

Subgroups — #14292815

Order 1920.

304 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 304 conjugacy classes of subgroups — too many to draw. Only its 96 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	C₅×#4679	C₂	1	1	yes	maximal, characteristic
2	C₃×#64971	C₂	1	1	yes	maximal, characteristic
3	#11747099	C₂	1	2	yes	maximal
4	#11746278	C₂	1	1	yes	maximal, characteristic
5	#11745855	C₂	1	2	yes	maximal
6	#10601757	—	3	1	no	maximal, Hall
7	C₄×C₅×Dic₆	C₂²	1	1	yes	characteristic
8	C₃×#4015	C₂²	1	2	yes	—
9	C₂×#6661	—	2	2	no	—
10	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
11	C₄⋊(C₃⋊#168)	C₂²	1	2	yes	—
12	C₅×#815	—	2	1	no	—
13	C₅×Dic₃⋊C₈	—	2	1	no	—
14	C₅×C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
15	C₃×#3950	—	2	1	no	—
16	C₃×#3950	—	2	1	no	—
17	#90970	—	2	2	no	—
18	#71379	—	5	1	no	maximal, Hall
19	C₅×#446	—	3	1	no	—
20	#65320	—	3	2	no	—
21	#64971	D₃	1	1	yes	characteristic
22	#64706	—	3	2	no	—
23	#64677	—	3	1	no	—
24	C₂×C₅×Dic₆	D₄	1	1	yes	characteristic
25	C₂×C₃×Dic₁₀	—	2	2	no	—
26	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
27	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
28	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
29	C₂×C₃⋊#168	—	2	2	no	—
30	C₅×C₄⋊Dic₃	D₄	1	1	yes	characteristic
31	C₅×Dic₃⋊C₄	—	2	1	no	—
32	C₄×C₅×Dic₃	—	2	1	no	—
33	C₂×C₅×C₃⋊C₈	—	2	1	no	—
34	C₃×C₄⋊Dic₅	—	2	2	no	—
35	C₃×C₄⋊Dic₅	—	2	2	no	—
36	#6661	—	4	4	no	—
37	C₃×#500	—	5	1	no	—
38	#5090	—	5	2	no	—
39	#4679	D₅	1	1	yes	characteristic
40	#4547	—	5	2	no	—
41	#4457	—	5	1	no	—
42	C₄×C₅×Q₈	—	3	1	no	—
43	C₂×#689	—	6	2	no	—
44	#4015	C₂×D₃	1	2	yes	—
45	C₅×C₄⋊C₈	—	3	1	no	—
46	C₅×C₄⋊C₈	—	6	1	no	—
47	C₅×#97	—	6	1	no	—
48	C₄×C₅×C₈	C₂×D₃	1	1	yes	characteristic
49	#3950	—	2	1	no	—
50	#3950	—	2	1	no	—
51	#3942	—	6	2	no	—
52	C₄⋊#168	—	3	2	no	—
53	#1694	—	15	1	no	Sylow
54	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
55	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
56	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
57	C₂×C₅×Dic₃	—	2	1	no	—
58	C₂×C₅×Dic₃	—	2	1	no	—
59	C₅×Dic₆	—	2	2	no	—
60	C₂×C₃×Dic₅	—	4	2	no	—
61	C₃×Dic₁₀	—	4	2	no	—
62	C₃×Dic₁₀	—	4	2	no	—
63	C₃×C₅×C₈	—	2	1	no	—
64	C₃×C₅×C₈	Q₁₆	1	2	yes	complemented
65	C₃⋊#168	—	4	2	no	—
66	C₅×C₃⋊C₈	—	4	1	no	—
67	C₃×C₄⋊Q₈	—	5	2	no	—
68	C₂×#216	—	10	2	no	—
69	C₄×Dic₆	C₂×D₅	1	1	yes	characteristic
70	C₃×#107	—	10	1	no	—
71	C₃×#107	—	10	1	no	—
72	C₃×C₄×C₈	C₂×D₅	1	1	yes	characteristic
73	#835	—	10	2	no	—
74	#815	—	2	1	no	—
75	Dic₃⋊C₈	—	2	1	no	—
76	C₄⋊(C₃⋊C₈)	—	5	2	no	—
77	C₄⋊(C₃⋊C₈)	C₂×D₅	1	1	yes	characteristic
78	C₂×C₅×Q₈	—	3	1	no	—
79	C₂×Dic₁₀	—	2	2	no	—
80	C₂×C₅×C₈	—	6	1	no	—
81	C₂×C₅×C₈	#67	1	1	yes	characteristic
82	C₂×C₅×C₈	#67	1	1	yes	characteristic
83	C₅×C₄⋊C₄	—	3	1	no	—
84	C₅×C₄⋊C₄	—	6	1	no	—
85	C₄²×C₅	—	6	1	no	—
86	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
87	#689	—	12	4	no	—
88	C₄⋊Dic₅	—	2	2	no	—
89	C₄⋊Dic₅	—	2	2	no	—
90	C₂×#168	—	6	2	no	—
91	#500	—	5	1	no	—
92	#462	—	15	2	no	—
93	#446	—	3	1	no	—
94	#334	—	15	1	no	—
95	#332	—	15	2	no	—
96	C₂²×C₃×C₅	#121	1	1	yes	characteristic
97	C₃×C₄×C₅	—	2	1	no	—
98	C₃×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
99	C₃×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
100	C₃×C₄×C₅	#137	1	1	yes	characteristic
101	C₃×Dic₅	—	8	2	no	—
102	C₅×Dic₃	—	4	1	no	—
103	C₅×Dic₃	—	4	1	no	—
104	C₂×C₃×Q₈	—	10	2	no	—
105	C₂×Dic₆	D₂₀	1	1	yes	characteristic
106	C₂×C₃×C₈	C₂²⋊D₅	1	1	yes	characteristic
107	C₂×C₃×C₈	C₂²⋊D₅	1	1	yes	characteristic
108	C₃×C₄⋊C₄	—	10	2	no	—
109	C₃×C₄⋊C₄	—	10	2	no	—
110	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
111	#216	—	20	4	no	—
112	C₄⋊Dic₃	D₂₀	1	1	yes	characteristic
113	Dic₃⋊C₄	—	2	1	no	—
114	C₄×Dic₃	—	2	1	no	—
115	C₂×C₃⋊C₈	—	2	1	no	—
116	C₂×C₃⋊C₈	—	10	2	no	—
117	C₅×Q₈	—	6	2	no	—
118	C₂×C₄×C₅	—	6	1	no	—
119	C₂×C₄×C₅	—	6	1	no	—
120	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
121	C₂×C₄×C₅	#237	1	1	yes	characteristic
122	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
123	C₂×Dic₅	—	4	2	no	—
124	Dic₁₀	—	4	2	no	—
125	Dic₁₀	—	4	2	no	—
126	C₅×C₈	—	2	1	no	—
127	C₅×C₈	—	12	1	no	—
128	C₅×C₈	#216	1	2	yes	complemented
129	#168	—	12	2	no	—
130	C₂×Q₁₆	—	30	2	no	—
131	C₄⋊Q₈	—	5	2	no	—
132	C₄×Q₈	—	3	1	no	—
133	#107	—	10	1	no	—
134	#107	—	10	1	no	—
135	C₄⋊C₈	—	3	1	no	—
136	C₄⋊C₈	—	6	1	no	—
137	C₄⋊C₈	—	15	2	no	—
138	#103	—	30	2	no	—
139	#97	—	6	1	no	—
140	C₄×C₈	D₃×D₅	1	1	yes	characteristic
141	C₂×C₃×C₅	#462	1	1	yes	characteristic
142	C₂×C₃×C₅	#469	1	1	yes	characteristic
143	C₂×C₃×C₅	#473	1	1	yes	characteristic
144	C₃×Q₈	—	20	2	no	—
145	C₃×Q₈	—	20	2	no	—
146	C₂×C₃×C₄	—	20	2	no	—
147	C₂×C₃×C₄	C₂×D₂₀	1	1	yes	characteristic
148	C₂×C₃×C₄	#709	1	1	yes	characteristic
149	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
150	C₂×Dic₃	—	2	1	no	—
151	C₂×Dic₃	—	2	1	no	—
152	Dic₆	—	2	2	no	—
153	C₃×C₈	—	2	1	no	—
154	C₃×C₈	#689	1	2	yes	complemented
155	C₃⋊C₈	—	4	1	no	—
156	C₃⋊C₈	—	20	2	no	—
157	C₂²×C₅	#938	1	1	yes	characteristic
158	C₄×C₅	—	2	1	no	—
159	C₄×C₅	—	12	1	no	—
160	C₄×C₅	—	12	1	no	—
161	C₄×C₅	#911	1	2	yes	—
162	C₄×C₅	#942	1	1	yes	characteristic
163	C₄×C₅	C₂×#216	1	1	yes	characteristic
164	Dic₅	—	8	2	no	—
165	C₂×Q₈	—	3	1	no	—
166	C₂×Q₈	—	10	2	no	—
167	Q₁₆	—	60	4	no	—
168	C₂×C₈	—	6	1	no	—
169	C₂×C₈	—	30	2	no	—
170	C₂×C₈	#1201	1	1	yes	characteristic
171	C₂×C₈	#1201	1	1	yes	characteristic
172	C₄⋊C₄	—	3	1	no	—
173	C₄⋊C₄	—	6	1	no	—
174	C₄⋊C₄	—	10	2	no	—
175	C₄⋊C₄	—	10	2	no	—
176	C₄²	—	6	1	no	—
177	C₄²	C₂×D₃×D₅	1	1	yes	characteristic
178	C₃×C₅	#1694	1	1	yes	characteristic, complemented, Hall
179	C₂²×C₃	#4076	1	1	yes	characteristic
180	C₃×C₄	—	2	1	no	—
181	C₃×C₄	—	40	2	no	—
182	C₃×C₄	#4054	1	2	yes	—
183	C₃×C₄	C₂×#689	1	1	yes	characteristic
184	C₃×C₄	#4097	1	1	yes	characteristic
185	Dic₃	—	4	1	no	—
186	Dic₃	—	4	1	no	—
187	C₂×C₅	#5090	1	1	yes	characteristic
188	C₂×C₅	#5159	1	1	yes	characteristic
189	C₂×C₅	#5192	1	1	yes	characteristic
190	Q₈	—	6	2	no	—
191	Q₈	—	20	2	no	—
192	Q₈	—	20	2	no	—
193	C₂×C₄	—	6	1	no	—
194	C₂×C₄	—	6	1	no	—
195	C₂×C₄	—	20	2	no	—
196	C₂×C₄	#6765	1	1	yes	characteristic
197	C₂×C₄	D₃×D₂₀	1	1	yes	characteristic
198	C₂×C₄	C₂×#1201	1	1	yes	characteristic
199	C₈	—	2	1	no	—
200	C₈	—	12	1	no	—
201	C₈	—	60	2	no	—
202	C₈	#6661	1	2	yes	complemented
203	C₂×C₃	#65320	1	1	yes	characteristic
204	C₂×C₃	#65424	1	1	yes	characteristic
205	C₂×C₃	#65432	1	1	yes	characteristic
206	C₅	#71379	1	1	yes	characteristic, complemented, Sylow
207	C₂²	#91456	1	1	yes	characteristic
208	C₄	—	2	1	no	—
209	C₄	—	12	1	no	—
210	C₄	—	12	1	no	—
211	C₄	—	40	2	no	—
212	C₄	#91254	1	2	yes	—
213	C₄	#91309	1	1	yes	characteristic
214	C₄	C₂×#6661	1	1	yes	characteristic
215	C₃	#10601757	1	1	yes	characteristic, complemented, Sylow
216	C₂	#11747099	1	1	yes	characteristic
217	C₂	#11747921	1	1	yes	characteristic
218	C₂	#11748094	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₅×#4679

C₂

yes

maximal, characteristic

C₃×#64971

C₂

yes

maximal, characteristic

#11747099

C₂

yes

maximal

#11746278

C₂

yes

maximal, characteristic

#11745855

C₂

yes

maximal

#10601757

—

maximal, Hall

C₄×C₅×Dic₆

C₂²

yes

characteristic

C₃×#4015

C₂²

yes

—

C₂×#6661

—

C₃×C₄×C₅×C₈

C₂²

yes

characteristic

C₄⋊(C₃⋊#168)

C₂²

yes

—

C₅×#815

—

C₅×Dic₃⋊C₈

—

C₅×C₄⋊(C₃⋊C₈)

C₂²

yes

—

—

—

—

—

—

D₃

yes

characteristic

#64706

—

#64677

—

C₂×C₅×Dic₆

D₄

yes

characteristic

C₂×C₃×Dic₁₀

—

C₂×C₃×C₅×C₈

D₄

yes

characteristic

C₂×C₃×C₅×C₈

D₄

yes

characteristic

C₃×C₄²×C₅

C₂³

yes

characteristic

C₂×C₃⋊#168

—

C₅×C₄⋊Dic₃

D₄

yes

characteristic

C₅×Dic₃⋊C₄

—

C₄×C₅×Dic₃

—

C₂×C₅×C₃⋊C₈

—

C₃×C₄⋊Dic₅

—

C₃×C₄⋊Dic₅

—

—

—

—

D₅

yes

characteristic

#4547

—

#4457

—

C₄×C₅×Q₈

—

C₂×#689

—

#4015

C₂×D₃

yes

—

C₅×C₄⋊C₈

—

C₅×C₄⋊C₈

—

C₅×#97

—

C₄×C₅×C₈

C₂×D₃

yes

—

—

—

—

—

C₂×C₃×C₄×C₅

C₂×D₄

yes

characteristic

C₂×C₃×C₄×C₅

C₂×D₄

yes

characteristic

C₂×C₃×C₄×C₅

#41

yes

characteristic

C₂×C₅×Dic₃

—

C₂×C₅×Dic₃

—

C₅×Dic₆

—

C₂×C₃×Dic₅

—

C₃×Dic₁₀

—

C₃×Dic₁₀

—

C₃×C₅×C₈

—

C₃×C₅×C₈

Q₁₆

yes

complemented

C₃⋊#168

—

C₅×C₃⋊C₈

—

C₃×C₄⋊Q₈

—

C₂×#216

—

C₄×Dic₆

C₂×D₅

yes

characteristic

C₃×#107

—

C₃×#107

—

C₃×C₄×C₈

C₂×D₅

yes

characteristic

#835

—

#815

—

Dic₃⋊C₈

—

C₄⋊(C₃⋊C₈)

—

C₄⋊(C₃⋊C₈)

C₂×D₅

yes

characteristic

C₂×C₅×Q₈

—

C₂×Dic₁₀

—

C₂×C₅×C₈

—

C₂×C₅×C₈

yes

C₂×C₅×C₈

yes

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₄²×C₅

—

C₄²×C₅

C₂²×D₃

yes

characteristic

#689

—

C₄⋊Dic₅

—

C₄⋊Dic₅

—

—

—

—

—

—

—

C₂²×C₃×C₅

#121

yes

characteristic

C₃×C₄×C₅

—

C₃×C₄×C₅

C₂×Q₁₆

yes

characteristic

C₃×C₄×C₅

AGL(1,ℤ/8)

yes

—

100

C₃×C₄×C₅

#137

yes

characteristic

101

C₃×Dic₅

—

102

C₅×Dic₃

—

103

C₅×Dic₃

—

104

C₂×C₃×Q₈

—

105

C₂×Dic₆

D₂₀

yes

characteristic

106

C₂×C₃×C₈

C₂²⋊D₅

yes

characteristic

107

C₂×C₃×C₈

C₂²⋊D₅

yes

characteristic

108

C₃×C₄⋊C₄

—

109

C₃×C₄⋊C₄

—

110

C₃×C₄²

C₂²×D₅

yes

characteristic

111

#216

—

112

C₄⋊Dic₃

D₂₀

yes

characteristic

113

Dic₃⋊C₄

—

114

C₄×Dic₃

—

115

C₂×C₃⋊C₈

—

116

C₂×C₃⋊C₈

—

117

C₅×Q₈

—

118

C₂×C₄×C₅

—

119

C₂×C₄×C₅

—

120

C₂×C₄×C₅

D₃×D₄

yes

characteristic

121

C₂×C₄×C₅

#237

yes

characteristic

122

C₂×C₄×C₅

C₂×#67

yes

characteristic

123

C₂×Dic₅

—

124

Dic₁₀

—

125

Dic₁₀

—

126

C₅×C₈

—

127

C₅×C₈

—

128

C₅×C₈

#216

yes

complemented

129

#168

—

130

C₂×Q₁₆

—

131

C₄⋊Q₈

—

132

C₄×Q₈

—

133

#107

—

134

#107

—

135

C₄⋊C₈

—

136

C₄⋊C₈

—

137

C₄⋊C₈

—

138

#103

—

139

#97

—

140

C₄×C₈

D₃×D₅

yes

characteristic

141

C₂×C₃×C₅

#462

yes

characteristic

142

C₂×C₃×C₅

#469

yes

characteristic

143

C₂×C₃×C₅

#473

yes

characteristic

144

C₃×Q₈

—

145

C₃×Q₈

—

146

C₂×C₃×C₄

—

147

C₂×C₃×C₄

C₂×D₂₀

yes

characteristic

148

C₂×C₃×C₄

#709

yes

characteristic

149

C₂×C₃×C₄

C₂×C₂²⋊D₅

yes

characteristic

150

C₂×Dic₃

—

151

C₂×Dic₃

—

152

Dic₆

—

153

C₃×C₈

—

154

C₃×C₈

#689

yes

complemented

155

C₃⋊C₈

—

156

C₃⋊C₈

—

157

C₂²×C₅

#938

yes

characteristic

158

C₄×C₅

—

159

C₄×C₅

—

160

C₄×C₅

—

161

C₄×C₅

#911

yes

—

162

C₄×C₅

#942

yes

characteristic

163

C₄×C₅

C₂×#216

yes

characteristic

164

Dic₅

—

165

C₂×Q₈

—

166

C₂×Q₈

—

167

Q₁₆

—

168

C₂×C₈

—

169

C₂×C₈

—

170

C₂×C₈

#1201

yes

characteristic

171

C₂×C₈

#1201

yes

characteristic

172

C₄⋊C₄

—

173

C₄⋊C₄

—

174

C₄⋊C₄

—

175

C₄⋊C₄

—

176

C₄²

—

177

C₄²

C₂×D₃×D₅

yes

characteristic

178

C₃×C₅

#1694

yes

characteristic, complemented, Hall

179

C₂²×C₃

#4076

yes

characteristic

180

C₃×C₄

—

181

C₃×C₄

—

182

C₃×C₄

#4054

yes

—

183

C₃×C₄

C₂×#689

yes

characteristic

184

C₃×C₄

#4097

yes

characteristic

185

Dic₃

—

186

Dic₃

—

187

C₂×C₅

#5090

yes

characteristic

188

C₂×C₅

#5159

yes

characteristic

189

C₂×C₅

#5192

yes

characteristic

190

Q₈

—

191

Q₈

—

192

Q₈

—

193

C₂×C₄

—

194

C₂×C₄

—

195

C₂×C₄

—

196

C₂×C₄

#6765

yes

characteristic

197

C₂×C₄

D₃×D₂₀

yes

characteristic

198

C₂×C₄

C₂×#1201

yes

characteristic

199

C₈

—

200

C₈

—

201

C₈

—

202

C₈

#6661

yes

complemented

203

C₂×C₃

#65320

yes

characteristic

204

C₂×C₃

#65424

yes

characteristic

205

C₂×C₃

#65432

yes

characteristic

206

C₅

#71379

yes

characteristic, complemented, Sylow

207

C₂²

#91456

yes

characteristic

208

C₄

—

209

C₄

—

210

C₄

—

211

C₄

—

212

C₄

#91254

yes

—

213

C₄

#91309

yes

characteristic

214

C₄

C₂×#6661

yes

characteristic

215

C₃

#10601757

yes

characteristic, complemented, Sylow

216

C₂

#11747099

yes

characteristic

217

C₂

#11747921

yes

characteristic

218

C₂

#11748094

yes

characteristic

	subgroup	G/H
center	C₂²	#91456
commutator, cosocle	C₃×C₄²×C₅	C₂³
Frattini	C₄²	C₂×D₃×D₅
Fitting	C₃×C₄×C₅×C₈	C₂²
socle, prime radical	C₂²×C₃×C₅	#121
hypercenter	C₄×C₈	D₃×D₅
hypocenter	C₃×C₅	#1694
LCS	G ▷ C₃×C₄²×C₅ ▷ C₂²×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈

subgroup

G/H

center

C₂²

#91456

commutator, cosocle

C₃×C₄²×C₅

C₂³

Frattini

C₄²

C₂×D₃×D₅

Fitting

C₃×C₄×C₅×C₈

C₂²

socle, prime radical

C₂²×C₃×C₅

#121

hypercenter

C₄×C₈

D₃×D₅

hypocenter

C₃×C₅

#1694

LCS

G ▷ C₃×C₄²×C₅ ▷ C₂²×C₃×C₅ ▷ C₃×C₅

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈

N	H ≅ G/N	type
64. C₃×C₅×C₈	Q₁₆	semidirect
128. C₅×C₈	#216	semidirect
154. C₃×C₈	#689	semidirect
178. C₃×C₅	#1694	semidirect
202. C₈	#6661	semidirect
206. C₅	#71379	semidirect
215. C₃	#10601757	semidirect

H ≅ G/N

type

64. C₃×C₅×C₈

Q₁₆

semidirect

128. C₅×C₈

#216

semidirect

154. C₃×C₈

#689

semidirect

178. C₃×C₅

#1694

semidirect

202. C₈

#6661

semidirect

206. C₅

#71379

semidirect

215. C₃

#10601757

semidirect

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

subgroup	index
C₅×#4679	2
C₃×#64971	2
#11747099	2
#11746278	2
#11745855	2
#10601757	3
#71379	5

subgroup

index

Subgroups — #14292815

Conjugacy classes of subgroups

Complements

Maximal subgroups