Group #14312434 — subgroups

Subgroups — #14312434

Order 1920.

316 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 104 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	#11751226	C₂	1	1	yes	maximal, characteristic
2	C₅×#5118	C₂	1	1	yes	maximal, characteristic
3	C₃×#64777	C₂	1	1	yes	maximal, characteristic
4	#11745993	C₂	1	2	yes	maximal
5	#11745974	C₂	1	2	yes	maximal
6	#10601319	—	3	1	no	maximal, Hall
7	C₂×C₃×C₅×M₁₆	C₂²	1	1	yes	characteristic
8	C₂×#6710	C₂²	1	2	yes	—
9	C₂×C₄×C₅×Dic₃	C₂²	1	1	yes	characteristic
10	C₂×C₅×#208	C₂²	1	1	yes	characteristic
11	C₃×#4072	C₂²	1	2	yes	—
12	#91103	—	2	2	no	—
13	C₅×Dic₃⋊C₈	—	2	2	no	—
14	#68701	—	5	1	no	maximal, Hall
15	C₅×#423	—	3	1	no	—
16	#65414	—	3	1	no	—
17	#64779	—	3	2	no	—
18	#64777	D₃	1	1	yes	characteristic
19	#64743	—	3	2	no	—
20	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
21	C₂²×C₅×Dic₃	—	2	1	no	—
22	C₃×C₅×M₁₆	—	2	2	no	—
23	C₂×C₃×C₅×C₈	D₄	1	2	yes	—
24	#6710	—	2	2	no	—
25	#6710	—	2	2	no	—
26	C₂×C₃⋊#168	—	2	2	no	—
27	C₄×C₅×Dic₃	—	2	1	no	—
28	C₄×C₅×Dic₃	Q₈	1	1	yes	characteristic
29	C₄×C₅×Dic₃	Q₈	1	1	yes	characteristic
30	C₅×#208	—	2	2	no	—
31	C₂×C₅×C₃⋊C₈	—	2	1	no	—
32	C₃×C₂²⋊Dic₅	—	2	2	no	—
33	C₃×C₄⋊Dic₅	—	2	2	no	—
34	C₃×C₄×Dic₅	—	2	2	no	—
35	#5125	—	5	1	no	—
36	#5118	D₅	1	1	yes	characteristic
37	C₃×H(3,ℤ/4)	—	5	1	no	—
38	#4533	—	5	2	no	—
39	#4532	—	5	2	no	—
40	C₂×C₅×M₁₆	—	3	1	no	—
41	C₂×C₅×M₁₆	C₂×D₃	1	1	yes	characteristic
42	C₂×C₄²×C₅	—	3	1	no	—
43	#4072	C₂×D₃	1	2	yes	—
44	C₂×#681	—	3	2	no	—
45	C₅×C₄⋊C₈	—	6	2	no	—
46	#3951	—	6	2	no	—
47	#2102	—	15	1	no	Sylow
48	C₂³×C₃×C₅	#41	1	1	yes	characteristic
49	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
50	C₂×C₃×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₃×C₄×C₅	#41	1	2	yes	—
52	C₂×C₃×C₄×C₅	#41	1	2	yes	—
53	C₂×C₅×Dic₃	—	2	1	no	—
54	C₂×C₅×Dic₃	—	2	1	no	—
55	C₂×C₅×Dic₃	—	4	2	no	—
56	C₂×C₃×Dic₅	—	4	2	no	—
57	C₃×C₅×C₈	—	2	2	no	—
58	C₃⋊#168	—	4	2	no	—
59	C₅×C₃⋊C₈	—	4	1	no	—
60	C₂×C₃×M₁₆	C₂×D₅	1	1	yes	characteristic
61	C₃×#117	—	5	2	no	—
62	C₂×C₄×Dic₃	C₂×D₅	1	1	yes	characteristic
63	C₂×#208	—	5	2	no	—
64	C₂×#208	C₂×D₅	1	1	yes	characteristic
65	#819	—	10	2	no	—
66	Dic₃⋊C₈	—	2	2	no	—
67	C₂²×C₄×C₅	—	6	1	no	—
68	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
69	C₅×M₁₆	—	2	2	no	—
70	C₅×M₁₆	—	6	2	no	—
71	C₂×C₅×C₈	—	6	1	no	—
72	C₂×C₅×C₈	D₁₂	1	2	yes	—
73	C₄²×C₅	—	3	1	no	—
74	C₄²×C₅	—	3	1	no	—
75	C₄²×C₅	—	6	1	no	—
76	C₂²⋊Dic₅	—	2	2	no	—
77	C₄⋊Dic₅	—	2	2	no	—
78	C₄×Dic₅	—	2	2	no	—
79	#681	—	6	2	no	—
80	#681	—	6	2	no	—
81	C₂×#168	—	6	2	no	—
82	#430	—	15	1	no	—
83	#423	—	3	1	no	—
84	#344	—	15	2	no	—
85	#339	—	15	2	no	—
86	H(3,ℤ/4)	—	5	1	no	—
87	C₂²×C₃×C₅	—	4	1	no	—
88	C₂²×C₃×C₅	C₂²⋊Q₈	1	1	yes	characteristic
89	C₂²×C₃×C₅	#126	1	2	yes	—
90	C₃×C₄×C₅	#123	1	1	yes	characteristic
91	C₃×C₄×C₅	#124	1	1	yes	characteristic
92	C₃×C₄×C₅	#125	1	2	yes	—
93	C₃×Dic₅	—	8	2	no	—
94	C₅×Dic₃	—	2	2	no	—
95	C₅×Dic₃	—	4	1	no	—
96	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
97	C₂²×Dic₃	—	2	1	no	—
98	C₃×M₁₆	—	2	2	no	—
99	C₂×C₃×C₈	C₂²⋊D₅	1	2	yes	—
100	C₃×C₄⋊C₄	—	10	2	no	—
101	C₃×C₂²⋊C₄	—	10	2	no	—
102	C₃×C₄²	—	10	2	no	—
103	C₄×Dic₃	—	2	1	no	—
104	C₄×Dic₃	Dic₁₀	1	1	yes	characteristic
105	C₄×Dic₃	Dic₁₀	1	1	yes	characteristic
106	#208	—	2	2	no	—
107	#208	—	10	2	no	—
108	#208	—	10	2	no	—
109	C₂×C₃⋊C₈	—	2	1	no	—
110	C₂×C₃⋊C₈	—	10	2	no	—
111	C₂³×C₅	#237	1	1	yes	characteristic
112	C₂×C₄×C₅	—	6	1	no	—
113	C₂×C₄×C₅	—	6	1	no	—
114	C₂×C₄×C₅	—	12	2	no	—
115	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
116	C₂×C₄×C₅	#235	1	2	yes	—
117	C₂×C₄×C₅	#237	1	2	yes	—
118	C₂×C₄×C₅	D₃×Q₈	1	1	yes	characteristic
119	C₂×Dic₅	—	4	2	no	—
120	C₅×C₈	—	2	2	no	—
121	C₅×C₈	—	12	1	no	—
122	#168	—	12	2	no	—
123	C₂×M₁₆	—	3	1	no	—
124	C₂×M₁₆	—	15	2	no	—
125	C₂×M₁₆	D₃×D₅	1	1	yes	characteristic
126	#117	—	5	2	no	—
127	C₂×C₄²	—	3	1	no	—
128	#108	—	30	2	no	—
129	C₄⋊C₈	—	6	2	no	—
130	C₂×C₃×C₅	—	2	1	no	—
131	C₂×C₃×C₅	—	2	2	no	—
132	C₂×C₃×C₅	#400	1	1	yes	characteristic
133	C₂³×C₃	#709	1	1	yes	characteristic
134	C₂×C₃×C₄	—	20	2	no	—
135	C₂×C₃×C₄	C₂×Dic₁₀	1	1	yes	characteristic
136	C₂×C₃×C₄	#711	1	2	yes	—
137	C₂×C₃×C₄	#713	1	2	yes	—
138	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
139	C₂×Dic₃	—	2	1	no	—
140	C₂×Dic₃	—	2	1	no	—
141	C₂×Dic₃	—	4	2	no	—
142	C₃×C₈	—	2	2	no	—
143	C₃⋊C₈	—	4	1	no	—
144	C₃⋊C₈	—	20	2	no	—
145	C₂²×C₅	—	4	1	no	—
146	C₂²×C₅	#879	1	2	yes	—
147	C₂²×C₅	#897	1	1	yes	characteristic
148	C₄×C₅	—	6	2	no	—
149	C₄×C₅	—	12	1	no	—
150	C₄×C₅	#875	1	1	yes	characteristic
151	C₄×C₅	#886	1	1	yes	characteristic
152	C₄×C₅	#889	1	2	yes	—
153	Dic₅	—	8	2	no	—
154	C₂²×C₄	—	6	1	no	—
155	C₂²×C₄	C₂×D₃×D₅	1	1	yes	characteristic
156	M₁₆	—	2	2	no	—
157	M₁₆	—	6	2	no	—
158	M₁₆	—	30	2	no	—
159	M₁₆	—	30	2	no	—
160	C₂×C₈	—	6	1	no	—
161	C₂×C₈	—	30	2	no	—
162	C₂×C₈	#1203	1	2	yes	—
163	C₄⋊C₄	—	10	2	no	—
164	C₂²⋊C₄	—	10	2	no	—
165	C₄²	—	3	1	no	—
166	C₄²	—	3	1	no	—
167	C₄²	—	6	1	no	—
168	C₄²	—	10	2	no	—
169	C₃×C₅	#2102	1	1	yes	characteristic, complemented, Hall
170	C₂²×C₃	—	4	1	no	—
171	C₂²×C₃	#4044	1	2	yes	—
172	C₂²×C₃	#4070	1	1	yes	characteristic
173	C₃×C₄	—	40	2	no	—
174	C₃×C₄	#4036	1	2	yes	—
175	C₃×C₄	#4081	1	1	yes	characteristic
176	C₃×C₄	#4093	1	1	yes	characteristic
177	Dic₃	—	2	2	no	—
178	Dic₃	—	4	1	no	—
179	C₂×C₅	—	2	2	no	—
180	C₂×C₅	—	2	1	no	—
181	C₂×C₅	#4659	1	1	yes	characteristic
182	C₂³	#6765	1	1	yes	characteristic
183	C₂×C₄	—	6	1	no	—
184	C₂×C₄	—	6	1	no	—
185	C₂×C₄	—	12	2	no	—
186	C₂×C₄	—	20	2	no	—
187	C₂×C₄	D₃×Dic₁₀	1	1	yes	characteristic
188	C₂×C₄	#6773	1	2	yes	—
189	C₂×C₄	#6780	1	2	yes	—
190	C₂×C₄	C₂×#1203	1	1	yes	characteristic
191	C₈	—	2	2	no	—
192	C₈	—	12	1	no	—
193	C₈	—	60	2	no	—
194	C₂×C₃	—	2	1	no	—
195	C₂×C₃	—	2	2	no	—
196	C₂×C₃	#65275	1	1	yes	characteristic
197	C₅	#68701	1	1	yes	characteristic, complemented, Sylow
198	C₂²	—	4	1	no	—
199	C₂²	#91362	1	2	yes	—
200	C₂²	#91365	1	1	yes	characteristic
201	C₄	—	6	2	no	—
202	C₄	—	12	1	no	—
203	C₄	—	40	2	no	—
204	C₄	#91343	1	2	yes	—
205	C₄	#91375	1	1	yes	characteristic
206	C₄	#91543	1	1	yes	characteristic
207	C₃	#10601319	1	1	yes	characteristic, complemented, Sylow
208	C₂	—	2	1	no	—
209	C₂	—	2	2	no	—
210	C₂	#11748596	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11751226

C₂

yes

maximal, characteristic

C₅×#5118

C₂

yes

maximal, characteristic

C₃×#64777

C₂

yes

maximal, characteristic

#11745993

C₂

yes

maximal

#11745974

C₂

yes

maximal

#10601319

—

maximal, Hall

C₂×C₃×C₅×M₁₆

C₂²

yes

characteristic

C₂×#6710

C₂²

yes

—

C₂×C₄×C₅×Dic₃

C₂²

yes

characteristic

C₂×C₅×#208

C₂²

yes

characteristic

C₃×#4072

C₂²

yes

—

#91103

—

C₅×Dic₃⋊C₈

—

—

—

—

—

D₃

yes

characteristic

#64743

—

C₂²×C₃×C₄×C₅

C₂³

yes

characteristic

C₂²×C₅×Dic₃

—

C₃×C₅×M₁₆

—

C₂×C₃×C₅×C₈

D₄

yes

—

#6710

—

#6710

—

C₂×C₃⋊#168

—

C₄×C₅×Dic₃

—

C₄×C₅×Dic₃

Q₈

yes

characteristic

C₄×C₅×Dic₃

Q₈

yes

characteristic

C₅×#208

—

C₂×C₅×C₃⋊C₈

—

C₃×C₂²⋊Dic₅

—

C₃×C₄⋊Dic₅

—

C₃×C₄×Dic₅

—

#5125

—

#5118

D₅

yes

—

—

—

C₂×C₅×M₁₆

—

C₂×C₅×M₁₆

C₂×D₃

yes

characteristic

C₂×C₄²×C₅

—

#4072

C₂×D₃

yes

—

C₂×#681

—

C₅×C₄⋊C₈

—

—

—

C₂³×C₃×C₅

yes

C₂×C₃×C₄×C₅

C₂×D₄

yes

characteristic

C₂×C₃×C₄×C₅

C₂×Q₈

yes

characteristic

C₂×C₃×C₄×C₅

#41

yes

—

C₂×C₃×C₄×C₅

#41

yes

—

C₂×C₅×Dic₃

—

C₂×C₅×Dic₃

—

C₂×C₅×Dic₃

—

C₂×C₃×Dic₅

—

C₃×C₅×C₈

—

C₃⋊#168

—

C₅×C₃⋊C₈

—

C₂×C₃×M₁₆

C₂×D₅

yes

characteristic

C₃×#117

—

C₂×C₄×Dic₃

C₂×D₅

yes

characteristic

C₂×#208

—

C₂×#208

C₂×D₅

yes

characteristic

#819

—

Dic₃⋊C₈

—

C₂²×C₄×C₅

—

C₂²×C₄×C₅

C₂²×D₃

yes

characteristic

C₅×M₁₆

—

C₅×M₁₆

—

C₂×C₅×C₈

—

C₂×C₅×C₈

D₁₂

yes

—

C₄²×C₅

—

C₄²×C₅

—

C₄²×C₅

—

C₂²⋊Dic₅

—

C₄⋊Dic₅

—

C₄×Dic₅

—

—

—

—

—

—

—

—

—

C₂²×C₃×C₅

—

C₂²×C₃×C₅

C₂²⋊Q₈

yes

characteristic

C₂²×C₃×C₅

#126

yes

—

C₃×C₄×C₅

yes

C₃×C₄×C₅

yes

C₃×C₄×C₅

yes

—

C₃×Dic₅

—

C₅×Dic₃

—

C₅×Dic₃

—

C₂²×C₃×C₄

C₂²×D₅

yes

characteristic

C₂²×Dic₃

—

C₃×M₁₆

—

C₂×C₃×C₈

C₂²⋊D₅

yes

—

100

C₃×C₄⋊C₄

—

101

C₃×C₂²⋊C₄

—

102

C₃×C₄²

—

103

C₄×Dic₃

—

104

C₄×Dic₃

Dic₁₀

yes

characteristic

105

C₄×Dic₃

Dic₁₀

yes

characteristic

106

#208

—

107

#208

—

108

#208

—

109

C₂×C₃⋊C₈

—

110

C₂×C₃⋊C₈

—

111

C₂³×C₅

#237

yes

characteristic

112

C₂×C₄×C₅

—

113

C₂×C₄×C₅

—

114

C₂×C₄×C₅

—

115

C₂×C₄×C₅

C₂×D₁₂

yes

characteristic

116

C₂×C₄×C₅

#235

yes

—

117

C₂×C₄×C₅

#237

yes

—

118

C₂×C₄×C₅

D₃×Q₈

yes

characteristic

119

C₂×Dic₅

—

120

C₅×C₈

—

121

C₅×C₈

—

122

#168

—

123

C₂×M₁₆

—

124

C₂×M₁₆

—

125

C₂×M₁₆

D₃×D₅

yes

characteristic

126

#117

—

127

C₂×C₄²

—

128

#108

—

129

C₄⋊C₈

—

130

C₂×C₃×C₅

—

131

C₂×C₃×C₅

—

132

C₂×C₃×C₅

#400

yes

characteristic

133

C₂³×C₃

#709

yes

characteristic

134

C₂×C₃×C₄

—

135

C₂×C₃×C₄

C₂×Dic₁₀

yes

characteristic

136

C₂×C₃×C₄

#711

yes

—

137

C₂×C₃×C₄

#713

yes

—

138

C₂×C₃×C₄

C₂×C₂²⋊D₅

yes

characteristic

139

C₂×Dic₃

—

140

C₂×Dic₃

—

141

C₂×Dic₃

—

142

C₃×C₈

—

143

C₃⋊C₈

—

144

C₃⋊C₈

—

145

C₂²×C₅

—

146

C₂²×C₅

#879

yes

—

147

C₂²×C₅

#897

yes

characteristic

148

C₄×C₅

—

149

C₄×C₅

—

150

C₄×C₅

#875

yes

characteristic

151

C₄×C₅

#886

yes

characteristic

152

C₄×C₅

#889

yes

—

153

Dic₅

—

154

C₂²×C₄

—

155

C₂²×C₄

C₂×D₃×D₅

yes

characteristic

156

M₁₆

—

157

M₁₆

—

158

M₁₆

—

159

M₁₆

—

160

C₂×C₈

—

161

C₂×C₈

—

162

C₂×C₈

#1203

yes

—

163

C₄⋊C₄

—

164

C₂²⋊C₄

—

165

C₄²

—

166

C₄²

—

167

C₄²

—

168

C₄²

—

169

C₃×C₅

#2102

yes

characteristic, complemented, Hall

170

C₂²×C₃

—

171

C₂²×C₃

#4044

yes

—

172

C₂²×C₃

#4070

yes

characteristic

173

C₃×C₄

—

174

C₃×C₄

#4036

yes

—

175

C₃×C₄

#4081

yes

characteristic

176

C₃×C₄

#4093

yes

characteristic

177

Dic₃

—

178

Dic₃

—

179

C₂×C₅

—

180

C₂×C₅

—

181

C₂×C₅

#4659

yes

characteristic

182

C₂³

#6765

yes

characteristic

183

C₂×C₄

—

184

C₂×C₄

—

185

C₂×C₄

—

186

C₂×C₄

—

187

C₂×C₄

D₃×Dic₁₀

yes

characteristic

188

C₂×C₄

#6773

yes

—

189

C₂×C₄

#6780

yes

—

190

C₂×C₄

C₂×#1203

yes

characteristic

191

C₈

—

192

C₈

—

193

C₈

—

194

C₂×C₃

—

195

C₂×C₃

—

196

C₂×C₃

#65275

yes

characteristic

197

C₅

#68701

yes

characteristic, complemented, Sylow

198

C₂²

—

199

C₂²

#91362

yes

—

200

C₂²

#91365

yes

characteristic

201

C₄

—

202

C₄

—

203

C₄

—

204

C₄

#91343

yes

—

205

C₄

#91375

yes

characteristic

206

C₄

#91543

yes

characteristic

207

C₃

#10601319

yes

characteristic, complemented, Sylow

208

C₂

—

209

C₂

—

210

C₂

#11748596

yes

characteristic

	subgroup	G/H
center	C₄	#91375
commutator, cosocle	C₂²×C₃×C₄×C₅	C₂³
Frattini	C₂²×C₄	C₂×D₃×D₅
Fitting	C₂×C₃×C₅×M₁₆	C₂²
socle	C₂×C₃×C₅	#400
prime radical	C₂³×C₃×C₅	#41
hypercenter	C₂×M₁₆	D₃×D₅
hypocenter	C₃×C₅	#2102
LCS	G ▷ C₂²×C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×M₁₆

subgroup

G/H

center

C₄

#91375

commutator, cosocle

C₂²×C₃×C₄×C₅

C₂³

Frattini

C₂²×C₄

C₂×D₃×D₅

Fitting

C₂×C₃×C₅×M₁₆

C₂²

C₂×C₃×C₅

C₂³×C₃×C₅

C₂×M₁₆

D₃×D₅

hypocenter

C₃×C₅

#2102

LCS

G ▷ C₂²×C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×M₁₆

N	H ≅ G/N	type
169. C₃×C₅	#2102	semidirect
197. C₅	#68701	semidirect
207. C₃	#10601319	semidirect

H ≅ G/N

type

169. C₃×C₅

#2102

semidirect

197. C₅

#68701

semidirect

207. C₃

#10601319

semidirect

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

subgroup	index
#11751226	2
C₅×#5118	2
C₃×#64777	2
#11745993	2
#11745974	2
#10601319	3
#68701	5

subgroup

index

Subgroups — #14312434

Conjugacy classes of subgroups

Complements

Maximal subgroups