Group #14298889 — subgroups

Subgroups — #14298889

Order 1920.

376 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 376 conjugacy classes of subgroups — too many to draw. Only its 131 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	#11751219	C₂	1	2	yes	maximal
2	C₂×C₈×Dic₁₅	C₂	1	1	yes	maximal, characteristic
3	C₃×C₅×#341	C₂	1	1	yes	maximal, characteristic
4	#11746379	C₂	1	2	yes	maximal
5	#11746373	C₂	1	1	yes	maximal, characteristic
6	#10599721	—	3	1	no	maximal, Hall
7	C₂×C₃×C₅×M₁₆	C₂²	1	2	yes	—
8	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
9	C₂×C₄×Dic₁₅	C₂²	1	1	yes	characteristic
10	C₂×#6710	C₂²	1	2	yes	—
11	C₂²×C₃⋊#168	C₂²	1	1	yes	characteristic
12	#91098	—	2	4	no	—
13	C₈×Dic₁₅	C₄	1	2	yes	—
14	C₈×Dic₁₅	C₄	1	2	yes	—
15	#68928	—	5	1	no	maximal, Hall
16	#65407	—	3	2	no	—
17	C₂×C₈×Dic₅	—	3	1	no	—
18	C₅×#341	D₃	1	1	yes	characteristic
19	#64778	—	3	2	no	—
20	#64772	—	3	1	no	—
21	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
22	C₂²×Dic₁₅	Q₈	1	2	yes	—
23	C₃×C₅×M₁₆	—	2	2	no	—
24	C₃×C₅×M₁₆	—	2	2	no	—
25	C₂×C₃×C₅×C₈	—	2	2	no	—
26	C₂×C₃×C₅×C₈	—	2	2	no	—
27	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
28	C₄×Dic₁₅	C₂×C₄	1	1	yes	characteristic
29	C₄×Dic₁₅	C₂×C₄	1	1	yes	characteristic
30	C₄×Dic₁₅	D₄	1	2	yes	—
31	#6710	—	2	2	no	—
32	#6710	—	2	2	no	—
33	C₂×C₃⋊#168	—	2	2	no	—
34	C₂×C₃⋊#168	—	2	2	no	—
35	C₂×C₃⋊#168	C₂×C₄	1	2	yes	—
36	#5118	—	5	2	no	—
37	C₂×C₈×Dic₃	—	5	1	no	—
38	C₃×#341	D₅	1	1	yes	characteristic
39	#4558	—	5	2	no	—
40	#4552	—	5	1	no	—
41	C₂×C₅×M₁₆	C₂×D₃	1	2	yes	—
42	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
43	C₂×C₄×Dic₅	—	3	1	no	—
44	C₂×#681	—	3	2	no	—
45	C₂²×#168	—	3	1	no	—
46	#3946	—	6	4	no	—
47	C₈×Dic₅	—	3	2	no	—
48	C₈×Dic₅	—	3	2	no	—
49	#1837	—	15	1	no	Sylow
50	C₂³×C₃×C₅	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
53	C₂×C₃×C₄×C₅	#41	1	2	yes	—
54	C₂×C₃×C₄×C₅	#41	1	2	yes	—
55	C₂×Dic₁₅	—	2	4	no	—
56	C₂×Dic₁₅	C₄⋊C₄	1	2	yes	—
57	C₂×Dic₁₅	C₄⋊C₄	1	2	yes	—
58	C₃×C₅×C₈	—	2	2	no	—
59	C₃×C₅×C₈	—	4	2	no	—
60	C₃⋊#168	—	2	2	no	—
61	C₃⋊#168	—	4	2	no	—
62	C₂×C₃×M₁₆	C₂×D₅	1	2	yes	—
63	C₂²×C₃×C₈	C₂×D₅	1	1	yes	characteristic
64	C₂×C₄×Dic₃	—	5	1	no	—
65	C₂×#208	—	5	2	no	—
66	C₂²×C₃⋊C₈	—	5	1	no	—
67	Dic₃⋊C₈	—	10	4	no	—
68	C₈×Dic₃	—	5	2	no	—
69	C₈×Dic₃	—	5	2	no	—
70	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
71	C₂²×Dic₅	—	3	2	no	—
72	C₅×M₁₆	—	2	2	no	—
73	C₅×M₁₆	—	2	2	no	—
74	C₂×C₅×C₈	—	2	2	no	—
75	C₂×C₅×C₈	—	2	2	no	—
76	C₂×C₅×C₈	C₄×D₃	1	2	yes	—
77	C₄×Dic₅	—	3	1	no	—
78	C₄×Dic₅	—	3	1	no	—
79	C₄×Dic₅	—	3	2	no	—
80	#681	—	6	2	no	—
81	#681	—	6	2	no	—
82	C₂×#168	—	3	2	no	—
83	C₂×#168	—	6	2	no	—
84	C₂×#168	—	6	2	no	—
85	#423	—	15	2	no	—
86	C₂×C₄×C₈	—	15	1	no	—
87	#341	—	15	2	no	—
88	#341	—	15	1	no	—
89	#341	D₁₅	1	1	yes	characteristic
90	C₂²×C₃×C₅	—	2	2	no	—
91	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
92	C₂²×C₃×C₅	#125	1	2	yes	—
93	C₃×C₄×C₅	#117	1	1	yes	characteristic
94	C₃×C₄×C₅	#117	1	1	yes	characteristic
95	C₃×C₄×C₅	#124	1	2	yes	—
96	Dic₁₅	—	2	2	no	—
97	Dic₁₅	#108	1	4	yes	—
98	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
99	C₂²×Dic₃	—	5	2	no	—
100	C₃×M₁₆	—	2	2	no	—
101	C₃×M₁₆	—	2	2	no	—
102	C₂×C₃×C₈	—	2	2	no	—
103	C₂×C₃×C₈	—	2	2	no	—
104	C₂×C₃×C₈	C₄×D₅	1	2	yes	—
105	C₄×Dic₃	—	5	1	no	—
106	C₄×Dic₃	—	5	1	no	—
107	C₄×Dic₃	—	5	2	no	—
108	#208	—	10	2	no	—
109	#208	—	10	2	no	—
110	C₂×C₃⋊C₈	—	5	2	no	—
111	C₂×C₃⋊C₈	—	10	2	no	—
112	C₂×C₃⋊C₈	—	10	2	no	—
113	C₂³×C₅	D₃×Q₈	1	1	yes	characteristic
114	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
115	C₂×C₄×C₅	#235	1	2	yes	—
116	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
117	C₂×C₄×C₅	#237	1	2	yes	—
118	C₂×Dic₅	—	3	2	no	—
119	C₂×Dic₅	—	3	2	no	—
120	C₂×Dic₅	—	6	4	no	—
121	C₅×C₈	—	2	2	no	—
122	C₅×C₈	—	4	2	no	—
123	#168	—	6	2	no	—
124	#168	—	12	2	no	—
125	C₂×M₁₆	—	15	2	no	—
126	C₂×M₁₆	C₂×D₁₅	1	2	yes	—
127	C₂²×C₈	—	15	1	no	—
128	C₂²×C₈	C₂×D₁₅	1	1	yes	characteristic
129	C₂×C₄²	—	15	1	no	—
130	C₄⋊C₈	—	30	4	no	—
131	C₄×C₈	—	15	2	no	—
132	C₄×C₈	—	15	2	no	—
133	C₂×C₃×C₅	—	2	2	no	—
134	C₂×C₃×C₅	#382	1	1	yes	characteristic
135	C₂×C₃×C₅	C₂×#108	1	2	yes	—
136	C₂³×C₃	Q₈×D₅	1	1	yes	characteristic
137	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
138	C₂×C₃×C₄	#709	1	2	yes	—
139	C₂×C₃×C₄	D₄×D₅	1	1	yes	characteristic
140	C₂×C₃×C₄	#711	1	2	yes	—
141	C₂×Dic₃	—	5	2	no	—
142	C₂×Dic₃	—	5	2	no	—
143	C₂×Dic₃	—	10	4	no	—
144	C₃×C₈	—	2	2	no	—
145	C₃×C₈	—	4	2	no	—
146	C₃⋊C₈	—	10	2	no	—
147	C₃⋊C₈	—	20	2	no	—
148	C₂²×C₅	—	2	2	no	—
149	C₂²×C₅	#889	1	2	yes	—
150	C₂²×C₅	D₃×C₄⋊C₄	1	1	yes	characteristic
151	C₄×C₅	#872	1	1	yes	characteristic
152	C₄×C₅	#877	1	1	yes	characteristic
153	C₄×C₅	#885	1	2	yes	—
154	Dic₅	—	3	4	no	—
155	Dic₅	—	6	2	no	—
156	C₂²×C₄	—	15	2	no	—
157	C₂²×C₄	C₂²×D₁₅	1	1	yes	characteristic
158	M₁₆	—	2	2	no	—
159	M₁₆	—	2	2	no	—
160	M₁₆	—	30	2	no	—
161	M₁₆	—	30	2	no	—
162	C₂×C₈	—	2	2	no	—
163	C₂×C₈	—	2	2	no	—
164	C₂×C₈	—	15	2	no	—
165	C₂×C₈	—	30	2	no	—
166	C₂×C₈	—	30	2	no	—
167	C₂×C₈	C₄×D₁₅	1	2	yes	—
168	C₄²	—	15	1	no	—
169	C₄²	—	15	1	no	—
170	C₄²	—	15	2	no	—
171	C₃×C₅	#1837	1	1	yes	characteristic, complemented, Hall
172	C₂²×C₃	—	2	2	no	—
173	C₂²×C₃	#4035	1	2	yes	—
174	C₂²×C₃	D₅×C₄⋊C₄	1	1	yes	characteristic
175	C₃×C₄	#4018	1	1	yes	characteristic
176	C₃×C₄	#4023	1	1	yes	characteristic
177	C₃×C₄	#4031	1	2	yes	—
178	Dic₃	—	5	4	no	—
179	Dic₃	—	10	2	no	—
180	C₂×C₅	—	2	2	no	—
181	C₂×C₅	#4650	1	1	yes	characteristic
182	C₂×C₅	D₃×#108	1	2	yes	—
183	C₂³	Q₈×D₁₅	1	1	yes	characteristic
184	C₂×C₄	—	15	2	no	—
185	C₂×C₄	—	15	2	no	—
186	C₂×C₄	—	30	4	no	—
187	C₂×C₄	C₂×C₄×D₁₅	1	1	yes	characteristic
188	C₂×C₄	#6817	1	2	yes	—
189	C₂×C₄	D₄×D₁₅	1	1	yes	characteristic
190	C₂×C₄	#6819	1	2	yes	—
191	C₈	—	2	2	no	—
192	C₈	—	4	2	no	—
193	C₈	—	30	2	no	—
194	C₈	—	60	2	no	—
195	C₂×C₃	—	2	2	no	—
196	C₂×C₃	#64939	1	1	yes	characteristic
197	C₂×C₃	D₅×#108	1	2	yes	—
198	C₅	#68928	1	1	yes	characteristic, complemented, Sylow
199	C₂²	—	2	2	no	—
200	C₂²	#91778	1	2	yes	—
201	C₂²	C₄⋊C₄×D₁₅	1	1	yes	characteristic
202	C₄	—	15	4	no	—
203	C₄	—	30	2	no	—
204	C₄	#91761	1	1	yes	characteristic
205	C₄	#91766	1	1	yes	characteristic
206	C₄	#91774	1	2	yes	—
207	C₃	#10599721	1	1	yes	characteristic, complemented, Sylow
208	C₂	—	2	2	no	—
209	C₂	#11750751	1	1	yes	characteristic
210	C₂	D₁₅×#108	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#11751219

C₂

yes

maximal

C₂×C₈×Dic₁₅

C₂

yes

maximal, characteristic

C₃×C₅×#341

C₂

yes

maximal, characteristic

#11746379

C₂

yes

maximal

#11746373

C₂

yes

maximal, characteristic

#10599721

—

maximal, Hall

C₂×C₃×C₅×M₁₆

C₂²

yes

—

C₂²×C₃×C₅×C₈

C₂²

yes

characteristic

C₂×C₄×Dic₁₅

C₂²

yes

characteristic

C₂×#6710

C₂²

yes

—

C₂²×C₃⋊#168

C₂²

yes

characteristic

#91098

—

C₈×Dic₁₅

C₄

yes

—

C₈×Dic₁₅

C₄

yes

—

#68928

—

maximal, Hall

#65407

—

C₂×C₈×Dic₅

—

C₅×#341

D₃

yes

characteristic

#64778

—

#64772

—

C₂²×C₃×C₄×C₅

C₂³

yes

characteristic

C₂²×Dic₁₅

Q₈

yes

—

C₃×C₅×M₁₆

—

C₃×C₅×M₁₆

—

C₂×C₃×C₅×C₈

—

C₂×C₃×C₅×C₈

—

C₂×C₃×C₅×C₈

C₂×C₄

yes

—

C₄×Dic₁₅

C₂×C₄

yes

characteristic

C₄×Dic₁₅

C₂×C₄

yes

characteristic

C₄×Dic₁₅

D₄

yes

—

#6710

—

#6710

—

C₂×C₃⋊#168

—

C₂×C₃⋊#168

—

C₂×C₃⋊#168

C₂×C₄

yes

—

#5118

—

C₂×C₈×Dic₃

—

C₃×#341

D₅

yes

characteristic

#4558

—

#4552

—

C₂×C₅×M₁₆

C₂×D₃

yes

—

C₂²×C₅×C₈

C₂×D₃

yes

characteristic

C₂×C₄×Dic₅

—

C₂×#681

—

C₂²×#168

—

#3946

—

C₈×Dic₅

—

C₈×Dic₅

—

#1837

—

Sylow

C₂³×C₃×C₅

C₂×Q₈

yes

characteristic

C₂×C₃×C₄×C₅

C₂²×C₄

yes

characteristic

C₂×C₃×C₄×C₅

C₂×D₄

yes

characteristic

C₂×C₃×C₄×C₅

#41

yes

—

C₂×C₃×C₄×C₅

#41

yes

—

C₂×Dic₁₅

—

C₂×Dic₁₅

C₄⋊C₄

yes

—

C₂×Dic₁₅

C₄⋊C₄

yes

—

C₃×C₅×C₈

—

C₃×C₅×C₈

—

C₃⋊#168

—

C₃⋊#168

—

C₂×C₃×M₁₆

C₂×D₅

yes

—

C₂²×C₃×C₈

C₂×D₅

yes

characteristic

C₂×C₄×Dic₃

—

C₂×#208

—

C₂²×C₃⋊C₈

—

Dic₃⋊C₈

—

C₈×Dic₃

—

C₈×Dic₃

—

C₂²×C₄×C₅

C₂²×D₃

yes

characteristic

C₂²×Dic₅

—

C₅×M₁₆

—

C₅×M₁₆

—

C₂×C₅×C₈

—

C₂×C₅×C₈

—

C₂×C₅×C₈

C₄×D₃

yes

—

C₄×Dic₅

—

C₄×Dic₅

—

C₄×Dic₅

—

—

—

—

—

—

—

C₂×C₄×C₈

—

#341

—

#341

—

#341

D₁₅

yes

characteristic

C₂²×C₃×C₅

—

C₂²×C₃×C₅

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₃×C₅

#125

yes

—

C₃×C₄×C₅

yes

C₃×C₄×C₅

yes

C₃×C₄×C₅

yes

—

Dic₁₅

—

Dic₁₅

#108

yes

—

C₂²×C₃×C₄

C₂²×D₅

yes

characteristic

C₂²×Dic₃

—

100

C₃×M₁₆

—

101

C₃×M₁₆

—

102

C₂×C₃×C₈

—

103

C₂×C₃×C₈

—

104

C₂×C₃×C₈

C₄×D₅

yes

—

105

C₄×Dic₃

—

106

C₄×Dic₃

—

107

C₄×Dic₃

—

108

#208

—

109

#208

—

110

C₂×C₃⋊C₈

—

111

C₂×C₃⋊C₈

—

112

C₂×C₃⋊C₈

—

113

C₂³×C₅

D₃×Q₈

yes

characteristic

114

C₂×C₄×C₅

C₂×C₄×D₃

yes

characteristic

115

C₂×C₄×C₅

#235

yes

—

116

C₂×C₄×C₅

D₃×D₄

yes

characteristic

117

C₂×C₄×C₅

#237

yes

—

118

C₂×Dic₅

—

119

C₂×Dic₅

—

120

C₂×Dic₅

—

121

C₅×C₈

—

122

C₅×C₈

—

123

#168

—

124

#168

—

125

C₂×M₁₆

—

126

C₂×M₁₆

C₂×D₁₅

yes

—

127

C₂²×C₈

—

128

C₂²×C₈

C₂×D₁₅

yes

characteristic

129

C₂×C₄²

—

130

C₄⋊C₈

—

131

C₄×C₈

—

132

C₄×C₈

—

133

C₂×C₃×C₅

—

134

C₂×C₃×C₅

#382

yes

characteristic

135

C₂×C₃×C₅

C₂×#108

yes

—

136

C₂³×C₃

Q₈×D₅

yes

characteristic

137

C₂×C₃×C₄

C₂×C₄×D₅

yes

characteristic

138

C₂×C₃×C₄

#709

yes

—

139

C₂×C₃×C₄

D₄×D₅

yes

characteristic

140

C₂×C₃×C₄

#711

yes

—

141

C₂×Dic₃

—

142

C₂×Dic₃

—

143

C₂×Dic₃

—

144

C₃×C₈

—

145

C₃×C₈

—

146

C₃⋊C₈

—

147

C₃⋊C₈

—

148

C₂²×C₅

—

149

C₂²×C₅

#889

yes

—

150

C₂²×C₅

D₃×C₄⋊C₄

yes

characteristic

151

C₄×C₅

#872

yes

characteristic

152

C₄×C₅

#877

yes

characteristic

153

C₄×C₅

#885

yes

—

154

Dic₅

—

155

Dic₅

—

156

C₂²×C₄

—

157

C₂²×C₄

C₂²×D₁₅

yes

characteristic

158

M₁₆

—

159

M₁₆

—

160

M₁₆

—

161

M₁₆

—

162

C₂×C₈

—

163

C₂×C₈

—

164

C₂×C₈

—

165

C₂×C₈

—

166

C₂×C₈

—

167

C₂×C₈

C₄×D₁₅

yes

—

168

C₄²

—

169

C₄²

—

170

C₄²

—

171

C₃×C₅

#1837

yes

characteristic, complemented, Hall

172

C₂²×C₃

—

173

C₂²×C₃

#4035

yes

—

174

C₂²×C₃

D₅×C₄⋊C₄

yes

characteristic

175

C₃×C₄

#4018

yes

characteristic

176

C₃×C₄

#4023

yes

characteristic

177

C₃×C₄

#4031

yes

—

178

Dic₃

—

179

Dic₃

—

180

C₂×C₅

—

181

C₂×C₅

#4650

yes

characteristic

182

C₂×C₅

D₃×#108

yes

—

183

C₂³

Q₈×D₁₅

yes

characteristic

184

C₂×C₄

—

185

C₂×C₄

—

186

C₂×C₄

—

187

C₂×C₄

C₂×C₄×D₁₅

yes

characteristic

188

C₂×C₄

#6817

yes

—

189

C₂×C₄

D₄×D₁₅

yes

characteristic

190

C₂×C₄

#6819

yes

—

191

C₈

—

192

C₈

—

193

C₈

—

194

C₈

—

195

C₂×C₃

—

196

C₂×C₃

#64939

yes

characteristic

197

C₂×C₃

D₅×#108

yes

—

198

C₅

#68928

yes

characteristic, complemented, Sylow

199

C₂²

—

200

C₂²

#91778

yes

—

201

C₂²

C₄⋊C₄×D₁₅

yes

characteristic

202

C₄

—

203

C₄

—

204

C₄

#91761

yes

characteristic

205

C₄

#91766

yes

characteristic

206

C₄

#91774

yes

—

207

C₃

#10599721

yes

characteristic, complemented, Sylow

208

C₂

—

209

C₂

#11750751

yes

characteristic

210

C₂

D₁₅×#108

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×D₁₅
commutator	C₂×C₃×C₄×C₅	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₅
Fitting	C₃×C₅×#341	C₂
socle	C₂²×C₃×C₅	C₂×C₄⋊C₄
prime radical	C₂³×C₃×C₅	C₂×Q₈
cosocle	C₂²×C₃×C₄×C₅	C₂³
hypercenter	#341	D₁₅
hypocenter	C₃×C₅	#1837
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #341

subgroup

G/H

center

C₂×C₄

D₄×D₁₅

commutator

C₂×C₃×C₄×C₅

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₅

Fitting

C₃×C₅×#341

C₂

socle

C₂²×C₃×C₅

C₂×C₄⋊C₄

prime radical

C₂³×C₃×C₅

C₂×Q₈

cosocle

C₂²×C₃×C₄×C₅

C₂³

D₁₅

C₃×C₅

G ▷ C₂×C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #341

N	H ≅ G/N	type
171. C₃×C₅	#1837	semidirect
198. C₅	#68928	semidirect
207. C₃	#10599721	semidirect

H ≅ G/N

type

171. C₃×C₅

#1837

semidirect

198. C₅

#68928

semidirect

207. C₃

#10599721

semidirect

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

subgroup	index
#11751219	2
C₂×C₈×Dic₁₅	2
C₃×C₅×#341	2
#11746379	2
#11746373	2
#10599721	3
#68928	5

subgroup

index

#11751219

C₂×C₈×Dic₁₅

C₃×C₅×#341

Subgroups — #14298889

Conjugacy classes of subgroups

Complements

Maximal subgroups