Group #14287411 — subgroups

Subgroups — #14287411

Order 1920.

304 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 304 conjugacy classes of subgroups — too many to draw. Only its 107 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	#11750932	C₂	1	2	yes	maximal
2	C₄×Dic₆₀	C₂	1	1	yes	maximal, characteristic
3	C₃×C₅×#335	C₂	1	1	yes	maximal, characteristic
4	#11746305	C₂	1	2	yes	maximal
5	C₈×C₃⋊#168	C₂	1	1	yes	maximal, characteristic
6	#10599357	—	3	1	no	maximal, Hall
7	C₂×Dic₆₀	C₄	1	1	yes	characteristic
8	C₄×Dic₃₀	C₂²	1	2	yes	—
9	C₃×C₅×C₄⋊C₈	C₂²	1	2	yes	—
10	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
11	#91102	C₄	1	1	yes	characteristic
12	#91100	C₄	1	2	yes	—
13	#91099	—	2	2	no	—
14	#91098	—	2	2	no	—
15	C₄×C₃⋊#168	C₂²	1	1	yes	characteristic
16	C₄×C₃⋊#168	C₂²	1	1	yes	characteristic
17	#68564	—	5	1	no	maximal, Hall
18	#65120	—	3	2	no	—
19	C₄×Dic₂₀	—	3	1	no	—
20	C₅×#335	D₃	1	1	yes	characteristic
21	#64704	—	3	2	no	—
22	C₈×#168	—	3	1	no	—
23	C₂×Dic₃₀	C₂×C₄	1	2	yes	—
24	C₂×C₃×C₅×C₈	—	2	2	no	—
25	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
26	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
27	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
28	C₄⋊Dic₁₅	C₂×C₄	1	2	yes	—
29	#6712	—	2	2	no	—
30	C₄×Dic₁₅	—	2	2	no	—
31	C₂×C₃⋊#168	—	2	1	no	—
32	C₂×C₃⋊#168	D₄	1	1	yes	characteristic
33	C₂×C₃⋊#168	D₄	1	1	yes	characteristic
34	Dic₆₀	—	2	2	no	—
35	#4831	—	5	2	no	—
36	C₄×Dic₁₂	—	5	1	no	—
37	C₃×#335	D₅	1	1	yes	characteristic
38	#4484	—	5	2	no	—
39	C₈×C₃⋊C₈	—	5	1	no	—
40	C₂×Dic₂₀	—	3	1	no	—
41	C₄×Dic₁₀	—	3	2	no	—
42	C₅×C₄⋊C₈	C₂×D₃	1	2	yes	—
43	C₄×C₅×C₈	C₂×D₃	1	1	yes	characteristic
44	#3950	—	3	1	no	—
45	#3948	—	3	2	no	—
46	#3947	—	6	2	no	—
47	#3946	—	6	2	no	—
48	C₄×#168	—	3	1	no	—
49	C₄×#168	—	3	1	no	—
50	#1591	—	15	1	no	Sylow
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	1	yes	characteristic
54	C₂×Dic₁₅	—	2	2	no	—
55	C₂×Dic₁₅	—	2	2	no	—
56	Dic₃₀	—	2	2	no	—
57	Dic₃₀	—	2	2	no	—
58	C₃×C₅×C₈	—	2	1	no	—
59	C₃×C₅×C₈	—	4	2	no	—
60	C₃×C₅×C₈	M₁₆	1	2	yes	—
61	C₃⋊#168	—	2	2	no	—
62	C₃⋊#168	—	2	1	no	—
63	C₃⋊#168	Q₁₆	1	2	yes	complemented
64	C₂×Dic₁₂	—	5	1	no	—
65	C₄×Dic₆	—	5	2	no	—
66	C₃×C₄⋊C₈	C₂×D₅	1	2	yes	—
67	C₃×C₄×C₈	C₂×D₅	1	1	yes	characteristic
68	#818	—	5	1	no	—
69	#816	—	5	2	no	—
70	#815	—	10	2	no	—
71	Dic₃⋊C₈	—	10	2	no	—
72	C₄×C₃⋊C₈	—	5	1	no	—
73	C₄×C₃⋊C₈	—	5	1	no	—
74	C₂×Dic₁₀	—	3	2	no	—
75	C₂×C₅×C₈	—	2	2	no	—
76	C₂×C₅×C₈	C₄×D₃	1	1	yes	characteristic
77	C₂×C₅×C₈	C₄×D₃	1	1	yes	characteristic
78	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
79	C₄⋊Dic₅	—	3	2	no	—
80	#683	—	6	2	no	—
81	C₄×Dic₅	—	6	2	no	—
82	C₂×#168	—	3	1	no	—
83	C₂×#168	—	3	1	no	—
84	C₂×#168	—	6	1	no	—
85	Dic₂₀	—	6	2	no	—
86	#446	—	15	2	no	—
87	C₄×Q₁₆	—	15	1	no	—
88	#335	D₁₅	1	1	yes	characteristic
89	Q₈⋊C₈	—	15	2	no	—
90	C₈²	—	15	1	no	—
91	C₂²×C₃×C₅	C₄×D₄	1	1	yes	characteristic
92	C₃×C₄×C₅	—	2	1	no	—
93	C₃×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
94	C₃×C₄×C₅	#131	1	1	yes	characteristic
95	C₃×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
96	C₃×C₄×C₅	#135	1	1	yes	characteristic
97	Dic₁₅	—	4	2	no	—
98	Dic₁₅	—	4	2	no	—
99	C₂×Dic₆	—	5	2	no	—
100	C₂×C₃×C₈	—	2	2	no	—
101	C₂×C₃×C₈	C₄×D₅	1	1	yes	characteristic
102	C₂×C₃×C₈	C₄×D₅	1	1	yes	characteristic
103	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
104	C₄⋊Dic₃	—	5	2	no	—
105	Dic₃⋊C₄	—	10	2	no	—
106	C₄×Dic₃	—	10	2	no	—
107	C₂×C₃⋊C₈	—	5	1	no	—
108	C₂×C₃⋊C₈	—	5	1	no	—
109	C₂×C₃⋊C₈	—	10	1	no	—
110	Dic₁₂	—	10	2	no	—
111	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
112	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
113	C₂×C₄×C₅	#239	1	1	yes	characteristic
114	C₂×Dic₅	—	6	2	no	—
115	C₂×Dic₅	—	6	2	no	—
116	Dic₁₀	—	6	2	no	—
117	Dic₁₀	—	6	2	no	—
118	C₅×C₈	—	2	1	no	—
119	C₅×C₈	—	4	2	no	—
120	C₅×C₈	ΓL(1,𝔽₂₅)	1	2	yes	—
121	#168	—	3	2	no	—
122	#168	—	6	2	no	—
123	#168	—	6	1	no	—
124	C₂×Q₁₆	—	15	1	no	—
125	C₄×Q₈	—	15	2	no	—
126	#107	—	15	1	no	—
127	C₄⋊C₈	—	30	2	no	—
128	C₄⋊C₈	C₂×D₁₅	1	2	yes	—
129	#103	—	15	2	no	—
130	#97	—	30	2	no	—
131	C₄×C₈	—	15	1	no	—
132	C₄×C₈	—	15	1	no	—
133	C₄×C₈	C₂×D₁₅	1	1	yes	characteristic
134	C₂×C₃×C₅	#436	1	1	yes	characteristic
135	C₂×C₃×C₅	C₄×Q₁₆	1	1	yes	characteristic
136	C₂×C₃×C₅	#443	1	1	yes	characteristic
137	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
138	C₂×C₃×C₄	D₄×D₅	1	1	yes	characteristic
139	C₂×C₃×C₄	#713	1	1	yes	characteristic
140	C₂×Dic₃	—	10	2	no	—
141	C₂×Dic₃	—	10	2	no	—
142	Dic₆	—	10	2	no	—
143	Dic₆	—	10	2	no	—
144	C₃×C₈	—	2	1	no	—
145	C₃×C₈	—	4	2	no	—
146	C₃×C₈	#676	1	2	yes	—
147	C₃⋊C₈	—	5	2	no	—
148	C₃⋊C₈	—	10	2	no	—
149	C₃⋊C₈	—	10	1	no	—
150	C₂²×C₅	#893	1	1	yes	characteristic
151	C₄×C₅	—	2	1	no	—
152	C₄×C₅	C₂×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
153	C₄×C₅	#907	1	1	yes	characteristic
154	C₄×C₅	#912	1	1	yes	characteristic
155	C₄×C₅	D₃×Q₁₆	1	1	yes	characteristic
156	Dic₅	—	12	2	no	—
157	Dic₅	—	12	2	no	—
158	C₂×Q₈	—	15	2	no	—
159	Q₁₆	—	30	2	no	—
160	C₂×C₈	—	2	2	no	—
161	C₂×C₈	—	15	1	no	—
162	C₂×C₈	—	15	1	no	—
163	C₂×C₈	—	30	1	no	—
164	C₂×C₈	C₄×D₁₅	1	1	yes	characteristic
165	C₂×C₈	C₄×D₁₅	1	1	yes	characteristic
166	C₄⋊C₄	—	15	2	no	—
167	C₄⋊C₄	—	30	2	no	—
168	C₄²	—	30	2	no	—
169	C₄²	C₂²×D₁₅	1	1	yes	characteristic
170	C₃×C₅	#1591	1	1	yes	characteristic, complemented, Hall
171	C₂²×C₃	#4039	1	1	yes	characteristic
172	C₃×C₄	—	2	1	no	—
173	C₃×C₄	C₂×#676	1	1	yes	characteristic
174	C₃×C₄	#4053	1	1	yes	characteristic
175	C₃×C₄	#4058	1	1	yes	characteristic
176	C₃×C₄	D₅×Q₁₆	1	1	yes	characteristic
177	Dic₃	—	20	2	no	—
178	Dic₃	—	20	2	no	—
179	C₂×C₅	#4838	1	1	yes	characteristic
180	C₂×C₅	#4873	1	1	yes	characteristic
181	C₂×C₅	#4895	1	1	yes	characteristic
182	Q₈	—	30	2	no	—
183	Q₈	—	30	2	no	—
184	C₂×C₄	—	30	2	no	—
185	C₂×C₄	—	30	2	no	—
186	C₂×C₄	C₂×C₄×D₁₅	1	1	yes	characteristic
187	C₂×C₄	D₄×D₁₅	1	1	yes	characteristic
188	C₂×C₄	#6821	1	1	yes	characteristic
189	C₈	—	2	1	no	—
190	C₈	—	4	2	no	—
191	C₈	—	15	2	no	—
192	C₈	—	30	2	no	—
193	C₈	—	30	1	no	—
194	C₈	#6705	1	2	yes	—
195	C₂×C₃	#65127	1	1	yes	characteristic
196	C₂×C₃	#65162	1	1	yes	characteristic
197	C₂×C₃	#65184	1	1	yes	characteristic
198	C₅	#68564	1	1	yes	characteristic, complemented, Sylow
199	C₂²	#91782	1	1	yes	characteristic
200	C₄	—	2	1	no	—
201	C₄	—	60	2	no	—
202	C₄	—	60	2	no	—
203	C₄	C₂×#6705	1	1	yes	characteristic
204	C₄	#91796	1	1	yes	characteristic
205	C₄	#91801	1	1	yes	characteristic
206	C₄	Q₁₆×D₁₅	1	1	yes	characteristic
207	C₃	#10599357	1	1	yes	characteristic, complemented, Sylow
208	C₂	#11750939	1	1	yes	characteristic
209	C₂	#11750974	1	1	yes	characteristic
210	C₂	#11750996	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11750932

C₂

yes

maximal

C₄×Dic₆₀

C₂

yes

maximal, characteristic

C₃×C₅×#335

C₂

yes

maximal, characteristic

#11746305

C₂

yes

maximal

C₈×C₃⋊#168

C₂

yes

maximal, characteristic

#10599357

—

maximal, Hall

C₂×Dic₆₀

C₄

yes

characteristic

C₄×Dic₃₀

C₂²

yes

—

C₃×C₅×C₄⋊C₈

C₂²

yes

—

C₃×C₄×C₅×C₈

C₂²

yes

characteristic

#91102

C₄

yes

characteristic

#91100

C₄

yes

—

#91099

—

#91098

—

C₄×C₃⋊#168

C₂²

yes

characteristic

C₄×C₃⋊#168

C₂²

yes

—

—

C₄×Dic₂₀

—

C₅×#335

D₃

yes

characteristic

#64704

—

C₈×#168

—

C₂×Dic₃₀

C₂×C₄

yes

—

C₂×C₃×C₅×C₈

—

C₂×C₃×C₅×C₈

C₂×C₄

yes

characteristic

C₂×C₃×C₅×C₈

C₂×C₄

yes

characteristic

C₃×C₄²×C₅

C₂³

yes

characteristic

C₄⋊Dic₁₅

C₂×C₄

yes

—

#6712

—

C₄×Dic₁₅

—

C₂×C₃⋊#168

—

C₂×C₃⋊#168

D₄

yes

characteristic

C₂×C₃⋊#168

D₄

yes

characteristic

Dic₆₀

—

#4831

—

C₄×Dic₁₂

—

C₃×#335

D₅

yes

characteristic

#4484

—

C₈×C₃⋊C₈

—

C₂×Dic₂₀

—

C₄×Dic₁₀

—

C₅×C₄⋊C₈

C₂×D₃

yes

—

C₄×C₅×C₈

C₂×D₃

yes

—

—

—

—

—

—

—

C₂×C₃×C₄×C₅

C₂²×C₄

yes

characteristic

C₂×C₃×C₄×C₅

C₂×D₄

yes

characteristic

C₂×C₃×C₄×C₅

#41

yes

characteristic

C₂×Dic₁₅

—

C₂×Dic₁₅

—

Dic₃₀

—

Dic₃₀

—

C₃×C₅×C₈

—

C₃×C₅×C₈

—

C₃×C₅×C₈

M₁₆

yes

—

C₃⋊#168

—

C₃⋊#168

—

C₃⋊#168

Q₁₆

yes

complemented

C₂×Dic₁₂

—

C₄×Dic₆

—

C₃×C₄⋊C₈

C₂×D₅

yes

—

C₃×C₄×C₈

C₂×D₅

yes

—

—

—

Dic₃⋊C₈

—

C₄×C₃⋊C₈

—

C₄×C₃⋊C₈

—

C₂×Dic₁₀

—

C₂×C₅×C₈

—

C₂×C₅×C₈

C₄×D₃

yes

characteristic

C₂×C₅×C₈

C₄×D₃

yes

characteristic

C₄²×C₅

C₂²×D₃

yes

characteristic

C₄⋊Dic₅

—

#683

—

C₄×Dic₅

—

C₂×#168

—

C₂×#168

—

C₂×#168

—

Dic₂₀

—

#446

—

C₄×Q₁₆

—

#335

D₁₅

yes

characteristic

Q₈⋊C₈

—

C₈²

—

C₂²×C₃×C₅

C₄×D₄

yes

characteristic

C₃×C₄×C₅

—

C₃×C₄×C₅

C₂×M₁₆

yes

characteristic

C₃×C₄×C₅

#131

yes

characteristic

C₃×C₄×C₅

C₂×Q₁₆

yes

characteristic

C₃×C₄×C₅

#135

yes

characteristic

Dic₁₅

—

Dic₁₅

—

C₂×Dic₆

—

100

C₂×C₃×C₈

—

101

C₂×C₃×C₈

C₄×D₅

yes

characteristic

102

C₂×C₃×C₈

C₄×D₅

yes

characteristic

103

C₃×C₄²

C₂²×D₅

yes

characteristic

104

C₄⋊Dic₃

—

105

Dic₃⋊C₄

—

106

C₄×Dic₃

—

107

C₂×C₃⋊C₈

—

108

C₂×C₃⋊C₈

—

109

C₂×C₃⋊C₈

—

110

Dic₁₂

—

111

C₂×C₄×C₅

C₂×C₄×D₃

yes

characteristic

112

C₂×C₄×C₅

D₃×D₄

yes

characteristic

113

C₂×C₄×C₅

#239

yes

characteristic

114

C₂×Dic₅

—

115

C₂×Dic₅

—

116

Dic₁₀

—

117

Dic₁₀

—

118

C₅×C₈

—

119

C₅×C₈

—

120

C₅×C₈

ΓL(1,𝔽₂₅)

yes

—

121

#168

—

122

#168

—

123

#168

—

124

C₂×Q₁₆

—

125

C₄×Q₈

—

126

#107

—

127

C₄⋊C₈

—

128

C₄⋊C₈

C₂×D₁₅

yes

—

129

#103

—

130

#97

—

131

C₄×C₈

—

132

C₄×C₈

—

133

C₄×C₈

C₂×D₁₅

yes

characteristic

134

C₂×C₃×C₅

#436

yes

characteristic

135

C₂×C₃×C₅

C₄×Q₁₆

yes

characteristic

136

C₂×C₃×C₅

#443

yes

characteristic

137

C₂×C₃×C₄

C₂×C₄×D₅

yes

characteristic

138

C₂×C₃×C₄

D₄×D₅

yes

characteristic

139

C₂×C₃×C₄

#713

yes

characteristic

140

C₂×Dic₃

—

141

C₂×Dic₃

—

142

Dic₆

—

143

Dic₆

—

144

C₃×C₈

—

145

C₃×C₈

—

146

C₃×C₈

#676

yes

—

147

C₃⋊C₈

—

148

C₃⋊C₈

—

149

C₃⋊C₈

—

150

C₂²×C₅

#893

yes

characteristic

151

C₄×C₅

—

152

C₄×C₅

C₂×ΓL(1,𝔽₂₅)

yes

characteristic

153

C₄×C₅

#907

yes

characteristic

154

C₄×C₅

#912

yes

characteristic

155

C₄×C₅

D₃×Q₁₆

yes

characteristic

156

Dic₅

—

157

Dic₅

—

158

C₂×Q₈

—

159

Q₁₆

—

160

C₂×C₈

—

161

C₂×C₈

—

162

C₂×C₈

—

163

C₂×C₈

—

164

C₂×C₈

C₄×D₁₅

yes

characteristic

165

C₂×C₈

C₄×D₁₅

yes

characteristic

166

C₄⋊C₄

—

167

C₄⋊C₄

—

168

C₄²

—

169

C₄²

C₂²×D₁₅

yes

characteristic

170

C₃×C₅

#1591

yes

characteristic, complemented, Hall

171

C₂²×C₃

#4039

yes

characteristic

172

C₃×C₄

—

173

C₃×C₄

C₂×#676

yes

characteristic

174

C₃×C₄

#4053

yes

characteristic

175

C₃×C₄

#4058

yes

characteristic

176

C₃×C₄

D₅×Q₁₆

yes

characteristic

177

Dic₃

—

178

Dic₃

—

179

C₂×C₅

#4838

yes

characteristic

180

C₂×C₅

#4873

yes

characteristic

181

C₂×C₅

#4895

yes

characteristic

182

Q₈

—

183

Q₈

—

184

C₂×C₄

—

185

C₂×C₄

—

186

C₂×C₄

C₂×C₄×D₁₅

yes

characteristic

187

C₂×C₄

D₄×D₁₅

yes

characteristic

188

C₂×C₄

#6821

yes

characteristic

189

C₈

—

190

C₈

—

191

C₈

—

192

C₈

—

193

C₈

—

194

C₈

#6705

yes

—

195

C₂×C₃

#65127

yes

characteristic

196

C₂×C₃

#65162

yes

characteristic

197

C₂×C₃

#65184

yes

characteristic

198

C₅

#68564

yes

characteristic, complemented, Sylow

199

C₂²

#91782

yes

characteristic

200

C₄

—

201

C₄

—

202

C₄

—

203

C₄

C₂×#6705

yes

characteristic

204

C₄

#91796

yes

characteristic

205

C₄

#91801

yes

characteristic

206

C₄

Q₁₆×D₁₅

yes

characteristic

207

C₃

#10599357

yes

characteristic, complemented, Sylow

208

C₂

#11750939

yes

characteristic

209

C₂

#11750974

yes

characteristic

210

C₂

#11750996

yes

characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₁₅
commutator	C₂×C₃×C₄×C₅	C₂²×C₄
Frattini	C₄²	C₂²×D₁₅
Fitting	C₃×C₅×#335	C₂
socle, prime radical	C₂²×C₃×C₅	C₄×D₄
cosocle	C₃×C₄²×C₅	C₂³
hypercenter	#335	D₁₅
hypocenter	C₃×C₅	#1591
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ #335

subgroup

G/H

center

C₂×C₄

D₄×D₁₅

commutator

C₂×C₃×C₄×C₅

C₂²×C₄

Frattini

C₄²

C₂²×D₁₅

Fitting

C₃×C₅×#335

C₂

socle, prime radical

C₂²×C₃×C₅

C₄×D₄

cosocle

C₃×C₄²×C₅

C₂³

D₁₅

C₃×C₅

G ▷ C₂×C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ #335

N	H ≅ G/N	type
63. C₃⋊#168	Q₁₆	semidirect
170. C₃×C₅	#1591	semidirect
198. C₅	#68564	semidirect
207. C₃	#10599357	semidirect

H ≅ G/N

type

63. C₃⋊#168

Q₁₆

semidirect

170. C₃×C₅

#1591

semidirect

198. C₅

#68564

semidirect

207. C₃

#10599357

semidirect

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

subgroup	index
#11750932	2
C₄×Dic₆₀	2
C₃×C₅×#335	2
#11746305	2
C₈×C₃⋊#168	2
#10599357	3
#68564	5

subgroup

index

#11750932

C₄×Dic₆₀

C₃×C₅×#335

#11746305

C₈×C₃⋊#168

#10599357

#68564

Subgroups — #14287411

Conjugacy classes of subgroups

Complements

Maximal subgroups