Group #14286415 — subgroups

Subgroups — #14286415

Order 1920.

320 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 320 conjugacy classes of subgroups — too many to draw. Only its 112 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	C₄×Dic₆₀	C₂	1	1	yes	maximal, characteristic
2	#11746851	C₂	1	2	yes	maximal
3	C₅×#4457	C₂	1	1	yes	maximal, characteristic
4	C₃×#64675	C₂	1	1	yes	maximal, characteristic
5	#11745845	C₂	1	2	yes	maximal
6	#10599310	—	3	1	no	maximal, Hall
7	C₂×Dic₆₀	C₄	1	1	yes	characteristic
8	C₄×Dic₃₀	C₂²	1	2	yes	—
9	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
10	#91102	C₄	1	1	yes	characteristic
11	#91100	C₄	1	2	yes	—
12	C₅×C₄⋊(C₃⋊C₈)	C₂²	1	2	yes	—
13	C₃×C₄×#168	C₂²	1	1	yes	characteristic
14	C₃×C₄×#168	C₂²	1	1	yes	characteristic
15	#90983	—	2	2	no	—
16	#90951	—	2	2	no	—
17	#68084	—	5	1	no	maximal, Hall
18	#65119	—	3	2	no	—
19	C₄×Dic₂₀	—	3	1	no	—
20	C₅×#334	—	3	1	no	—
21	#64704	—	3	2	no	—
22	#64675	D₃	1	1	yes	characteristic
23	C₂×Dic₃₀	C₂×C₄	1	2	yes	—
24	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
25	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
26	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
27	C₄⋊Dic₁₅	C₂×C₄	1	2	yes	—
28	#6712	—	2	2	no	—
29	C₄×Dic₁₅	—	2	2	no	—
30	Dic₆₀	C₈	1	4	yes	complemented
31	C₂×C₅×C₃⋊C₈	—	2	2	no	—
32	C₂×C₃×#168	—	2	1	no	—
33	C₂×C₃×#168	D₄	1	2	yes	—
34	C₄×Dic₁₂	—	5	1	no	—
35	C₈×Dic₆	—	5	2	no	—
36	C₃×#322	—	5	1	no	—
37	#4457	D₅	1	1	yes	characteristic
38	#4456	—	5	2	no	—
39	C₂×Dic₂₀	—	3	1	no	—
40	C₄×Dic₁₀	—	3	2	no	—
41	C₅×C₄⋊C₈	—	3	2	no	—
42	C₄×C₅×C₈	C₂×D₃	1	1	yes	characteristic
43	#3950	—	3	1	no	—
44	#3948	—	3	2	no	—
45	#3946	—	6	2	no	—
46	C₈×Dic₅	—	6	2	no	—
47	C₄×#168	C₂×D₃	1	1	yes	characteristic
48	C₄×#168	C₂×D₃	1	1	yes	characteristic
49	#1579	—	15	1	no	Sylow
50	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
52	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
53	C₂×Dic₁₅	—	2	2	no	—
54	C₂×Dic₁₅	—	2	2	no	—
55	Dic₃₀	—	2	2	no	—
56	Dic₃₀	C₂×C₈	1	4	yes	—
57	C₃×C₅×C₈	—	2	1	no	—
58	C₃×C₅×C₈	C₂×C₈	1	2	yes	—
59	C₃×#168	—	2	2	no	—
60	C₃×#168	—	4	1	no	—
61	C₅×C₃⋊C₈	—	4	2	no	—
62	C₂×Dic₁₂	—	5	1	no	—
63	C₄×Dic₆	—	5	2	no	—
64	C₃×C₄×C₈	—	5	1	no	—
65	C₃×C₄×C₈	—	5	1	no	—
66	C₃×C₄×C₈	C₂×D₅	1	1	yes	characteristic
67	#818	—	5	1	no	—
68	#816	—	5	2	no	—
69	Dic₃⋊C₈	—	10	2	no	—
70	C₈×Dic₃	—	10	2	no	—
71	C₄⋊(C₃⋊C₈)	C₂×D₅	1	2	yes	—
72	C₂×Dic₁₀	—	3	2	no	—
73	C₂×C₅×C₈	—	6	2	no	—
74	C₂×C₅×C₈	C₄×D₃	1	1	yes	characteristic
75	C₂×C₅×C₈	C₄×D₃	1	1	yes	characteristic
76	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
77	C₄⋊Dic₅	—	3	2	no	—
78	#683	—	6	2	no	—
79	C₄×Dic₅	—	6	2	no	—
80	C₂×#168	—	2	1	no	—
81	C₂×#168	D₁₂	1	2	yes	—
82	Dic₂₀	—	3	4	no	—
83	C₈×Q₈	—	15	2	no	—
84	C₄×Q₁₆	—	15	1	no	—
85	#334	—	3	1	no	—
86	Q₈⋊C₈	—	15	2	no	—
87	#322	—	5	1	no	—
88	C₂²×C₃×C₅	C₄×D₄	1	1	yes	characteristic
89	C₃×C₄×C₅	—	2	1	no	—
90	C₃×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
91	C₃×C₄×C₅	#131	1	1	yes	characteristic
92	C₃×C₄×C₅	#137	1	2	yes	—
93	Dic₁₅	—	2	4	no	—
94	Dic₁₅	—	4	2	no	—
95	C₂×Dic₆	—	5	2	no	—
96	C₂×C₃×C₈	—	5	2	no	—
97	C₂×C₃×C₈	—	10	1	no	—
98	C₂×C₃×C₈	C₄×D₅	1	1	yes	characteristic
99	C₂×C₃×C₈	C₄×D₅	1	1	yes	characteristic
100	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
101	C₄⋊Dic₃	—	5	2	no	—
102	Dic₃⋊C₄	—	10	2	no	—
103	C₄×Dic₃	—	10	2	no	—
104	C₂×C₃⋊C₈	—	2	2	no	—
105	Dic₁₂	—	5	4	no	—
106	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
107	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
108	C₂×C₄×C₅	#235	1	1	yes	characteristic
109	C₂×Dic₅	—	6	2	no	—
110	C₂×Dic₅	—	6	2	no	—
111	Dic₁₀	—	3	4	no	—
112	Dic₁₀	—	6	2	no	—
113	C₅×C₈	—	2	1	no	—
114	C₅×C₈	—	12	2	no	—
115	C₅×C₈	D₃×C₈	1	2	yes	—
116	#168	—	2	2	no	—
117	#168	—	4	1	no	—
118	C₂×Q₁₆	—	15	1	no	—
119	C₄×Q₈	—	15	2	no	—
120	#107	—	15	1	no	—
121	C₄⋊C₈	—	3	2	no	—
122	C₄⋊C₈	—	30	2	no	—
123	#103	—	15	2	no	—
124	C₄×C₈	—	5	1	no	—
125	C₄×C₈	—	5	1	no	—
126	C₄×C₈	—	30	2	no	—
127	C₄×C₈	D₃×D₅	1	1	yes	characteristic
128	C₂×C₃×C₅	C₈×D₄	1	1	yes	characteristic
129	C₂×C₃×C₅	#441	1	1	yes	characteristic
130	C₂×C₃×C₅	#444	1	1	yes	characteristic
131	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
132	C₂×C₃×C₄	D₄×D₅	1	1	yes	characteristic
133	C₂×C₃×C₄	#713	1	1	yes	characteristic
134	C₂×Dic₃	—	10	2	no	—
135	C₂×Dic₃	—	10	2	no	—
136	Dic₆	—	5	4	no	—
137	Dic₆	—	10	2	no	—
138	C₃×C₈	—	2	1	no	—
139	C₃×C₈	—	10	2	no	—
140	C₃×C₈	—	20	1	no	—
141	C₃×C₈	C₈×D₅	1	2	yes	—
142	C₃⋊C₈	—	4	2	no	—
143	C₂²×C₅	C₄×D₁₂	1	1	yes	characteristic
144	C₄×C₅	—	2	1	no	—
145	C₄×C₅	C₂×D₃×C₈	1	1	yes	characteristic
146	C₄×C₅	#901	1	1	yes	characteristic
147	C₄×C₅	#909	1	2	yes	—
148	Dic₅	—	6	4	no	—
149	Dic₅	—	12	2	no	—
150	C₂×Q₈	—	15	2	no	—
151	Q₁₆	—	15	4	no	—
152	C₂×C₈	—	5	2	no	—
153	C₂×C₈	—	6	2	no	—
154	C₂×C₈	—	10	1	no	—
155	C₂×C₈	#1200	1	1	yes	characteristic
156	C₂×C₈	#1200	1	1	yes	characteristic
157	C₄⋊C₄	—	15	2	no	—
158	C₄⋊C₄	—	30	2	no	—
159	C₄²	—	30	2	no	—
160	C₄²	C₂×D₃×D₅	1	1	yes	characteristic
161	C₃×C₅	#1579	1	1	yes	characteristic, complemented, Hall
162	C₂²×C₃	#4039	1	1	yes	characteristic
163	C₃×C₄	—	2	1	no	—
164	C₃×C₄	C₂×C₈×D₅	1	1	yes	characteristic
165	C₃×C₄	#4053	1	1	yes	characteristic
166	C₃×C₄	#4061	1	2	yes	—
167	Dic₃	—	10	4	no	—
168	Dic₃	—	20	2	no	—
169	C₂×C₅	C₈×D₁₂	1	1	yes	characteristic
170	C₂×C₅	#4716	1	1	yes	characteristic
171	C₂×C₅	#4717	1	1	yes	characteristic
172	Q₈	—	15	4	no	—
173	Q₈	—	30	2	no	—
174	C₂×C₄	—	30	2	no	—
175	C₂×C₄	—	30	2	no	—
176	C₂×C₄	#6773	1	1	yes	characteristic
177	C₂×C₄	D₅×D₁₂	1	1	yes	characteristic
178	C₂×C₄	C₂×#1200	1	1	yes	characteristic
179	C₈	—	2	1	no	—
180	C₈	—	10	2	no	—
181	C₈	—	12	2	no	—
182	C₈	—	20	1	no	—
183	C₈	#6648	1	2	yes	—
184	C₂×C₃	#65123	1	1	yes	characteristic
185	C₂×C₃	#65142	1	1	yes	characteristic
186	C₂×C₃	#65183	1	1	yes	characteristic
187	C₅	#68084	1	1	yes	characteristic, complemented, Sylow
188	C₂²	#91418	1	1	yes	characteristic
189	C₄	—	2	1	no	—
190	C₄	—	30	4	no	—
191	C₄	—	60	2	no	—
192	C₄	#91260	1	2	yes	—
193	C₄	C₂×#6648	1	1	yes	characteristic
194	C₄	#91290	1	1	yes	characteristic
195	C₃	#10599310	1	1	yes	characteristic, complemented, Sylow
196	C₂	#11746988	1	1	yes	characteristic
197	C₂	#11747854	1	1	yes	characteristic
198	C₂	#11748065	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×Dic₆₀

C₂

yes

maximal, characteristic

#11746851

C₂

yes

maximal

C₅×#4457

C₂

yes

maximal, characteristic

C₃×#64675

C₂

yes

maximal, characteristic

#11745845

C₂

yes

maximal

#10599310

—

maximal, Hall

C₂×Dic₆₀

C₄

yes

characteristic

C₄×Dic₃₀

C₂²

yes

—

C₃×C₄×C₅×C₈

C₂²

yes

characteristic

#91102

C₄

yes

characteristic

#91100

C₄

yes

—

C₅×C₄⋊(C₃⋊C₈)

C₂²

yes

—

C₃×C₄×#168

C₂²

yes

characteristic

C₃×C₄×#168

C₂²

yes

—

—

—

—

C₄×Dic₂₀

—

C₅×#334

—

#64704

—

#64675

D₃

yes

characteristic

C₂×Dic₃₀

C₂×C₄

yes

—

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

—

Dic₆₀

C₈

yes

complemented

C₂×C₅×C₃⋊C₈

—

C₂×C₃×#168

—

C₂×C₃×#168

D₄

yes

—

C₄×Dic₁₂

—

C₈×Dic₆

—

C₃×#322

—

#4457

D₅

yes

characteristic

#4456

—

C₂×Dic₂₀

—

C₄×Dic₁₀

—

C₅×C₄⋊C₈

—

C₄×C₅×C₈

C₂×D₃

yes

—

—

—

C₈×Dic₅

—

C₄×#168

C₂×D₃

yes

characteristic

C₄×#168

C₂×D₃

yes

characteristic

#1579

—

Sylow

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

C₂×C₈

yes

—

C₃×C₅×C₈

—

C₃×C₅×C₈

C₂×C₈

yes

—

C₃×#168

—

C₃×#168

—

C₅×C₃⋊C₈

—

C₂×Dic₁₂

—

C₄×Dic₆

—

C₃×C₄×C₈

—

C₃×C₄×C₈

—

C₃×C₄×C₈

C₂×D₅

yes

characteristic

#818

—

#816

—

Dic₃⋊C₈

—

C₈×Dic₃

—

C₄⋊(C₃⋊C₈)

C₂×D₅

yes

—

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

D₁₂

yes

—

Dic₂₀

—

C₈×Q₈

—

C₄×Q₁₆

—

#334

—

Q₈⋊C₈

—

#322

—

C₂²×C₃×C₅

C₄×D₄

yes

characteristic

C₃×C₄×C₅

—

C₃×C₄×C₅

C₂²×C₈

yes

C₃×C₄×C₅

yes

C₃×C₄×C₅

yes

—

Dic₁₅

—

Dic₁₅

—

C₂×Dic₆

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₂×C₃×C₈

C₄×D₅

yes

characteristic

C₂×C₃×C₈

C₄×D₅

yes

characteristic

100

C₃×C₄²

C₂²×D₅

yes

characteristic

101

C₄⋊Dic₃

—

102

Dic₃⋊C₄

—

103

C₄×Dic₃

—

104

C₂×C₃⋊C₈

—

105

Dic₁₂

—

106

C₂×C₄×C₅

C₂×C₄×D₃

yes

characteristic

107

C₂×C₄×C₅

C₂×D₁₂

yes

characteristic

108

C₂×C₄×C₅

#235

yes

characteristic

109

C₂×Dic₅

—

110

C₂×Dic₅

—

111

Dic₁₀

—

112

Dic₁₀

—

113

C₅×C₈

—

114

C₅×C₈

—

115

C₅×C₈

D₃×C₈

yes

—

116

#168

—

117

#168

—

118

C₂×Q₁₆

—

119

C₄×Q₈

—

120

#107

—

121

C₄⋊C₈

—

122

C₄⋊C₈

—

123

#103

—

124

C₄×C₈

—

125

C₄×C₈

—

126

C₄×C₈

—

127

C₄×C₈

D₃×D₅

yes

characteristic

128

C₂×C₃×C₅

C₈×D₄

yes

characteristic

129

C₂×C₃×C₅

#441

yes

characteristic

130

C₂×C₃×C₅

#444

yes

characteristic

131

C₂×C₃×C₄

C₂×C₄×D₅

yes

characteristic

132

C₂×C₃×C₄

D₄×D₅

yes

characteristic

133

C₂×C₃×C₄

#713

yes

characteristic

134

C₂×Dic₃

—

135

C₂×Dic₃

—

136

Dic₆

—

137

Dic₆

—

138

C₃×C₈

—

139

C₃×C₈

—

140

C₃×C₈

—

141

C₃×C₈

C₈×D₅

yes

—

142

C₃⋊C₈

—

143

C₂²×C₅

C₄×D₁₂

yes

characteristic

144

C₄×C₅

—

145

C₄×C₅

C₂×D₃×C₈

yes

characteristic

146

C₄×C₅

#901

yes

characteristic

147

C₄×C₅

#909

yes

—

148

Dic₅

—

149

Dic₅

—

150

C₂×Q₈

—

151

Q₁₆

—

152

C₂×C₈

—

153

C₂×C₈

—

154

C₂×C₈

—

155

C₂×C₈

#1200

yes

characteristic

156

C₂×C₈

#1200

yes

characteristic

157

C₄⋊C₄

—

158

C₄⋊C₄

—

159

C₄²

—

160

C₄²

C₂×D₃×D₅

yes

characteristic

161

C₃×C₅

#1579

yes

characteristic, complemented, Hall

162

C₂²×C₃

#4039

yes

characteristic

163

C₃×C₄

—

164

C₃×C₄

C₂×C₈×D₅

yes

characteristic

165

C₃×C₄

#4053

yes

characteristic

166

C₃×C₄

#4061

yes

—

167

Dic₃

—

168

Dic₃

—

169

C₂×C₅

C₈×D₁₂

yes

characteristic

170

C₂×C₅

#4716

yes

characteristic

171

C₂×C₅

#4717

yes

characteristic

172

Q₈

—

173

Q₈

—

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

#6773

yes

characteristic

177

C₂×C₄

D₅×D₁₂

yes

characteristic

178

C₂×C₄

C₂×#1200

yes

characteristic

179

C₈

—

180

C₈

—

181

C₈

—

182

C₈

—

183

C₈

#6648

yes

—

184

C₂×C₃

#65123

yes

characteristic

185

C₂×C₃

#65142

yes

characteristic

186

C₂×C₃

#65183

yes

characteristic

187

C₅

#68084

yes

characteristic, complemented, Sylow

188

C₂²

#91418

yes

characteristic

189

C₄

—

190

C₄

—

191

C₄

—

192

C₄

#91260

yes

—

193

C₄

C₂×#6648

yes

characteristic

194

C₄

#91290

yes

characteristic

195

C₃

#10599310

yes

characteristic, complemented, Sylow

196

C₂

#11746988

yes

characteristic

197

C₂

#11747854

yes

characteristic

198

C₂

#11748065

yes

characteristic

	subgroup	G/H
center	C₂×C₄	D₅×D₁₂
commutator	C₃×C₄×C₅	C₂²×C₈
Frattini	C₄²	C₂×D₃×D₅
Fitting	C₃×C₄×C₅×C₈	C₂²
socle, prime radical	C₂²×C₃×C₅	C₄×D₄
cosocle	C₃×C₄²×C₅	C₂³
hypercenter	C₄×C₈	D₃×D₅
hypocenter	C₃×C₅	#1579
LCS	G ▷ C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈

subgroup

G/H

center

C₂×C₄

D₅×D₁₂

commutator

C₃×C₄×C₅

C₂²×C₈

Frattini

C₄²

C₂×D₃×D₅

Fitting

C₃×C₄×C₅×C₈

C₂²

socle, prime radical

C₂²×C₃×C₅

C₄×D₄

cosocle

C₃×C₄²×C₅

C₂³

hypercenter

C₄×C₈

D₃×D₅

hypocenter

C₃×C₅

#1579

LCS

G ▷ C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈

N	H ≅ G/N	type
30. Dic₆₀	C₈	semidirect
161. C₃×C₅	#1579	semidirect
187. C₅	#68084	semidirect
195. C₃	#10599310	semidirect

H ≅ G/N

type

30. Dic₆₀

C₈

semidirect

161. C₃×C₅

#1579

semidirect

187. C₅

#68084

semidirect

195. C₃

#10599310

semidirect

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

subgroup	index
C₄×Dic₆₀	2
#11746851	2
C₅×#4457	2
C₃×#64675	2
#11745845	2
#10599310	3
#68084	5

subgroup

index

C₄×Dic₆₀

Subgroups — #14286415

Conjugacy classes of subgroups

Complements

Maximal subgroups