Group #14286391 — subgroups

Subgroups — #14286391

Order 1920.

320 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 320 conjugacy classes of subgroups — too many to draw. Only its 115 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	#11752362	C₂	1	2	yes	maximal
2	C₃×#65162	C₂	1	1	yes	maximal, characteristic
3	#11746752	C₂	1	1	yes	maximal, characteristic
4	#11746750	C₂	1	1	yes	maximal, characteristic
5	#11746743	C₂	1	2	yes	maximal
6	#10604061	—	3	1	no	maximal, Hall
7	C₂×C₃×Dic₂₀	C₄	1	1	yes	characteristic
8	C₃×#4033	C₂²	1	2	yes	—
9	#91233	—	2	2	no	—
10	C₄⋊(C₃⋊#170)	C₂²	1	2	yes	—
11	C₄×C₃⋊#170	—	2	2	no	—
12	C₄×C₃⋊#170	C₂²	1	1	yes	characteristic
13	C₄×C₃⋊#170	C₂²	1	1	yes	characteristic
14	C₃×C₅×#107	C₄	1	1	yes	characteristic
15	C₃×C₈×Dic₅	C₂²	1	1	yes	characteristic
16	C₃×#3942	C₄	1	2	yes	—
17	#70902	—	5	1	no	maximal, Hall
18	#65703	—	3	2	no	—
19	#65162	D₃	1	1	yes	characteristic
20	#64881	—	3	1	no	—
21	#64879	—	3	1	no	—
22	#64872	—	3	2	no	—
23	C₂×C₃×Dic₁₀	C₂×C₄	1	2	yes	—
24	C₂×C₃⋊#170	—	2	2	no	—
25	C₂×C₃⋊#170	—	2	1	no	—
26	C₂×C₃⋊#170	D₄	1	2	yes	—
27	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
28	C₃×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
29	C₃×#683	—	2	2	no	—
30	C₃×C₄×Dic₅	—	2	2	no	—
31	C₃×C₄×Dic₅	C₂³	1	1	yes	characteristic
32	C₂×C₃×#168	C₂×C₄	1	1	yes	characteristic
33	C₃×Dic₂₀	C₈	1	4	yes	complemented
34	C₃×C₄×Q₁₆	—	5	1	no	—
35	Q₈×C₃⋊C₈	—	5	2	no	—
36	#4535	—	5	2	no	—
37	#4457	—	5	1	no	—
38	#4455	—	5	1	no	—
39	C₂×Dic₂₀	Dic₃	1	1	yes	characteristic
40	#4033	C₂×D₃	1	2	yes	—
41	#4004	—	6	2	no	—
42	C₄⋊#170	—	3	2	no	—
43	C₄×#170	—	3	1	no	—
44	C₄×#170	—	3	1	no	—
45	C₄×#170	—	6	2	no	—
46	C₅×#107	Dic₃	1	1	yes	characteristic
47	C₈×Dic₅	C₂×D₃	1	1	yes	characteristic
48	#3942	Dic₃	1	2	yes	—
49	#1579	—	15	1	no	Sylow
50	C₂×C₃×C₄×C₅	—	2	2	no	—
51	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₃×Dic₅	—	2	2	no	—
53	C₂×C₃×Dic₅	C₂×D₄	1	1	yes	characteristic
54	C₂×C₃×Dic₅	#41	1	1	yes	characteristic
55	C₃×Dic₁₀	—	2	2	no	—
56	C₃×Dic₁₀	C₂×C₈	1	4	yes	—
57	C₃⋊#170	—	2	2	no	—
58	C₃⋊#170	—	4	1	no	—
59	C₃⋊#170	—	4	2	no	—
60	C₃×C₅×C₈	C₂×C₈	1	2	yes	—
61	C₃×#168	—	2	1	no	—
62	C₂×C₃×Q₁₆	—	5	1	no	—
63	C₃×C₄×Q₈	—	5	2	no	—
64	C₃×#107	AGL(1,𝔽₅)	1	1	yes	characteristic
65	C₃×#103	—	5	2	no	—
66	C₃×C₄×C₈	—	5	1	no	—
67	C₄⋊(C₃⋊C₈)	—	5	2	no	—
68	C₄⋊(C₃⋊C₈)	—	10	2	no	—
69	C₄×C₃⋊C₈	—	5	1	no	—
70	C₄×C₃⋊C₈	—	5	1	no	—
71	C₄×C₃⋊C₈	—	10	2	no	—
72	C₂×Dic₁₀	C₂×Dic₃	1	2	yes	—
73	C₂×#170	—	3	2	no	—
74	C₂×#170	—	6	2	no	—
75	C₂×#170	—	6	1	no	—
76	C₂×C₅×C₈	C₂×Dic₃	1	1	yes	characteristic
77	C₅×C₄⋊C₄	C₂×Dic₃	1	2	yes	—
78	#683	—	2	2	no	—
79	C₄×Dic₅	—	2	2	no	—
80	C₄×Dic₅	C₂²×D₃	1	1	yes	characteristic
81	C₂×#168	C₂×Dic₃	1	1	yes	characteristic
82	Dic₂₀	C₃⋊C₈	1	4	yes	complemented
83	C₈×Q₈	—	15	2	no	—
84	C₄×Q₁₆	—	5	1	no	—
85	#334	—	15	1	no	—
86	Q₈⋊C₈	—	15	2	no	—
87	#322	—	15	1	no	—
88	C₂²×C₃×C₅	C₄×D₄	1	1	yes	characteristic
89	C₃×C₄×C₅	—	4	2	no	—
90	C₃×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
91	C₃×C₄×C₅	#131	1	1	yes	characteristic
92	C₃×Dic₅	—	2	4	no	—
93	C₃×Dic₅	—	2	1	no	—
94	C₃×Dic₅	#137	1	2	yes	—
95	C₂×C₃×Q₈	—	5	2	no	—
96	C₃×Q₁₆	—	5	4	no	—
97	C₂×C₃×C₈	—	5	1	no	—
98	C₂×C₃×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
99	C₃×C₄⋊C₄	—	10	2	no	—
100	C₃×C₄⋊C₄	C₂×AGL(1,𝔽₅)	1	2	yes	—
101	C₃×C₄²	—	5	1	no	—
102	C₃×C₄²	—	10	2	no	—
103	C₂×C₃⋊C₈	—	5	2	no	—
104	C₂×C₃⋊C₈	—	10	1	no	—
105	C₂×C₃⋊C₈	—	10	2	no	—
106	C₂×C₄×C₅	—	2	2	no	—
107	C₂×C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
108	C₂×Dic₅	—	2	2	no	—
109	C₂×Dic₅	D₃×D₄	1	1	yes	characteristic
110	C₂×Dic₅	#237	1	1	yes	characteristic
111	Dic₁₀	—	2	2	no	—
112	Dic₁₀	C₂×C₃⋊C₈	1	4	yes	—
113	#170	—	6	2	no	—
114	#170	—	12	1	no	—
115	#170	—	12	2	no	—
116	C₅×C₈	C₂×C₃⋊C₈	1	2	yes	—
117	#168	—	2	1	no	—
118	C₂×Q₁₆	—	5	1	no	—
119	C₄×Q₈	—	5	2	no	—
120	#107	ΓL(1,𝔽₁₆)	1	1	yes	characteristic
121	C₄⋊C₈	—	15	2	no	—
122	C₄⋊C₈	—	30	2	no	—
123	#103	—	5	2	no	—
124	C₄×C₈	—	5	1	no	—
125	C₄×C₈	—	15	1	no	—
126	C₄×C₈	—	15	1	no	—
127	C₄×C₈	—	30	2	no	—
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	C₃×Q₈	—	5	4	no	—
132	C₃×Q₈	—	10	2	no	—
133	C₂×C₃×C₄	—	2	2	no	—
134	C₂×C₃×C₄	—	5	1	no	—
135	C₂×C₃×C₄	—	5	1	no	—
136	C₂×C₃×C₄	—	10	2	no	—
137	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
138	C₃×C₈	—	10	1	no	—
139	C₃×C₈	D₅⋊C₈	1	2	yes	—
140	C₃⋊C₈	—	10	2	no	—
141	C₃⋊C₈	—	20	2	no	—
142	C₃⋊C₈	—	20	1	no	—
143	C₂²×C₅	D₄×Dic₃	1	1	yes	characteristic
144	C₄×C₅	—	4	2	no	—
145	C₄×C₅	C₂²×C₃⋊C₈	1	1	yes	characteristic
146	C₄×C₅	#948	1	1	yes	characteristic
147	Dic₅	—	2	1	no	—
148	Dic₅	—	2	4	no	—
149	Dic₅	#918	1	2	yes	—
150	C₂×Q₈	—	5	2	no	—
151	Q₁₆	—	5	4	no	—
152	C₂×C₈	—	5	1	no	—
153	C₂×C₈	—	15	2	no	—
154	C₂×C₈	—	30	1	no	—
155	C₂×C₈	—	30	2	no	—
156	C₂×C₈	C₂×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
157	C₄⋊C₄	—	10	2	no	—
158	C₄⋊C₄	C₂×ΓL(1,𝔽₁₆)	1	2	yes	—
159	C₄²	—	5	1	no	—
160	C₄²	—	10	2	no	—
161	C₃×C₅	#1579	1	1	yes	characteristic, complemented, Hall
162	C₂²×C₃	D₄×AGL(1,𝔽₅)	1	1	yes	characteristic
163	C₃×C₄	—	4	2	no	—
164	C₃×C₄	—	5	2	no	—
165	C₃×C₄	—	10	4	no	—
166	C₃×C₄	—	10	1	no	—
167	C₃×C₄	C₂×D₅⋊C₈	1	1	yes	characteristic
168	C₃×C₄	#4133	1	1	yes	characteristic
169	C₂×C₅	D₄×C₃⋊C₈	1	1	yes	characteristic
170	C₂×C₅	#5184	1	1	yes	characteristic
171	C₂×C₅	#5197	1	1	yes	characteristic
172	Q₈	—	5	4	no	—
173	Q₈	—	10	2	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	5	1	no	—
176	C₂×C₄	—	5	1	no	—
177	C₂×C₄	—	10	2	no	—
178	C₂×C₄	C₂²×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
179	C₈	—	10	1	no	—
180	C₈	—	30	2	no	—
181	C₈	—	60	2	no	—
182	C₈	—	60	1	no	—
183	C₈	D₅⋊(C₃⋊C₈)	1	2	yes	—
184	C₂×C₃	#65702	1	1	yes	characteristic
185	C₂×C₃	#65733	1	1	yes	characteristic
186	C₂×C₃	#65739	1	1	yes	characteristic
187	C₅	#70902	1	1	yes	characteristic, complemented, Sylow
188	C₂²	D₄×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
189	C₄	—	4	2	no	—
190	C₄	—	5	2	no	—
191	C₄	—	10	4	no	—
192	C₄	—	10	1	no	—
193	C₄	C₂×D₅⋊(C₃⋊C₈)	1	1	yes	characteristic
194	C₄	#91992	1	1	yes	characteristic
195	C₃	#10604061	1	1	yes	characteristic, complemented, Sylow
196	C₂	#11752361	1	1	yes	characteristic
197	C₂	#11752392	1	1	yes	characteristic
198	C₂	#11752398	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11752362

C₂

yes

maximal

C₃×#65162

C₂

yes

maximal, characteristic

#11746752

C₂

yes

maximal, characteristic

#11746750

C₂

yes

maximal, characteristic

#11746743

C₂

yes

maximal

#10604061

—

maximal, Hall

C₂×C₃×Dic₂₀

C₄

yes

characteristic

C₃×#4033

C₂²

yes

—

#91233

—

C₄⋊(C₃⋊#170)

C₂²

yes

—

C₄×C₃⋊#170

—

C₄×C₃⋊#170

C₂²

yes

characteristic

C₄×C₃⋊#170

C₂²

yes

characteristic

C₃×C₅×#107

C₄

yes

characteristic

C₃×C₈×Dic₅

C₂²

yes

characteristic

C₃×#3942

C₄

yes

—

—

—

D₃

yes

—

—

—

C₂×C₃×Dic₁₀

C₂×C₄

yes

—

C₂×C₃⋊#170

—

C₂×C₃⋊#170

—

C₂×C₃⋊#170

D₄

yes

—

C₂×C₃×C₅×C₈

C₂×C₄

yes

characteristic

C₃×C₅×C₄⋊C₄

C₂×C₄

yes

—

C₃×#683

—

C₃×C₄×Dic₅

—

C₃×C₄×Dic₅

C₂³

yes

characteristic

C₂×C₃×#168

C₂×C₄

yes

characteristic

C₃×Dic₂₀

C₈

yes

complemented

C₃×C₄×Q₁₆

—

Q₈×C₃⋊C₈

—

#4535

—

#4457

—

#4455

—

C₂×Dic₂₀

Dic₃

yes

characteristic

#4033

C₂×D₃

yes

—

—

—

—

—

—

Dic₃

yes

characteristic

C₈×Dic₅

C₂×D₃

yes

characteristic

#3942

Dic₃

yes

—

#1579

—

Sylow

C₂×C₃×C₄×C₅

—

C₂×C₃×C₄×C₅

C₂²×C₄

yes

characteristic

C₂×C₃×Dic₅

—

C₂×C₃×Dic₅

C₂×D₄

yes

characteristic

C₂×C₃×Dic₅

#41

yes

characteristic

C₃×Dic₁₀

—

C₃×Dic₁₀

C₂×C₈

yes

—

C₃⋊#170

—

C₃⋊#170

—

C₃⋊#170

—

C₃×C₅×C₈

C₂×C₈

yes

—

C₃×#168

—

C₂×C₃×Q₁₆

—

C₃×C₄×Q₈

—

yes

—

C₃×C₄×C₈

—

C₄⋊(C₃⋊C₈)

—

C₄⋊(C₃⋊C₈)

—

C₄×C₃⋊C₈

—

C₄×C₃⋊C₈

—

C₄×C₃⋊C₈

—

C₂×Dic₁₀

C₂×Dic₃

yes

—

C₂×#170

—

C₂×#170

—

C₂×#170

—

C₂×C₅×C₈

C₂×Dic₃

yes

characteristic

C₅×C₄⋊C₄

C₂×Dic₃

yes

—

#683

—

C₄×Dic₅

—

C₄×Dic₅

C₂²×D₃

yes

characteristic

C₂×#168

C₂×Dic₃

yes

characteristic

Dic₂₀

C₃⋊C₈

yes

complemented

C₈×Q₈

—

C₄×Q₁₆

—

#334

—

Q₈⋊C₈

—

#322

—

C₂²×C₃×C₅

C₄×D₄

yes

characteristic

C₃×C₄×C₅

—

C₃×C₄×C₅

C₂²×C₈

yes

characteristic

C₃×C₄×C₅

#131

yes

characteristic

C₃×Dic₅

—

C₃×Dic₅

—

C₃×Dic₅

#137

yes

—

C₂×C₃×Q₈

—

C₃×Q₁₆

—

C₂×C₃×C₈

—

C₂×C₃×C₈

C₂×AGL(1,𝔽₅)

yes

characteristic

C₃×C₄⋊C₄

—

100

C₃×C₄⋊C₄

C₂×AGL(1,𝔽₅)

yes

—

101

C₃×C₄²

—

102

C₃×C₄²

—

103

C₂×C₃⋊C₈

—

104

C₂×C₃⋊C₈

—

105

C₂×C₃⋊C₈

—

106

C₂×C₄×C₅

—

107

C₂×C₄×C₅

C₂²×Dic₃

yes

characteristic

108

C₂×Dic₅

—

109

C₂×Dic₅

D₃×D₄

yes

characteristic

110

C₂×Dic₅

#237

yes

characteristic

111

Dic₁₀

—

112

Dic₁₀

C₂×C₃⋊C₈

yes

—

113

#170

—

114

#170

—

115

#170

—

116

C₅×C₈

C₂×C₃⋊C₈

yes

—

117

#168

—

118

C₂×Q₁₆

—

119

C₄×Q₈

—

120

#107

ΓL(1,𝔽₁₆)

yes

characteristic

121

C₄⋊C₈

—

122

C₄⋊C₈

—

123

#103

—

124

C₄×C₈

—

125

C₄×C₈

—

126

C₄×C₈

—

127

C₄×C₈

—

128

C₂×C₃×C₅

C₈×D₄

yes

characteristic

129

C₂×C₃×C₅

#441

yes

characteristic

130

C₂×C₃×C₅

#444

yes

characteristic

131

C₃×Q₈

—

132

C₃×Q₈

—

133

C₂×C₃×C₄

—

134

C₂×C₃×C₄

—

135

C₂×C₃×C₄

—

136

C₂×C₃×C₄

—

137

C₂×C₃×C₄

C₂²×AGL(1,𝔽₅)

yes

characteristic

138

C₃×C₈

—

139

C₃×C₈

D₅⋊C₈

yes

—

140

C₃⋊C₈

—

141

C₃⋊C₈

—

142

C₃⋊C₈

—

143

C₂²×C₅

D₄×Dic₃

yes

characteristic

144

C₄×C₅

—

145

C₄×C₅

C₂²×C₃⋊C₈

yes

characteristic

146

C₄×C₅

#948

yes

characteristic

147

Dic₅

—

148

Dic₅

—

149

Dic₅

#918

yes

—

150

C₂×Q₈

—

151

Q₁₆

—

152

C₂×C₈

—

153

C₂×C₈

—

154

C₂×C₈

—

155

C₂×C₈

—

156

C₂×C₈

C₂×ΓL(1,𝔽₁₆)

yes

characteristic

157

C₄⋊C₄

—

158

C₄⋊C₄

C₂×ΓL(1,𝔽₁₆)

yes

—

159

C₄²

—

160

C₄²

—

161

C₃×C₅

#1579

yes

characteristic, complemented, Hall

162

C₂²×C₃

D₄×AGL(1,𝔽₅)

yes

characteristic

163

C₃×C₄

—

164

C₃×C₄

—

165

C₃×C₄

—

166

C₃×C₄

—

167

C₃×C₄

C₂×D₅⋊C₈

yes

characteristic

168

C₃×C₄

#4133

yes

characteristic

169

C₂×C₅

D₄×C₃⋊C₈

yes

characteristic

170

C₂×C₅

#5184

yes

characteristic

171

C₂×C₅

#5197

yes

characteristic

172

Q₈

—

173

Q₈

—

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

C₂²×ΓL(1,𝔽₁₆)

yes

characteristic

179

C₈

—

180

C₈

—

181

C₈

—

182

C₈

—

183

C₈

D₅⋊(C₃⋊C₈)

yes

—

184

C₂×C₃

#65702

yes

characteristic

185

C₂×C₃

#65733

yes

characteristic

186

C₂×C₃

#65739

yes

characteristic

187

C₅

#70902

yes

characteristic, complemented, Sylow

188

C₂²

D₄×ΓL(1,𝔽₁₆)

yes

characteristic

189

C₄

—

190

C₄

—

191

C₄

—

192

C₄

—

193

C₄

C₂×D₅⋊(C₃⋊C₈)

yes

characteristic

194

C₄

#91992

yes

characteristic

195

C₃

#10604061

yes

characteristic, complemented, Sylow

196

C₂

#11752361

yes

characteristic

197

C₂

#11752392

yes

characteristic

198

C₂

#11752398

yes

characteristic

	subgroup	G/H
center	C₂²	D₄×ΓL(1,𝔽₁₆)
commutator	C₃×C₄×C₅	C₂²×C₈
Frattini	C₂×C₄	C₂²×ΓL(1,𝔽₁₆)
Fitting	C₃×C₅×#107	C₄
socle, prime radical	C₂²×C₃×C₅	C₄×D₄
cosocle	C₃×C₄×Dic₅	C₂³
hypercenter	#107	ΓL(1,𝔽₁₆)
hypocenter	C₃×C₅	#1579
LCS	G ▷ C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #107

subgroup

G/H

center

C₂²

D₄×ΓL(1,𝔽₁₆)

commutator

C₃×C₄×C₅

C₂²×C₈

Frattini

C₂×C₄

C₂²×ΓL(1,𝔽₁₆)

Fitting

C₃×C₅×#107

C₄

socle, prime radical

C₂²×C₃×C₅

C₄×D₄

cosocle

C₃×C₄×Dic₅

C₂³

ΓL(1,𝔽₁₆)

C₃×C₅

G ▷ C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ #107

N	H ≅ G/N	type
33. C₃×Dic₂₀	C₈	semidirect
82. Dic₂₀	C₃⋊C₈	semidirect
161. C₃×C₅	#1579	semidirect
187. C₅	#70902	semidirect
195. C₃	#10604061	semidirect

H ≅ G/N

type

33. C₃×Dic₂₀

C₈

semidirect

82. Dic₂₀

C₃⋊C₈

semidirect

161. C₃×C₅

#1579

semidirect

187. C₅

#70902

semidirect

195. C₃

#10604061

semidirect

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

subgroup	index
#11752362	2
C₃×#65162	2
#11746752	2
#11746750	2
#11746743	2
#10604061	3
#70902	5

subgroup

index

Subgroups — #14286391

Conjugacy classes of subgroups

Complements

Maximal subgroups