Group #14298923 — subgroups

Subgroups — #14298923

Order 1920.

360 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 121 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	#11752341	C₂	1	1	yes	maximal, characteristic
2	#11752340	C₂	1	1	yes	maximal, characteristic
3	C₃×D₅×#97	C₂	1	1	yes	maximal, characteristic
4	#11746767	C₂	1	4	yes	maximal
5	#10603889	—	3	1	no	maximal, Hall
6	C₂×#6758	C₂²	1	2	yes	—
7	C₂×#6758	C₂²	1	2	yes	—
8	C₂×C₃×C₈×D₅	C₂²	1	2	yes	—
9	C₃×C₄²×D₅	C₂²	1	1	yes	characteristic
10	#91233	—	2	2	no	—
11	C₄⋊(C₃⋊#170)	—	2	2	no	—
12	C₃×C₅×#97	C₄	1	1	yes	characteristic
13	C₃×#3947	C₄	1	2	yes	—
14	C₃×#3935	C₄	1	1	yes	characteristic
15	#69841	—	5	1	no	maximal, Hall
16	#65682	—	3	1	no	—
17	#65681	—	3	1	no	—
18	D₅×#97	D₃	1	1	yes	characteristic
19	#64896	—	3	4	no	—
20	C₂×C₃×C₄×D₅	Q₈	1	2	yes	—
21	C₂×C₃×C₄×D₅	C₂³	1	1	yes	characteristic
22	C₂×C₃⋊#170	—	2	2	no	—
23	C₂×C₃⋊#170	—	2	2	no	—
24	#6758	—	2	4	no	—
25	#6758	—	2	4	no	—
26	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
27	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
28	C₃×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
29	C₃×C₄×Dic₅	D₄	1	2	yes	—
30	C₂×C₃×#168	C₂×C₄	1	2	yes	—
31	C₃×C₈×D₅	—	2	2	no	—
32	C₃×C₈×D₅	—	2	2	no	—
33	C₂×C₃×#97	—	5	1	no	—
34	#4924	—	5	1	no	—
35	#4924	—	5	1	no	—
36	#4552	—	5	4	no	—
37	C₂×#700	—	3	2	no	—
38	C₂×#700	—	3	2	no	—
39	C₂×C₈×D₅	C₂×D₃	1	2	yes	—
40	C₄²×D₅	C₂×D₃	1	1	yes	characteristic
41	#4004	—	6	2	no	—
42	C₄⋊#170	—	6	2	no	—
43	C₅×#97	Dic₃	1	1	yes	characteristic
44	#3947	Dic₃	1	2	yes	—
45	#3935	Dic₃	1	1	yes	characteristic
46	#1838	—	15	1	no	Sylow
47	C₂²×C₃×D₅	C₂×Q₈	1	1	yes	characteristic
48	C₂×C₃×C₄×C₅	C₄⋊C₄	1	2	yes	—
49	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₃×Dic₅	C₄⋊C₄	1	2	yes	—
51	C₂×C₃×Dic₅	C₂×D₄	1	1	yes	characteristic
52	C₃×C₄×D₅	—	4	2	no	—
53	C₃×C₄×D₅	#41	1	2	yes	—
54	C₃×C₄×D₅	#41	1	2	yes	—
55	C₃⋊#170	—	4	2	no	—
56	C₃⋊#170	—	4	2	no	—
57	C₃×C₅×C₈	—	2	2	no	—
58	C₃×#168	—	2	2	no	—
59	C₂²×C₃×C₈	—	5	2	no	—
60	C₂×C₃×C₄²	—	5	1	no	—
61	C₂×#208	—	5	2	no	—
62	C₂×#208	—	5	2	no	—
63	C₃×#97	—	5	2	no	—
64	C₃×#97	—	5	1	no	—
65	C₃×#97	AGL(1,𝔽₅)	1	1	yes	characteristic
66	C₄⋊(C₃⋊C₈)	—	10	2	no	—
67	C₄⋊(C₃⋊C₈)	—	10	2	no	—
68	C₂×C₄×D₅	Dic₆	1	2	yes	—
69	C₂×C₄×D₅	C₂²×D₃	1	1	yes	characteristic
70	C₂×#170	—	6	2	no	—
71	C₂×#170	—	6	2	no	—
72	#700	—	6	4	no	—
73	#700	—	6	4	no	—
74	C₂×C₅×C₈	C₂×Dic₃	1	2	yes	—
75	C₄²×C₅	C₂×Dic₃	1	1	yes	characteristic
76	C₄×Dic₅	D₁₂	1	2	yes	—
77	C₄×Dic₅	C₂×Dic₃	1	1	yes	characteristic
78	C₂×#168	C₂×Dic₃	1	2	yes	—
79	C₈×D₅	—	2	2	no	—
80	C₈×D₅	—	2	2	no	—
81	#423	—	15	1	no	—
82	#423	—	15	1	no	—
83	C₂×#97	—	5	1	no	—
84	#341	—	15	4	no	—
85	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
86	C₂×C₃×D₅	—	2	2	no	—
87	C₂×C₃×D₅	#125	1	2	yes	—
88	C₃×C₄×C₅	—	2	2	no	—
89	C₃×C₄×C₅	#117	1	2	yes	—
90	C₃×Dic₅	—	2	2	no	—
91	C₃×Dic₅	#124	1	1	yes	characteristic
92	C₃×Dic₅	#124	1	1	yes	characteristic
93	C₂²×C₃×C₄	—	5	1	no	—
94	C₂²×C₃×C₄	—	5	2	no	—
95	C₂×C₃×C₈	—	5	2	no	—
96	C₂×C₃×C₈	—	10	2	no	—
97	C₂×C₃×C₈	—	10	2	no	—
98	C₂×C₃×C₈	C₂×AGL(1,𝔽₅)	1	2	yes	—
99	C₃×C₄²	—	5	2	no	—
100	C₃×C₄²	—	5	1	no	—
101	C₃×C₄²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
102	#208	—	10	4	no	—
103	#208	—	10	4	no	—
104	C₂×C₃⋊C₈	—	10	2	no	—
105	C₂×C₃⋊C₈	—	10	2	no	—
106	C₂²×D₅	C₂×Dic₆	1	1	yes	characteristic
107	C₂×C₄×C₅	C₄⋊Dic₃	1	2	yes	—
108	C₂×C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
109	C₂×Dic₅	C₄⋊Dic₃	1	2	yes	—
110	C₂×Dic₅	C₂×D₁₂	1	1	yes	characteristic
111	C₄×D₅	—	4	2	no	—
112	C₄×D₅	#235	1	2	yes	—
113	C₄×D₅	#235	1	2	yes	—
114	#170	—	12	2	no	—
115	#170	—	12	2	no	—
116	C₅×C₈	—	2	2	no	—
117	#168	—	2	2	no	—
118	C₂×M₁₆	—	15	2	no	—
119	C₂×M₁₆	—	15	2	no	—
120	C₂²×C₈	—	5	2	no	—
121	C₂×C₄²	—	5	1	no	—
122	C₄⋊C₈	—	30	2	no	—
123	C₄⋊C₈	—	30	2	no	—
124	#97	—	5	2	no	—
125	#97	—	5	1	no	—
126	#97	ΓL(1,𝔽₁₆)	1	1	yes	characteristic
127	C₂×C₃×C₅	#382	1	1	yes	characteristic
128	C₂×C₃×C₅	#430	1	2	yes	—
129	C₃×D₅	—	2	2	no	—
130	C₂³×C₃	—	5	1	no	—
131	C₂×C₃×C₄	—	5	1	no	—
132	C₂×C₃×C₄	—	5	2	no	—
133	C₂×C₃×C₄	—	5	2	no	—
134	C₂×C₃×C₄	—	5	2	no	—
135	C₂×C₃×C₄	—	20	2	no	—
136	C₂×C₃×C₄	C₄⋊AGL(1,𝔽₅)	1	2	yes	—
137	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
138	C₃×C₈	—	2	2	no	—
139	C₃×C₈	—	10	2	no	—
140	C₃⋊C₈	—	20	2	no	—
141	C₃⋊C₈	—	20	2	no	—
142	C₂²×C₅	C₂×C₄⋊Dic₃	1	1	yes	characteristic
143	C₂×D₅	—	2	2	no	—
144	C₂×D₅	#870	1	2	yes	—
145	C₄×C₅	—	2	2	no	—
146	C₄×C₅	#926	1	2	yes	—
147	Dic₅	—	2	2	no	—
148	Dic₅	#875	1	1	yes	characteristic
149	Dic₅	#875	1	1	yes	characteristic
150	C₂²×C₄	—	5	1	no	—
151	C₂²×C₄	—	5	2	no	—
152	M₁₆	—	30	4	no	—
153	M₁₆	—	30	4	no	—
154	C₂×C₈	—	5	2	no	—
155	C₂×C₈	—	10	2	no	—
156	C₂×C₈	—	10	2	no	—
157	C₂×C₈	—	30	2	no	—
158	C₂×C₈	—	30	2	no	—
159	C₂×C₈	C₂×ΓL(1,𝔽₁₆)	1	2	yes	—
160	C₄²	—	5	2	no	—
161	C₄²	—	5	1	no	—
162	C₄²	C₂×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
163	C₃×C₅	#1838	1	1	yes	characteristic, complemented, Hall
164	C₂²×C₃	—	5	2	no	—
165	C₂²×C₃	—	10	2	no	—
166	C₂²×C₃	C₂×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
167	C₃×C₄	—	2	2	no	—
168	C₃×C₄	—	5	1	no	—
169	C₃×C₄	—	5	1	no	—
170	C₃×C₄	—	10	2	no	—
171	C₃×C₄	#4130	1	2	yes	—
172	C₂×C₅	#4936	1	1	yes	characteristic
173	C₂×C₅	#5125	1	2	yes	—
174	D₅	—	2	2	no	—
175	C₂³	—	5	1	no	—
176	C₂×C₄	—	5	1	no	—
177	C₂×C₄	—	5	2	no	—
178	C₂×C₄	—	5	2	no	—
179	C₂×C₄	—	5	2	no	—
180	C₂×C₄	—	20	2	no	—
181	C₂×C₄	C₄⋊ΓL(1,𝔽₁₆)	1	2	yes	—
182	C₂×C₄	C₂²×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
183	C₈	—	2	2	no	—
184	C₈	—	10	2	no	—
185	C₈	—	60	2	no	—
186	C₈	—	60	2	no	—
187	C₂×C₃	—	10	2	no	—
188	C₂×C₃	#65689	1	1	yes	characteristic
189	C₂×C₃	#65729	1	2	yes	—
190	C₅	#69841	1	1	yes	characteristic, complemented, Sylow
191	C₂²	—	5	2	no	—
192	C₂²	—	10	2	no	—
193	C₂²	C₂×C₄⋊ΓL(1,𝔽₁₆)	1	1	yes	characteristic
194	C₄	—	2	2	no	—
195	C₄	—	5	1	no	—
196	C₄	—	5	1	no	—
197	C₄	—	10	2	no	—
198	C₄	#91989	1	2	yes	—
199	C₃	#10603889	1	1	yes	characteristic, complemented, Sylow
200	C₂	—	10	2	no	—
201	C₂	#11752348	1	1	yes	characteristic
202	C₂	#11752388	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#11752341

C₂

yes

maximal, characteristic

#11752340

C₂

yes

maximal, characteristic

C₃×D₅×#97

C₂

yes

maximal, characteristic

#11746767

C₂

yes

—

C₂²

yes

—

C₂×#6758

C₂²

yes

—

C₂×C₃×C₈×D₅

C₂²

yes

—

C₃×C₄²×D₅

C₂²

yes

characteristic

#91233

—

C₄⋊(C₃⋊#170)

—

C₃×C₅×#97

C₄

yes

characteristic

C₃×#3947

C₄

yes

—

C₃×#3935

C₄

yes

—

—

—

D₅×#97

D₃

yes

characteristic

#64896

—

C₂×C₃×C₄×D₅

Q₈

yes

—

C₂×C₃×C₄×D₅

C₂³

yes

characteristic

C₂×C₃⋊#170

—

C₂×C₃⋊#170

—

#6758

—

#6758

—

C₂×C₃×C₅×C₈

C₂×C₄

yes

—

C₃×C₄²×C₅

C₂×C₄

yes

characteristic

C₃×C₄×Dic₅

C₂×C₄

yes

characteristic

C₃×C₄×Dic₅

D₄

yes

—

C₂×C₃×#168

C₂×C₄

yes

—

C₃×C₈×D₅

—

C₃×C₈×D₅

—

C₂×C₃×#97

—

—

—

—

—

—

C₂×C₈×D₅

C₂×D₃

yes

—

C₄²×D₅

C₂×D₃

yes

characteristic

#4004

—

C₄⋊#170

—

C₅×#97

Dic₃

yes

characteristic

#3947

Dic₃

yes

—

#3935

Dic₃

yes

characteristic

#1838

—

Sylow

C₂²×C₃×D₅

C₂×Q₈

yes

characteristic

C₂×C₃×C₄×C₅

C₄⋊C₄

yes

—

C₂×C₃×C₄×C₅

C₂²×C₄

yes

characteristic

C₂×C₃×Dic₅

C₄⋊C₄

yes

—

C₂×C₃×Dic₅

C₂×D₄

yes

characteristic

C₃×C₄×D₅

—

C₃×C₄×D₅

#41

yes

—

C₃×C₄×D₅

#41

yes

—

C₃⋊#170

—

C₃⋊#170

—

C₃×C₅×C₈

—

C₃×#168

—

C₂²×C₃×C₈

—

C₂×C₃×C₄²

—

C₂×#208

—

C₂×#208

—

C₃×#97

—

C₃×#97

—

C₃×#97

AGL(1,𝔽₅)

yes

characteristic

C₄⋊(C₃⋊C₈)

—

C₄⋊(C₃⋊C₈)

—

C₂×C₄×D₅

Dic₆

yes

—

C₂×C₄×D₅

C₂²×D₃

yes

—

—

—

—

C₂×C₅×C₈

C₂×Dic₃

yes

—

C₄²×C₅

C₂×Dic₃

yes

characteristic

C₄×Dic₅

D₁₂

yes

—

C₄×Dic₅

C₂×Dic₃

yes

characteristic

C₂×#168

C₂×Dic₃

yes

—

C₈×D₅

—

C₈×D₅

—

#423

—

#423

—

C₂×#97

—

#341

—

C₂²×C₃×C₅

C₂×C₄⋊C₄

yes

characteristic

C₂×C₃×D₅

—

C₂×C₃×D₅

#125

yes

—

C₃×C₄×C₅

—

C₃×C₄×C₅

#117

yes

—

C₃×Dic₅

—

C₃×Dic₅

yes

C₃×Dic₅

yes

C₂²×C₃×C₄

—

C₂²×C₃×C₄

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₂×C₃×C₈

C₂×AGL(1,𝔽₅)

yes

—

C₃×C₄²

—

100

C₃×C₄²

—

101

C₃×C₄²

C₂×AGL(1,𝔽₅)

yes

characteristic

102

#208

—

103

#208

—

104

C₂×C₃⋊C₈

—

105

C₂×C₃⋊C₈

—

106

C₂²×D₅

C₂×Dic₆

yes

characteristic

107

C₂×C₄×C₅

C₄⋊Dic₃

yes

—

108

C₂×C₄×C₅

C₂²×Dic₃

yes

characteristic

109

C₂×Dic₅

C₄⋊Dic₃

yes

—

110

C₂×Dic₅

C₂×D₁₂

yes

characteristic

111

C₄×D₅

—

112

C₄×D₅

#235

yes

—

113

C₄×D₅

#235

yes

—

114

#170

—

115

#170

—

116

C₅×C₈

—

117

#168

—

118

C₂×M₁₆

—

119

C₂×M₁₆

—

120

C₂²×C₈

—

121

C₂×C₄²

—

122

C₄⋊C₈

—

123

C₄⋊C₈

—

124

#97

—

125

#97

—

126

#97

ΓL(1,𝔽₁₆)

yes

characteristic

127

C₂×C₃×C₅

#382

yes

characteristic

128

C₂×C₃×C₅

#430

yes

—

129

C₃×D₅

—

130

C₂³×C₃

—

131

C₂×C₃×C₄

—

132

C₂×C₃×C₄

—

133

C₂×C₃×C₄

—

134

C₂×C₃×C₄

—

135

C₂×C₃×C₄

—

136

C₂×C₃×C₄

C₄⋊AGL(1,𝔽₅)

yes

—

137

C₂×C₃×C₄

C₂²×AGL(1,𝔽₅)

yes

characteristic

138

C₃×C₈

—

139

C₃×C₈

—

140

C₃⋊C₈

—

141

C₃⋊C₈

—

142

C₂²×C₅

C₂×C₄⋊Dic₃

yes

characteristic

143

C₂×D₅

—

144

C₂×D₅

#870

yes

—

145

C₄×C₅

—

146

C₄×C₅

#926

yes

—

147

Dic₅

—

148

Dic₅

#875

yes

characteristic

149

Dic₅

#875

yes

characteristic

150

C₂²×C₄

—

151

C₂²×C₄

—

152

M₁₆

—

153

M₁₆

—

154

C₂×C₈

—

155

C₂×C₈

—

156

C₂×C₈

—

157

C₂×C₈

—

158

C₂×C₈

—

159

C₂×C₈

C₂×ΓL(1,𝔽₁₆)

yes

—

160

C₄²

—

161

C₄²

—

162

C₄²

C₂×ΓL(1,𝔽₁₆)

yes

characteristic

163

C₃×C₅

#1838

yes

characteristic, complemented, Hall

164

C₂²×C₃

—

165

C₂²×C₃

—

166

C₂²×C₃

C₂×C₄⋊AGL(1,𝔽₅)

yes

characteristic

167

C₃×C₄

—

168

C₃×C₄

—

169

C₃×C₄

—

170

C₃×C₄

—

171

C₃×C₄

#4130

yes

—

172

C₂×C₅

#4936

yes

characteristic

173

C₂×C₅

#5125

yes

—

174

D₅

—

175

C₂³

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

—

181

C₂×C₄

C₄⋊ΓL(1,𝔽₁₆)

yes

—

182

C₂×C₄

C₂²×ΓL(1,𝔽₁₆)

yes

characteristic

183

C₈

—

184

C₈

—

185

C₈

—

186

C₈

—

187

C₂×C₃

—

188

C₂×C₃

#65689

yes

characteristic

189

C₂×C₃

#65729

yes

—

190

C₅

#69841

yes

characteristic, complemented, Sylow

191

C₂²

—

192

C₂²

—

193

C₂²

C₂×C₄⋊ΓL(1,𝔽₁₆)

yes

characteristic

194

C₄

—

195

C₄

—

196

C₄

—

197

C₄

—

198

C₄

#91989

yes

—

199

C₃

#10603889

yes

characteristic, complemented, Sylow

200

C₂

—

201

C₂

#11752348

yes

characteristic

202

C₂

#11752388

yes

—

	subgroup	G/H
center	C₂²	C₂×C₄⋊ΓL(1,𝔽₁₆)
commutator	C₂×C₃×C₄×C₅	C₂²×C₄
Frattini	C₂×C₄	C₂²×ΓL(1,𝔽₁₆)
Fitting	C₃×C₅×#97	C₄
socle	C₂²×C₃×C₅	C₂×C₄⋊C₄
prime radical	C₂²×C₃×D₅	C₂×Q₈
cosocle	C₂×C₃×C₄×D₅	C₂³
hypercenter	#97	ΓL(1,𝔽₁₆)
hypocenter	C₃×C₅	#1838
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #97

subgroup

G/H

center

C₂²

C₂×C₄⋊ΓL(1,𝔽₁₆)

commutator

C₂×C₃×C₄×C₅

C₂²×C₄

Frattini

C₂×C₄

C₂²×ΓL(1,𝔽₁₆)

Fitting

C₃×C₅×#97

C₄

socle

C₂²×C₃×C₅

C₂×C₄⋊C₄

prime radical

C₂²×C₃×D₅

C₂×Q₈

cosocle

C₂×C₃×C₄×D₅

C₂³

ΓL(1,𝔽₁₆)

C₃×C₅

G ▷ C₂×C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #97

N	H ≅ G/N	type
163. C₃×C₅	#1838	semidirect
190. C₅	#69841	semidirect
199. C₃	#10603889	semidirect

H ≅ G/N

type

163. C₃×C₅

#1838

semidirect

190. C₅

#69841

semidirect

199. C₃

#10603889

semidirect

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

subgroup	index
#11752341	2
#11752340	2
C₃×D₅×#97	2
#11746767	2
#10603889	3
#69841	5

subgroup

index

C₃×D₅×#97

Subgroups — #14298923

Conjugacy classes of subgroups

Complements

Maximal subgroups