Group #14294955 — subgroups

Subgroups — #14294955

Order 1920.

480 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 480 conjugacy classes of subgroups — too many to draw. Only its 150 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	#11752345	C₂	1	1	yes	maximal, characteristic
2	C₃×#65676	C₂	1	1	yes	maximal, characteristic
3	C₄×D₅×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#11746517	C₂	1	4	yes	maximal
5	#10603840	—	3	1	no	maximal, Hall
6	C₂×C₄×ΓL(1,𝔽₁₆)	C₂²	1	2	yes	—
7	C₂×C₃×D₅⋊C₈	C₂²	1	2	yes	—
8	C₃×C₄²×D₅	C₂²	1	1	yes	characteristic
9	C₂×D₅×C₃⋊C₈	C₂²	1	2	yes	—
10	#91235	C₄	1	4	yes	—
11	C₃×#4002	C₄	1	4	yes	complemented
12	C₄×C₃⋊#168	C₄	1	1	yes	characteristic, complemented
13	C₄×C₅×C₃⋊C₈	C₄	1	1	yes	characteristic, complemented
14	Dic₅×C₃⋊C₈	C₄	1	2	yes	complemented
15	#69806	—	5	1	no	maximal, Hall
16	#65686	—	3	1	no	—
17	#65676	D₃	1	1	yes	characteristic
18	C₄×C₈×D₅	—	3	1	no	—
19	(C₂×AGL(1,𝔽₅))⋊C₈	—	3	4	no	—
20	C₂²×ΓL(1,𝔽₁₆)	C₂×C₄	1	4	yes	—
21	C₂×C₃×C₄×D₅	C₂×C₄	1	2	yes	—
22	C₂×C₃×C₄×D₅	C₂³	1	1	yes	characteristic
23	C₄×ΓL(1,𝔽₁₆)	—	2	4	no	—
24	C₂×C₃×#170	C₂×C₄	1	4	yes	—
25	C₃×D₅⋊C₈	—	2	4	no	—
26	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
27	C₂×C₃⋊#168	C₂×C₄	1	2	yes	—
28	C₂×C₅×C₃⋊C₈	C₂×C₄	1	2	yes	—
29	C₃×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
30	C₃×C₄×Dic₅	C₂×C₄	1	2	yes	—
31	D₅×C₃⋊C₈	—	2	4	no	—
32	C₂×C₃×#97	—	5	1	no	—
33	#4932	—	5	1	no	—
34	C₂×C₄×C₃⋊C₈	—	5	1	no	—
35	#4550	—	5	4	no	—
36	C₂×C₄×AGL(1,𝔽₅)	—	3	2	no	—
37	C₂×D₅⋊C₈	C₂×D₃	1	2	yes	—
38	C₂×C₈×D₅	—	3	2	no	—
39	C₄²×D₅	C₂×D₃	1	1	yes	characteristic
40	(C₂×AGL(1,𝔽₅))⋊C₄	—	3	4	no	—
41	#4002	Dic₃	1	4	yes	complemented
42	C₄×C₅×C₈	—	3	1	no	—
43	C₈×Dic₅	—	3	2	no	—
44	C₄×#168	—	3	1	no	—
45	#1746	—	15	1	no	Sylow
46	C₂²×C₃×D₅	C₂²×C₄	1	1	yes	characteristic
47	C₂×ΓL(1,𝔽₁₆)	—	2	4	no	—
48	C₂×ΓL(1,𝔽₁₆)	—	2	8	no	—
49	C₂×C₃×C₄×C₅	C₄²	1	2	yes	—
50	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₃×Dic₅	C₄²	1	2	yes	—
52	C₂×C₃×Dic₅	C₂²×C₄	1	1	yes	characteristic
53	C₃×C₄×D₅	—	2	4	no	—
54	C₃×C₄×D₅	#41	1	4	yes	—
55	C₃×#170	—	2	4	no	—
56	C₃⋊#168	—	2	2	no	—
57	C₅×C₃⋊C₈	—	2	2	no	—
58	C₂²×C₃×C₈	—	5	2	no	—
59	C₂×C₃×C₄²	—	5	1	no	—
60	C₂×C₄×Dic₃	—	5	2	no	—
61	C₂²×C₃⋊C₈	—	5	2	no	—
62	C₃×#97	—	5	4	no	—
63	C₃⋊#95	—	5	4	no	—
64	C₄×C₃⋊C₈	—	5	1	no	—
65	C₄×C₃⋊C₈	—	5	2	no	—
66	C₄×C₃⋊C₈	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
67	C₂²×AGL(1,𝔽₅)	—	3	4	no	—
68	C₂×C₄×D₅	C₄×D₃	1	2	yes	—
69	C₂×C₄×D₅	C₂²×D₃	1	1	yes	characteristic
70	C₂×#170	C₂×Dic₃	1	4	yes	—
71	C₄×AGL(1,𝔽₅)	—	6	4	no	—
72	D₅⋊C₈	—	2	4	no	—
73	C₂×C₅×C₈	—	3	2	no	—
74	C₄²×C₅	C₄×D₃	1	1	yes	characteristic
75	C₄×Dic₅	C₄×D₃	1	1	yes	characteristic
76	C₄×Dic₅	C₂×Dic₃	1	2	yes	—
77	C₂×#168	—	3	2	no	—
78	C₈×D₅	—	6	4	no	—
79	C₂×#97	—	5	1	no	—
80	C₂×C₄×C₈	—	15	1	no	—
81	#376	—	15	1	no	—
82	#336	—	15	4	no	—
83	C₂²×C₃×C₅	C₂×C₄²	1	1	yes	characteristic
84	C₂×C₃×D₅	#117	1	2	yes	—
85	C₂×C₃×D₅	#131	1	4	yes	—
86	ΓL(1,𝔽₁₆)	—	4	4	no	—
87	C₃×C₄×C₅	—	2	2	no	—
88	C₃×C₄×C₅	#117	1	2	yes	—
89	C₃×Dic₅	—	2	2	no	—
90	C₃×Dic₅	#117	1	2	yes	—
91	C₂²×C₃×C₄	—	5	1	no	—
92	C₂²×C₃×C₄	—	5	2	no	—
93	C₂²×Dic₃	—	5	4	no	—
94	C₂×C₃×C₈	—	5	4	no	—
95	C₂×C₃×C₈	—	10	4	no	—
96	C₃×C₄²	—	5	2	no	—
97	C₃×C₄²	—	5	1	no	—
98	C₃×C₄²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
99	C₄×Dic₃	—	10	4	no	—
100	C₂×C₃⋊C₈	—	5	2	no	—
101	C₂×C₃⋊C₈	—	10	4	no	—
102	C₂×C₃⋊C₈	C₂×AGL(1,𝔽₅)	1	2	yes	—
103	C₂²×D₅	C₂×C₄×D₃	1	1	yes	characteristic
104	C₂×AGL(1,𝔽₅)	—	6	4	no	—
105	C₂×AGL(1,𝔽₅)	—	6	8	no	—
106	C₂×C₄×C₅	C₄×Dic₃	1	2	yes	—
107	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
108	C₂×Dic₅	C₄×Dic₃	1	2	yes	—
109	C₂×Dic₅	C₂²×Dic₃	1	1	yes	characteristic
110	C₄×D₅	—	2	4	no	—
111	C₄×D₅	#235	1	4	yes	—
112	#170	—	2	4	no	—
113	C₅×C₈	—	6	2	no	—
114	#168	—	6	2	no	—
115	C₂²×C₈	—	5	2	no	—
116	C₂²×C₈	—	15	2	no	—
117	C₂×C₄²	—	5	1	no	—
118	C₂×C₄²	—	15	2	no	—
119	#97	—	5	4	no	—
120	C₄×C₈	—	3	1	no	—
121	C₄×C₈	—	15	2	no	—
122	C₄×C₈	—	15	1	no	—
123	#95	—	15	4	no	—
124	C₂×C₃×C₅	#376	1	1	yes	characteristic
125	C₂×C₃×C₅	#405	1	2	yes	—
126	C₃×D₅	#433	1	4	yes	—
127	C₂³×C₃	—	5	1	no	—
128	C₂×C₃×C₄	—	5	1	no	—
129	C₂×C₃×C₄	—	5	4	no	—
130	C₂×C₃×C₄	—	5	2	no	—
131	C₂×C₃×C₄	—	10	4	no	—
132	C₂×C₃×C₄	C₄×AGL(1,𝔽₅)	1	2	yes	—
133	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
134	C₂×Dic₃	—	10	4	no	—
135	C₂×Dic₃	—	10	8	no	—
136	C₃×C₈	—	10	4	no	—
137	C₃⋊C₈	—	2	2	no	—
138	C₃⋊C₈	—	10	2	no	—
139	C₂²×C₅	C₂×C₄×Dic₃	1	1	yes	characteristic
140	C₂×D₅	#872	1	2	yes	—
141	C₂×D₅	#907	1	4	yes	—
142	AGL(1,𝔽₅)	—	12	4	no	—
143	C₄×C₅	—	2	2	no	—
144	C₄×C₅	#872	1	2	yes	—
145	Dic₅	—	2	2	no	—
146	Dic₅	#926	1	2	yes	—
147	C₂²×C₄	—	5	1	no	—
148	C₂²×C₄	—	5	2	no	—
149	C₂²×C₄	—	15	4	no	—
150	C₂×C₈	—	3	2	no	—
151	C₂×C₈	—	5	4	no	—
152	C₂×C₈	—	10	4	no	—
153	C₂×C₈	—	15	2	no	—
154	C₂×C₈	—	30	4	no	—
155	C₄²	—	5	2	no	—
156	C₄²	—	5	1	no	—
157	C₄²	—	30	4	no	—
158	C₄²	D₃×AGL(1,𝔽₅)	1	1	yes	characteristic
159	C₃×C₅	#1746	1	1	yes	characteristic, complemented, Hall
160	C₂²×C₃	—	5	2	no	—
161	C₂²×C₃	—	5	4	no	—
162	C₂²×C₃	C₂×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
163	C₃×C₄	—	2	2	no	—
164	C₃×C₄	—	5	2	no	—
165	C₃×C₄	—	10	2	no	—
166	C₃×C₄	#4130	1	2	yes	—
167	Dic₃	—	20	4	no	—
168	C₂×C₅	#4932	1	1	yes	characteristic
169	C₂×C₅	#5122	1	2	yes	—
170	D₅	#4709	1	4	yes	—
171	C₂³	—	5	1	no	—
172	C₂×C₄	—	5	1	no	—
173	C₂×C₄	—	5	4	no	—
174	C₂×C₄	—	5	2	no	—
175	C₂×C₄	—	10	4	no	—
176	C₂×C₄	—	30	4	no	—
177	C₂×C₄	—	30	8	no	—
178	C₂×C₄	Dic₃×AGL(1,𝔽₅)	1	2	yes	—
179	C₂×C₄	C₂×D₃×AGL(1,𝔽₅)	1	1	yes	characteristic
180	C₈	—	6	2	no	—
181	C₈	—	10	4	no	—
182	C₈	—	30	2	no	—
183	C₂×C₃	—	5	4	no	—
184	C₂×C₃	#65686	1	1	yes	characteristic
185	C₂×C₃	#65718	1	2	yes	—
186	C₅	#69806	1	1	yes	characteristic, complemented, Sylow
187	C₂²	—	5	4	no	—
188	C₂²	—	5	2	no	—
189	C₂²	C₂×Dic₃×AGL(1,𝔽₅)	1	1	yes	characteristic
190	C₄	—	2	2	no	—
191	C₄	—	5	2	no	—
192	C₄	—	10	2	no	—
193	C₄	—	60	4	no	—
194	C₄	#91909	1	2	yes	—
195	C₃	#10603840	1	1	yes	characteristic, complemented, Sylow
196	C₂	—	5	4	no	—
197	C₂	#11751696	1	2	yes	—
198	C₂	#11751748	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11752345

C₂

yes

maximal, characteristic

C₃×#65676

C₂

yes

maximal, characteristic

C₄×D₅×C₃⋊C₈

C₂

yes

maximal, characteristic

#11746517

C₂

yes

maximal

#10603840

—

maximal, Hall

C₂×C₄×ΓL(1,𝔽₁₆)

C₂²

yes

—

C₂×C₃×D₅⋊C₈

C₂²

yes

—

C₃×C₄²×D₅

C₂²

yes

characteristic

C₂×D₅×C₃⋊C₈

C₂²

yes

—

#91235

C₄

yes

—

C₃×#4002

C₄

yes

complemented

C₄×C₃⋊#168

C₄

yes

characteristic, complemented

C₄×C₅×C₃⋊C₈

C₄

yes

characteristic, complemented

Dic₅×C₃⋊C₈

C₄

yes

—

—

D₃

yes

characteristic

C₄×C₈×D₅

—

(C₂×AGL(1,𝔽₅))⋊C₈

—

C₂²×ΓL(1,𝔽₁₆)

C₂×C₄

yes

—

C₂×C₃×C₄×D₅

C₂×C₄

yes

—

C₂×C₃×C₄×D₅

C₂³

yes

characteristic

C₄×ΓL(1,𝔽₁₆)

—

C₂×C₃×#170

C₂×C₄

yes

—

C₃×D₅⋊C₈

—

C₃×C₄²×C₅

C₂×C₄

yes

characteristic

C₂×C₃⋊#168

C₂×C₄

yes

—

C₂×C₅×C₃⋊C₈

C₂×C₄

yes

—

C₃×C₄×Dic₅

C₂×C₄

yes

characteristic

C₃×C₄×Dic₅

C₂×C₄

yes

—

D₅×C₃⋊C₈

—

C₂×C₃×#97

—

#4932

—

C₂×C₄×C₃⋊C₈

—

#4550

—

C₂×C₄×AGL(1,𝔽₅)

—

C₂×D₅⋊C₈

C₂×D₃

yes

—

C₂×C₈×D₅

—

C₄²×D₅

C₂×D₃

yes

characteristic

(C₂×AGL(1,𝔽₅))⋊C₄

—

#4002

Dic₃

yes

complemented

C₄×C₅×C₈

—

C₈×Dic₅

—

C₄×#168

—

#1746

—

Sylow

C₂²×C₃×D₅

C₂²×C₄

yes

characteristic

C₂×ΓL(1,𝔽₁₆)

—

C₂×ΓL(1,𝔽₁₆)

—

C₂×C₃×C₄×C₅

C₄²

yes

—

C₂×C₃×C₄×C₅

C₂²×C₄

yes

characteristic

C₂×C₃×Dic₅

C₄²

yes

—

C₂×C₃×Dic₅

C₂²×C₄

yes

characteristic

C₃×C₄×D₅

—

C₃×C₄×D₅

#41

yes

—

C₃×#170

—

C₃⋊#168

—

C₅×C₃⋊C₈

—

C₂²×C₃×C₈

—

C₂×C₃×C₄²

—

C₂×C₄×Dic₃

—

C₂²×C₃⋊C₈

—

C₃×#97

—

C₃⋊#95

—

C₄×C₃⋊C₈

—

C₄×C₃⋊C₈

—

C₄×C₃⋊C₈

AGL(1,𝔽₅)

yes

characteristic, complemented

C₂²×AGL(1,𝔽₅)

—

C₂×C₄×D₅

C₄×D₃

yes

—

C₂×C₄×D₅

C₂²×D₃

yes

characteristic

C₂×#170

C₂×Dic₃

yes

—

C₄×AGL(1,𝔽₅)

—

D₅⋊C₈

—

C₂×C₅×C₈

—

C₄²×C₅

C₄×D₃

yes

characteristic

C₄×Dic₅

C₄×D₃

yes

characteristic

C₄×Dic₅

C₂×Dic₃

yes

—

C₂×#168

—

C₈×D₅

—

C₂×#97

—

C₂×C₄×C₈

—

#376

—

#336

—

C₂²×C₃×C₅

C₂×C₄²

yes

characteristic

C₂×C₃×D₅

#117

yes

—

C₂×C₃×D₅

#131

yes

—

ΓL(1,𝔽₁₆)

—

C₃×C₄×C₅

—

C₃×C₄×C₅

#117

yes

—

C₃×Dic₅

—

C₃×Dic₅

#117

yes

—

C₂²×C₃×C₄

—

C₂²×C₃×C₄

—

C₂²×Dic₃

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₃×C₄²

—

C₃×C₄²

—

C₃×C₄²

C₂×AGL(1,𝔽₅)

yes

characteristic

C₄×Dic₃

—

100

C₂×C₃⋊C₈

—

101

C₂×C₃⋊C₈

—

102

C₂×C₃⋊C₈

C₂×AGL(1,𝔽₅)

yes

—

103

C₂²×D₅

C₂×C₄×D₃

yes

characteristic

104

C₂×AGL(1,𝔽₅)

—

105

C₂×AGL(1,𝔽₅)

—

106

C₂×C₄×C₅

C₄×Dic₃

yes

—

107

C₂×C₄×C₅

C₂×C₄×D₃

yes

characteristic

108

C₂×Dic₅

C₄×Dic₃

yes

—

109

C₂×Dic₅

C₂²×Dic₃

yes

characteristic

110

C₄×D₅

—

111

C₄×D₅

#235

yes

—

112

#170

—

113

C₅×C₈

—

114

#168

—

115

C₂²×C₈

—

116

C₂²×C₈

—

117

C₂×C₄²

—

118

C₂×C₄²

—

119

#97

—

120

C₄×C₈

—

121

C₄×C₈

—

122

C₄×C₈

—

123

#95

—

124

C₂×C₃×C₅

#376

yes

characteristic

125

C₂×C₃×C₅

#405

yes

—

126

C₃×D₅

#433

yes

—

127

C₂³×C₃

—

128

C₂×C₃×C₄

—

129

C₂×C₃×C₄

—

130

C₂×C₃×C₄

—

131

C₂×C₃×C₄

—

132

C₂×C₃×C₄

C₄×AGL(1,𝔽₅)

yes

—

133

C₂×C₃×C₄

C₂²×AGL(1,𝔽₅)

yes

characteristic

134

C₂×Dic₃

—

135

C₂×Dic₃

—

136

C₃×C₈

—

137

C₃⋊C₈

—

138

C₃⋊C₈

—

139

C₂²×C₅

C₂×C₄×Dic₃

yes

characteristic

140

C₂×D₅

#872

yes

—

141

C₂×D₅

#907

yes

—

142

AGL(1,𝔽₅)

—

143

C₄×C₅

—

144

C₄×C₅

#872

yes

—

145

Dic₅

—

146

Dic₅

#926

yes

—

147

C₂²×C₄

—

148

C₂²×C₄

—

149

C₂²×C₄

—

150

C₂×C₈

—

151

C₂×C₈

—

152

C₂×C₈

—

153

C₂×C₈

—

154

C₂×C₈

—

155

C₄²

—

156

C₄²

—

157

C₄²

—

158

C₄²

D₃×AGL(1,𝔽₅)

yes

characteristic

159

C₃×C₅

#1746

yes

characteristic, complemented, Hall

160

C₂²×C₃

—

161

C₂²×C₃

—

162

C₂²×C₃

C₂×C₄×AGL(1,𝔽₅)

yes

characteristic

163

C₃×C₄

—

164

C₃×C₄

—

165

C₃×C₄

—

166

C₃×C₄

#4130

yes

—

167

Dic₃

—

168

C₂×C₅

#4932

yes

characteristic

169

C₂×C₅

#5122

yes

—

170

D₅

#4709

yes

—

171

C₂³

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

Dic₃×AGL(1,𝔽₅)

yes

—

179

C₂×C₄

C₂×D₃×AGL(1,𝔽₅)

yes

characteristic

180

C₈

—

181

C₈

—

182

C₈

—

183

C₂×C₃

—

184

C₂×C₃

#65686

yes

characteristic

185

C₂×C₃

#65718

yes

—

186

C₅

#69806

yes

characteristic, complemented, Sylow

187

C₂²

—

188

C₂²

—

189

C₂²

C₂×Dic₃×AGL(1,𝔽₅)

yes

characteristic

190

C₄

—

191

C₄

—

192

C₄

—

193

C₄

—

194

C₄

#91909

yes

—

195

C₃

#10603840

yes

characteristic, complemented, Sylow

196

C₂

—

197

C₂

#11751696

yes

—

198

C₂

#11751748

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂×D₃×AGL(1,𝔽₅)
commutator, socle	C₂²×C₃×C₅	C₂×C₄²
Fitting	C₃×C₄²×C₅	C₂×C₄
prime radical	C₂²×C₃×D₅	C₂²×C₄
cosocle	C₂×C₃×C₄×D₅	C₂³
hypercenter	C₄²	D₃×AGL(1,𝔽₅)
hypocenter	C₃×C₅	#1746
LCS	G ▷ C₂²×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₄²

subgroup

G/H

center, Frattini

C₂×C₄

C₂×D₃×AGL(1,𝔽₅)

commutator, socle

C₂²×C₃×C₅

C₂×C₄²

Fitting

C₃×C₄²×C₅

C₂×C₄

prime radical

C₂²×C₃×D₅

C₂²×C₄

cosocle

C₂×C₃×C₄×D₅

C₂³

C₄²

C₃×C₅

G ▷ C₂²×C₃×C₅ ▷ C₃×C₅

UCS

C₁ ◁ C₂×C₄ ◁ C₄²

N	H ≅ G/N	type
11. C₃×#4002	C₄	semidirect
12. C₄×C₃⋊#168	C₄	semidirect
13. C₄×C₅×C₃⋊C₈	C₄	semidirect
14. Dic₅×C₃⋊C₈	C₄	semidirect
41. #4002	Dic₃	semidirect
66. C₄×C₃⋊C₈	AGL(1,𝔽₅)	semidirect
159. C₃×C₅	#1746	semidirect
186. C₅	#69806	semidirect
195. C₃	#10603840	semidirect

H ≅ G/N

type

11. C₃×#4002

C₄

semidirect

12. C₄×C₃⋊#168

C₄

semidirect

13. C₄×C₅×C₃⋊C₈

C₄

semidirect

14. Dic₅×C₃⋊C₈

C₄

semidirect

41. #4002

Dic₃

semidirect

66. C₄×C₃⋊C₈

AGL(1,𝔽₅)

semidirect

159. C₃×C₅

#1746

semidirect

186. C₅

#69806

semidirect

195. C₃

#10603840

semidirect

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

subgroup	index
#11752345	2
C₃×#65676	2
C₄×D₅×C₃⋊C₈	2
#11746517	2
#10603840	3
#69806	5

subgroup

index

C₄×D₅×C₃⋊C₈

Subgroups — #14294955

Conjugacy classes of subgroups

Complements

Maximal subgroups