Group #14298405 — subgroups

Subgroups — #14298405

Order 1920.

544 conjugacy classes of subgroups; only the 175 normal subgroups are shown.

normal subgroups only

This group has 544 conjugacy classes of subgroups — too many to draw. Only its 175 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	C₂×C₃×C₅×#117	C₂	1	1	yes	maximal, characteristic
2	#11751308	C₂	1	2	yes	maximal
3	#11746351	C₂	1	4	yes	maximal, complemented
4	#10602382	—	3	1	no	maximal, Hall
5	C₂³×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
6	C₃×C₅×#117	—	2	4	no	—
7	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
8	C₂×C₃×C₅×C₂²⋊C₄	C₄	1	2	yes	—
9	C₂×C₃×C₄²×C₅	C₄	1	2	yes	—
10	C₂×#6710	C₂²	1	4	yes	—
11	#91114	—	2	4	no	—
12	#71589	—	5	1	no	maximal, Hall
13	C₂×C₅×#117	D₃	1	1	yes	characteristic
14	#65496	—	3	2	no	—
15	#64750	—	3	4	no	—
16	C₂⁴×C₃×C₅	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₃×C₄×C₅	—	2	4	no	—
18	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₃×C₄×C₅	C₂×C₄	1	4	yes	—
20	C₂²×C₃×C₄×C₅	D₄	1	4	yes	—
21	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
22	C₃×C₅×C₄⋊C₄	—	2	4	no	—
23	C₃×C₅×C₂²⋊C₄	—	2	4	no	—
24	C₃×C₄²×C₅	—	2	4	no	—
25	#6710	—	2	8	no	—
26	C₂×C₃⋊#168	—	2	4	no	—
27	C₂×C₃×#117	D₅	1	1	yes	characteristic
28	#5207	—	5	2	no	—
29	#4530	—	5	4	no	—
30	C₂³×C₄×C₅	C₂×D₃	1	1	yes	characteristic
31	C₅×#117	—	2	4	no	—
32	C₂×C₅×C₄⋊C₄	C₂×D₃	1	2	yes	—
33	C₂×C₅×C₂²⋊C₄	Dic₃	1	2	yes	—
34	C₂×C₄²×C₅	Dic₃	1	2	yes	—
35	C₂×#681	—	3	4	no	—
36	C₂²⋊#168	—	6	4	no	—
37	#1822	—	15	1	no	Sylow
38	C₂³×C₃×C₅	—	2	2	no	—
39	C₂³×C₃×C₅	—	4	2	no	—
40	C₂³×C₃×C₅	C₂²⋊C₄	1	2	yes	—
41	C₂³×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₃×C₄×C₅	—	2	2	no	—
43	C₂×C₃×C₄×C₅	—	2	8	no	—
44	C₂×C₃×C₄×C₅	—	2	4	no	—
45	C₂×C₃×C₄×C₅	—	4	4	no	—
46	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
47	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
48	C₂×C₃×C₄×C₅	#41	1	4	yes	—
49	C₃⋊#168	—	4	4	no	—
50	C₂³×C₃×C₄	C₂×D₅	1	1	yes	characteristic
51	C₃×#117	—	2	4	no	—
52	C₂×C₃×C₄⋊C₄	C₂×D₅	1	2	yes	—
53	C₂×C₃×C₂²⋊C₄	Dic₅	1	2	yes	—
54	C₂×C₃×C₄²	Dic₅	1	2	yes	—
55	C₂×#208	—	5	4	no	—
56	#830	—	10	4	no	—
57	C₂⁴×C₅	C₂×Dic₃	1	1	yes	characteristic
58	C₂²×C₄×C₅	—	2	4	no	—
59	C₂²×C₄×C₅	C₂×Dic₃	1	1	yes	characteristic
60	C₂²×C₄×C₅	C₂×Dic₃	1	4	yes	—
61	C₂²×C₄×C₅	#67	1	4	yes	—
62	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
63	C₅×C₄⋊C₄	—	2	4	no	—
64	C₅×C₂²⋊C₄	—	2	4	no	—
65	C₄²×C₅	—	2	4	no	—
66	#681	—	6	8	no	—
67	C₂×#168	—	6	4	no	—
68	C₂×#117	D₁₅	1	1	yes	characteristic
69	#407	—	15	2	no	—
70	#343	—	15	4	no	—
71	C₂²×C₃×C₅	—	2	2	no	—
72	C₂²×C₃×C₅	—	2	2	no	—
73	C₂²×C₃×C₅	—	2	4	no	—
74	C₂²×C₃×C₅	—	4	4	no	—
75	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
76	C₂²×C₃×C₅	#117	1	2	yes	—
77	C₃×C₄×C₅	—	2	2	no	—
78	C₃×C₄×C₅	—	4	4	no	—
79	C₃×C₄×C₅	#123	1	4	yes	—
80	C₂⁴×C₃	C₂×Dic₅	1	1	yes	characteristic
81	C₂²×C₃×C₄	—	2	4	no	—
82	C₂²×C₃×C₄	C₂×Dic₅	1	1	yes	characteristic
83	C₂²×C₃×C₄	C₂×Dic₅	1	4	yes	—
84	C₂²×C₃×C₄	C₂²⋊D₅	1	4	yes	—
85	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
86	C₃×C₄⋊C₄	—	2	4	no	—
87	C₃×C₂²⋊C₄	—	2	4	no	—
88	C₃×C₄²	—	2	4	no	—
89	#208	—	10	8	no	—
90	C₂×C₃⋊C₈	—	10	4	no	—
91	C₂³×C₅	—	2	2	no	—
92	C₂³×C₅	—	4	2	no	—
93	C₂³×C₅	#217	1	2	yes	—
94	C₂³×C₅	C₂²×Dic₃	1	1	yes	characteristic
95	C₂×C₄×C₅	—	2	2	no	—
96	C₂×C₄×C₅	—	2	8	no	—
97	C₂×C₄×C₅	—	2	4	no	—
98	C₂×C₄×C₅	—	4	4	no	—
99	C₂×C₄×C₅	#217	1	2	yes	—
100	C₂×C₄×C₅	#235	1	4	yes	—
101	C₂×C₄×C₅	C₂×#67	1	2	yes	—
102	#168	—	12	4	no	—
103	C₂³×C₄	C₂×D₁₅	1	1	yes	characteristic
104	C₂×M₁₆	—	15	4	no	—
105	#117	—	2	4	no	—
106	C₂×C₄⋊C₄	C₂×D₁₅	1	2	yes	—
107	C₂×C₂²⋊C₄	Dic₁₅	1	2	yes	—
108	C₂×C₄²	Dic₁₅	1	2	yes	—
109	C₂²⋊C₈	—	30	4	no	—
110	C₂×C₃×C₅	—	2	2	no	—
111	C₂×C₃×C₅	—	4	2	no	—
112	C₂×C₃×C₅	#381	1	1	yes	characteristic
113	C₂×C₃×C₅	#413	1	2	yes	—
114	C₂³×C₃	—	2	2	no	—
115	C₂³×C₃	—	4	2	no	—
116	C₂³×C₃	C₂²⋊Dic₅	1	2	yes	—
117	C₂³×C₃	C₂²×Dic₅	1	1	yes	characteristic
118	C₂×C₃×C₄	—	2	2	no	—
119	C₂×C₃×C₄	—	2	8	no	—
120	C₂×C₃×C₄	—	2	4	no	—
121	C₂×C₃×C₄	—	4	4	no	—
122	C₂×C₃×C₄	C₂²⋊Dic₅	1	2	yes	—
123	C₂×C₃×C₄	#709	1	4	yes	—
124	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	2	yes	—
125	C₃⋊C₈	—	20	4	no	—
126	C₂²×C₅	—	2	2	no	—
127	C₂²×C₅	—	2	2	no	—
128	C₂²×C₅	—	2	4	no	—
129	C₂²×C₅	—	4	4	no	—
130	C₂²×C₅	#926	1	2	yes	—
131	C₂²×C₅	C₂×#217	1	1	yes	characteristic
132	C₄×C₅	—	2	2	no	—
133	C₄×C₅	—	4	4	no	—
134	C₄×C₅	#929	1	4	yes	—
135	C₂⁴	C₂×Dic₁₅	1	1	yes	characteristic
136	C₂²×C₄	—	2	4	no	—
137	C₂²×C₄	C₂×Dic₁₅	1	1	yes	characteristic
138	C₂²×C₄	C₂×Dic₁₅	1	4	yes	—
139	C₂²×C₄	#1220	1	4	yes	—
140	C₂²×C₄	C₂²×D₁₅	1	1	yes	characteristic
141	M₁₆	—	30	8	no	—
142	C₂×C₈	—	30	4	no	—
143	C₄⋊C₄	—	2	4	no	—
144	C₂²⋊C₄	—	2	4	no	—
145	C₄²	—	2	4	no	—
146	C₃×C₅	#1822	1	1	yes	characteristic, complemented, Hall
147	C₂²×C₃	—	2	2	no	—
148	C₂²×C₃	—	2	2	no	—
149	C₂²×C₃	—	2	4	no	—
150	C₂²×C₃	—	4	4	no	—
151	C₂²×C₃	#4072	1	2	yes	—
152	C₂²×C₃	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
153	C₃×C₄	—	2	2	no	—
154	C₃×C₄	—	4	4	no	—
155	C₃×C₄	#4075	1	4	yes	—
156	C₂×C₅	—	2	2	no	—
157	C₂×C₅	—	4	2	no	—
158	C₂×C₅	#5211	1	1	yes	characteristic
159	C₂×C₅	#5237	1	2	yes	—
160	C₂³	—	2	2	no	—
161	C₂³	—	4	2	no	—
162	C₂³	#6719	1	2	yes	—
163	C₂³	C₂²×Dic₁₅	1	1	yes	characteristic
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	2	8	no	—
166	C₂×C₄	—	2	4	no	—
167	C₂×C₄	—	4	4	no	—
168	C₂×C₄	#6719	1	2	yes	—
169	C₂×C₄	#6817	1	4	yes	—
170	C₂×C₄	C₂×#1220	1	2	yes	—
171	C₈	—	60	4	no	—
172	C₂×C₃	—	2	2	no	—
173	C₂×C₃	—	4	2	no	—
174	C₂×C₃	#65500	1	1	yes	characteristic
175	C₂×C₃	#65526	1	2	yes	—
176	C₅	#71589	1	1	yes	characteristic, complemented, Sylow
177	C₂²	—	2	2	no	—
178	C₂²	—	2	2	no	—
179	C₂²	—	2	4	no	—
180	C₂²	—	4	4	no	—
181	C₂²	#91815	1	2	yes	—
182	C₂²	C₂×#6719	1	1	yes	characteristic
183	C₄	—	2	2	no	—
184	C₄	—	4	4	no	—
185	C₄	#91818	1	4	yes	—
186	C₃	#10602382	1	1	yes	characteristic, complemented, Sylow
187	C₂	—	2	2	no	—
188	C₂	—	4	2	no	—
189	C₂	#11751312	1	1	yes	characteristic
190	C₂	#11751338	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₃×C₅×#117

C₂

yes

maximal, characteristic

#11751308

C₂

yes

maximal

#11746351

C₂

yes

maximal, complemented

#10602382

—

maximal, Hall

C₂³×C₃×C₄×C₅

C₂²

yes

characteristic

C₃×C₅×#117

—

C₂×C₃×C₅×C₄⋊C₄

C₂²

yes

—

C₂×C₃×C₅×C₂²⋊C₄

C₄

yes

—

C₂×C₃×C₄²×C₅

C₄

yes

—

C₂×#6710

C₂²

yes

—

#91114

—

#71589

—

maximal, Hall

C₂×C₅×#117

D₃

yes

characteristic

#65496

—

#64750

—

C₂⁴×C₃×C₅

C₂×C₄

yes

characteristic

C₂²×C₃×C₄×C₅

—

C₂²×C₃×C₄×C₅

C₂×C₄

yes

characteristic

C₂²×C₃×C₄×C₅

C₂×C₄

yes

—

C₂²×C₃×C₄×C₅

D₄

yes

—

C₂²×C₃×C₄×C₅

C₂³

yes

characteristic

C₃×C₅×C₄⋊C₄

—

C₃×C₅×C₂²⋊C₄

—

C₃×C₄²×C₅

—

#6710

—

C₂×C₃⋊#168

—

C₂×C₃×#117

D₅

yes

characteristic

#5207

—

#4530

—

C₂³×C₄×C₅

C₂×D₃

yes

characteristic

C₅×#117

—

C₂×C₅×C₄⋊C₄

C₂×D₃

yes

—

C₂×C₅×C₂²⋊C₄

Dic₃

yes

—

C₂×C₄²×C₅

Dic₃

yes

—

C₂×#681

—

C₂²⋊#168

—

#1822

—

Sylow

C₂³×C₃×C₅

—

C₂³×C₃×C₅

—

C₂³×C₃×C₅

C₂²⋊C₄

yes

—

C₂³×C₃×C₅

C₂²×C₄

yes

characteristic

C₂×C₃×C₄×C₅

—

C₂×C₃×C₄×C₅

—

C₂×C₃×C₄×C₅

—

C₂×C₃×C₄×C₅

—

C₂×C₃×C₄×C₅

C₂²⋊C₄

yes

—

C₂×C₃×C₄×C₅

C₂×D₄

yes

—

C₂×C₃×C₄×C₅

#41

yes

—

C₃⋊#168

—

C₂³×C₃×C₄

C₂×D₅

yes

characteristic

C₃×#117

—

C₂×C₃×C₄⋊C₄

C₂×D₅

yes

—

C₂×C₃×C₂²⋊C₄

Dic₅

yes

—

C₂×C₃×C₄²

Dic₅

yes

—

C₂×#208

—

#830

—

C₂⁴×C₅

C₂×Dic₃

yes

characteristic

C₂²×C₄×C₅

—

C₂²×C₄×C₅

C₂×Dic₃

yes

characteristic

C₂²×C₄×C₅

C₂×Dic₃

yes

—

C₂²×C₄×C₅

#67

yes

—

C₂²×C₄×C₅

C₂²×D₃

yes

characteristic

C₅×C₄⋊C₄

—

C₅×C₂²⋊C₄

—

C₄²×C₅

—

#681

—

C₂×#168

—

C₂×#117

D₁₅

yes

characteristic

#407

—

#343

—

C₂²×C₃×C₅

—

C₂²×C₃×C₅

—

C₂²×C₃×C₅

—

C₂²×C₃×C₅

—

C₂²×C₃×C₅

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₃×C₅

#117

yes

—

C₃×C₄×C₅

—

C₃×C₄×C₅

—

C₃×C₄×C₅

#123

yes

—

C₂⁴×C₃

C₂×Dic₅

yes

characteristic

C₂²×C₃×C₄

—

C₂²×C₃×C₄

C₂×Dic₅

yes

characteristic

C₂²×C₃×C₄

C₂×Dic₅

yes

—

C₂²×C₃×C₄

C₂²⋊D₅

yes

—

C₂²×C₃×C₄

C₂²×D₅

yes

characteristic

C₃×C₄⋊C₄

—

C₃×C₂²⋊C₄

—

C₃×C₄²

—

#208

—

C₂×C₃⋊C₈

—

C₂³×C₅

—

C₂³×C₅

—

C₂³×C₅

#217

yes

—

C₂³×C₅

C₂²×Dic₃

yes

characteristic

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

#217

yes

—

100

C₂×C₄×C₅

#235

yes

—

101

C₂×C₄×C₅

C₂×#67

yes

—

102

#168

—

103

C₂³×C₄

C₂×D₁₅

yes

characteristic

104

C₂×M₁₆

—

105

#117

—

106

C₂×C₄⋊C₄

C₂×D₁₅

yes

—

107

C₂×C₂²⋊C₄

Dic₁₅

yes

—

108

C₂×C₄²

Dic₁₅

yes

—

109

C₂²⋊C₈

—

110

C₂×C₃×C₅

—

111

C₂×C₃×C₅

—

112

C₂×C₃×C₅

#381

yes

characteristic

113

C₂×C₃×C₅

#413

yes

—

114

C₂³×C₃

—

115

C₂³×C₃

—

116

C₂³×C₃

C₂²⋊Dic₅

yes

—

117

C₂³×C₃

C₂²×Dic₅

yes

characteristic

118

C₂×C₃×C₄

—

119

C₂×C₃×C₄

—

120

C₂×C₃×C₄

—

121

C₂×C₃×C₄

—

122

C₂×C₃×C₄

C₂²⋊Dic₅

yes

—

123

C₂×C₃×C₄

#709

yes

—

124

C₂×C₃×C₄

C₂×C₂²⋊D₅

yes

—

125

C₃⋊C₈

—

126

C₂²×C₅

—

127

C₂²×C₅

—

128

C₂²×C₅

—

129

C₂²×C₅

—

130

C₂²×C₅

#926

yes

—

131

C₂²×C₅

C₂×#217

yes

characteristic

132

C₄×C₅

—

133

C₄×C₅

—

134

C₄×C₅

#929

yes

—

135

C₂⁴

C₂×Dic₁₅

yes

characteristic

136

C₂²×C₄

—

137

C₂²×C₄

C₂×Dic₁₅

yes

characteristic

138

C₂²×C₄

C₂×Dic₁₅

yes

—

139

C₂²×C₄

#1220

yes

—

140

C₂²×C₄

C₂²×D₁₅

yes

characteristic

141

M₁₆

—

142

C₂×C₈

—

143

C₄⋊C₄

—

144

C₂²⋊C₄

—

145

C₄²

—

146

C₃×C₅

#1822

yes

characteristic, complemented, Hall

147

C₂²×C₃

—

148

C₂²×C₃

—

149

C₂²×C₃

—

150

C₂²×C₃

—

151

C₂²×C₃

#4072

yes

—

152

C₂²×C₃

C₂×C₂²⋊Dic₅

yes

characteristic

153

C₃×C₄

—

154

C₃×C₄

—

155

C₃×C₄

#4075

yes

—

156

C₂×C₅

—

157

C₂×C₅

—

158

C₂×C₅

#5211

yes

characteristic

159

C₂×C₅

#5237

yes

—

160

C₂³

—

161

C₂³

—

162

C₂³

#6719

yes

—

163

C₂³

C₂²×Dic₁₅

yes

characteristic

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

#6719

yes

—

169

C₂×C₄

#6817

yes

—

170

C₂×C₄

C₂×#1220

yes

—

171

C₈

—

172

C₂×C₃

—

173

C₂×C₃

—

174

C₂×C₃

#65500

yes

characteristic

175

C₂×C₃

#65526

yes

—

176

C₅

#71589

yes

characteristic, complemented, Sylow

177

C₂²

—

178

C₂²

—

179

C₂²

—

180

C₂²

—

181

C₂²

#91815

yes

—

182

C₂²

C₂×#6719

yes

characteristic

183

C₄

—

184

C₄

—

185

C₄

#91818

yes

—

186

C₃

#10602382

yes

characteristic, complemented, Sylow

187

C₂

—

188

C₂

—

189

C₂

#11751312

yes

characteristic

190

C₂

#11751338

yes

—

	subgroup	G/H
center	C₂²	C₂×#6719
commutator	C₂³×C₃×C₅	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₅
Fitting	C₂×C₃×C₅×#117	C₂
socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₃×C₅	C₂×C₄
cosocle	C₂²×C₃×C₄×C₅	C₂³
hypercenter	C₂×#117	D₁₅
hypocenter	C₃×C₅	#1822
LCS	G ▷ C₂³×C₃×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#117

subgroup

G/H

center

C₂²

C₂×#6719

commutator

C₂³×C₃×C₅

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₅

Fitting

C₂×C₃×C₅×#117

C₂

socle

C₂²×C₃×C₅

C₂×C₂²⋊C₄

prime radical

C₂⁴×C₃×C₅

C₂×C₄

cosocle

C₂²×C₃×C₄×C₅

C₂³

D₁₅

C₃×C₅

G ▷ C₂³×C₃×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅

UCS

C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#117

N	H ≅ G/N	type
3. #11746351	C₂	semidirect
146. C₃×C₅	#1822	semidirect
176. C₅	#71589	semidirect
186. C₃	#10602382	semidirect

H ≅ G/N

type

3. #11746351

C₂

semidirect

146. C₃×C₅

#1822

semidirect

176. C₅

#71589

semidirect

186. C₃

#10602382

semidirect

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

subgroup	index
C₂×C₃×C₅×#117	2
#11751308	2
#11746351	2
#10602382	3
#71589	5

subgroup

index

C₂×C₃×C₅×#117

Subgroups — #14298405

Conjugacy classes of subgroups

Complements

Maximal subgroups