Group #14284012 — subgroups

Subgroups — #14284012

Order 1920.

452 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 452 conjugacy classes of subgroups — too many to draw. Only its 159 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	C₃×C₅×#533	C₂	1	1	yes	maximal, characteristic
2	C₄×#6710	C₂	1	2	yes	maximal
3	#11746364	C₂	1	4	yes	maximal, complemented
4	#10602614	—	3	1	no	maximal, Hall
5	C₃×C₅×C₄⋊Q₈	C₄	1	2	yes	—
6	C₃×C₅×#125	C₂²	1	2	yes	—
7	C₃×C₅×C₂²⋊Q₈	—	2	2	no	—
8	C₃×C₄×C₅×Q₈	—	2	2	no	—
9	C₃×C₅×#117	C₄	1	2	yes	—
10	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
11	C₂×#6710	—	2	2	no	—
12	#91087	C₂²	1	4	yes	—
13	C₄×C₃⋊#168	—	2	2	no	—
14	#71821	—	5	1	no	maximal, Hall
15	C₅×#533	D₃	1	1	yes	characteristic
16	C₄×#681	—	3	2	no	—
17	#64763	—	3	4	no	—
18	C₂×C₃×C₅×Q₈	—	2	2	no	—
19	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
21	C₃×C₅×C₄⋊C₄	—	2	2	no	—
22	C₃×C₅×C₄⋊C₄	—	2	4	no	—
23	C₃×C₅×C₄⋊C₄	C₂×C₄	1	4	yes	—
24	C₃×C₅×C₂²⋊C₄	—	2	2	no	—
25	C₃×C₄²×C₅	—	2	2	no	—
26	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
27	C₃×C₄²×C₅	D₄	1	2	yes	—
28	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
29	#6710	—	2	8	no	—
30	C₂×C₃⋊#168	—	2	4	no	—
31	C₃×#533	D₅	1	1	yes	characteristic
32	C₄×#208	—	5	2	no	—
33	#4543	—	5	4	no	—
34	C₅×C₄⋊Q₈	Dic₃	1	2	yes	—
35	C₅×#125	C₂×D₃	1	2	yes	—
36	C₅×C₂²⋊Q₈	—	2	2	no	—
37	C₄×C₅×Q₈	—	2	2	no	—
38	C₅×#117	Dic₃	1	2	yes	—
39	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
40	C₂×#681	—	6	2	no	—
41	#3935	—	3	4	no	—
42	C₄×#168	—	6	2	no	—
43	#1526	—	15	1	no	Sylow
44	C₂³×C₃×C₅	C₂²⋊C₄	1	1	yes	characteristic
45	C₃×C₅×Q₈	—	4	4	no	—
46	C₂×C₃×C₄×C₅	—	2	2	no	—
47	C₂×C₃×C₄×C₅	—	2	4	no	—
48	C₂×C₃×C₄×C₅	—	2	4	no	—
49	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
50	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
51	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
53	C₃⋊#168	—	2	8	no	—
54	C₃×C₄⋊Q₈	Dic₅	1	2	yes	—
55	C₃×#125	C₂×D₅	1	2	yes	—
56	C₃×C₂²⋊Q₈	—	2	2	no	—
57	C₃×C₄×Q₈	—	2	2	no	—
58	C₃×#117	Dic₅	1	2	yes	—
59	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
60	C₂×#208	—	10	2	no	—
61	#803	—	5	4	no	—
62	C₄×C₃⋊C₈	—	10	2	no	—
63	C₂×C₅×Q₈	—	2	2	no	—
64	C₂²×C₄×C₅	C₂×Dic₃	1	1	yes	characteristic
65	C₂²×C₄×C₅	#67	1	2	yes	—
66	C₅×C₄⋊C₄	—	2	2	no	—
67	C₅×C₄⋊C₄	—	2	4	no	—
68	C₅×C₄⋊C₄	C₂×Dic₃	1	4	yes	—
69	C₅×C₂²⋊C₄	—	2	2	no	—
70	C₄²×C₅	—	2	2	no	—
71	C₄²×C₅	C₂×Dic₃	1	1	yes	characteristic
72	C₄²×C₅	#67	1	2	yes	—
73	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
74	#681	—	6	8	no	—
75	C₂×#168	—	6	4	no	—
76	#533	D₁₅	1	1	yes	characteristic
77	C₄×M₁₆	—	15	2	no	—
78	#330	—	15	4	no	—
79	C₂²×C₃×C₅	—	2	1	no	—
80	C₂²×C₃×C₅	—	2	2	no	—
81	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
82	C₃×C₄×C₅	—	2	1	no	—
83	C₃×C₄×C₅	—	2	2	no	—
84	C₃×C₄×C₅	—	4	4	no	—
85	C₃×C₄×C₅	#101	1	2	yes	—
86	C₃×C₄×C₅	#104	1	4	yes	—
87	C₂×C₃×Q₈	—	2	2	no	—
88	C₂²×C₃×C₄	C₂×Dic₅	1	1	yes	characteristic
89	C₂²×C₃×C₄	C₂²⋊D₅	1	2	yes	—
90	C₃×C₄⋊C₄	—	2	2	no	—
91	C₃×C₄⋊C₄	—	2	4	no	—
92	C₃×C₄⋊C₄	C₂×Dic₅	1	4	yes	—
93	C₃×C₂²⋊C₄	—	2	2	no	—
94	C₃×C₄²	—	2	2	no	—
95	C₃×C₄²	C₂×Dic₅	1	1	yes	characteristic
96	C₃×C₄²	C₂²⋊D₅	1	2	yes	—
97	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
98	#208	—	10	8	no	—
99	C₂×C₃⋊C₈	—	10	4	no	—
100	C₂³×C₅	#217	1	1	yes	characteristic
101	C₅×Q₈	—	4	4	no	—
102	C₂×C₄×C₅	—	2	2	no	—
103	C₂×C₄×C₅	—	2	4	no	—
104	C₂×C₄×C₅	—	2	4	no	—
105	C₂×C₄×C₅	#217	1	2	yes	—
106	C₂×C₄×C₅	#217	1	1	yes	characteristic
107	C₂×C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
108	C₂×C₄×C₅	C₂×#67	1	2	yes	—
109	#168	—	6	8	no	—
110	C₂×M₁₆	—	30	2	no	—
111	C₄⋊Q₈	Dic₁₅	1	2	yes	—
112	#125	C₂×D₁₅	1	2	yes	—
113	C₂²⋊Q₈	—	2	2	no	—
114	C₄×Q₈	—	2	2	no	—
115	#117	Dic₁₅	1	2	yes	—
116	C₂×C₄²	C₂×D₁₅	1	1	yes	characteristic
117	#97	—	15	4	no	—
118	C₄×C₈	—	30	2	no	—
119	C₂×C₃×C₅	—	4	1	no	—
120	C₂×C₃×C₅	C₂×#101	1	1	yes	characteristic
121	C₂×C₃×C₅	C₂×#104	1	2	yes	—
122	C₂³×C₃	C₂²⋊Dic₅	1	1	yes	characteristic
123	C₃×Q₈	—	4	4	no	—
124	C₂×C₃×C₄	—	2	2	no	—
125	C₂×C₃×C₄	—	2	4	no	—
126	C₂×C₃×C₄	—	2	4	no	—
127	C₂×C₃×C₄	C₂²⋊Dic₅	1	2	yes	—
128	C₂×C₃×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
129	C₂×C₃×C₄	C₂²×Dic₅	1	1	yes	characteristic
130	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	2	yes	—
131	C₃⋊C₈	—	10	8	no	—
132	C₂²×C₅	—	2	1	no	—
133	C₂²×C₅	—	2	2	no	—
134	C₂²×C₅	C₂×#217	1	1	yes	characteristic
135	C₄×C₅	—	2	1	no	—
136	C₄×C₅	—	2	2	no	—
137	C₄×C₅	—	4	4	no	—
138	C₄×C₅	#836	1	2	yes	—
139	C₄×C₅	#837	1	4	yes	—
140	C₂×Q₈	—	2	2	no	—
141	C₂²×C₄	C₂×Dic₁₅	1	1	yes	characteristic
142	C₂²×C₄	#1220	1	2	yes	—
143	M₁₆	—	30	8	no	—
144	C₂×C₈	—	30	4	no	—
145	C₄⋊C₄	—	2	2	no	—
146	C₄⋊C₄	—	2	4	no	—
147	C₄⋊C₄	C₂×Dic₁₅	1	4	yes	—
148	C₂²⋊C₄	—	2	2	no	—
149	C₄²	—	2	2	no	—
150	C₄²	C₂×Dic₁₅	1	1	yes	characteristic
151	C₄²	#1220	1	2	yes	—
152	C₄²	C₂²×D₁₅	1	1	yes	characteristic
153	C₃×C₅	#1526	1	1	yes	characteristic, complemented, Hall
154	C₂²×C₃	—	2	1	no	—
155	C₂²×C₃	—	2	2	no	—
156	C₂²×C₃	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
157	C₃×C₄	—	2	1	no	—
158	C₃×C₄	—	2	2	no	—
159	C₃×C₄	—	4	4	no	—
160	C₃×C₄	#3968	1	2	yes	—
161	C₃×C₄	#3969	1	4	yes	—
162	C₂×C₅	—	4	1	no	—
163	C₂×C₅	C₂×#836	1	1	yes	characteristic
164	C₂×C₅	C₂×#837	1	2	yes	—
165	C₂³	#6719	1	1	yes	characteristic
166	Q₈	—	4	4	no	—
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	2	4	no	—
169	C₂×C₄	—	2	4	no	—
170	C₂×C₄	#6719	1	2	yes	—
171	C₂×C₄	#6719	1	1	yes	characteristic
172	C₂×C₄	C₂²×Dic₁₅	1	1	yes	characteristic
173	C₂×C₄	C₂×#1220	1	2	yes	—
174	C₈	—	30	8	no	—
175	C₂×C₃	—	4	1	no	—
176	C₂×C₃	C₂×#3968	1	1	yes	characteristic
177	C₂×C₃	C₂×#3969	1	2	yes	—
178	C₅	#71821	1	1	yes	characteristic, complemented, Sylow
179	C₂²	—	2	1	no	—
180	C₂²	—	2	2	no	—
181	C₂²	C₂×#6719	1	1	yes	characteristic
182	C₄	—	2	1	no	—
183	C₄	—	2	2	no	—
184	C₄	—	4	4	no	—
185	C₄	#91120	1	2	yes	—
186	C₄	#91121	1	4	yes	—
187	C₃	#10602614	1	1	yes	characteristic, complemented, Sylow
188	C₂	—	4	1	no	—
189	C₂	C₂×#91120	1	1	yes	characteristic
190	C₂	C₂×#91121	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×C₅×#533

C₂

yes

maximal, characteristic

C₄×#6710

C₂

yes

maximal

#11746364

C₂

yes

maximal, complemented

#10602614

—

maximal, Hall

C₃×C₅×C₄⋊Q₈

C₄

yes

—

C₃×C₅×#125

C₂²

yes

—

C₃×C₅×C₂²⋊Q₈

—

C₃×C₄×C₅×Q₈

—

C₃×C₅×#117

C₄

yes

—

C₂×C₃×C₄²×C₅

C₂²

yes

characteristic

C₂×#6710

—

#91087

C₂²

yes

—

C₄×C₃⋊#168

—

#71821

—

maximal, Hall

C₅×#533

D₃

yes

characteristic

C₄×#681

—

#64763

—

C₂×C₃×C₅×Q₈

—

C₂²×C₃×C₄×C₅

C₂×C₄

yes

characteristic

C₂²×C₃×C₄×C₅

D₄

yes

—

C₃×C₅×C₄⋊C₄

—

C₃×C₅×C₄⋊C₄

—

C₃×C₅×C₄⋊C₄

C₂×C₄

yes

—

C₃×C₅×C₂²⋊C₄

—

C₃×C₄²×C₅

—

C₃×C₄²×C₅

C₂×C₄

yes

characteristic

C₃×C₄²×C₅

D₄

yes

—

C₃×C₄²×C₅

C₂³

yes

characteristic

#6710

—

C₂×C₃⋊#168

—

C₃×#533

D₅

yes

characteristic

C₄×#208

—

#4543

—

C₅×C₄⋊Q₈

Dic₃

yes

—

C₅×#125

C₂×D₃

yes

—

C₅×C₂²⋊Q₈

—

C₄×C₅×Q₈

—

C₅×#117

Dic₃

yes

—

C₂×C₄²×C₅

C₂×D₃

yes

—

—

—

—

C₂³×C₃×C₅

C₂²⋊C₄

yes

characteristic

C₃×C₅×Q₈

—

C₂×C₃×C₄×C₅

—

C₂×C₃×C₄×C₅

—

C₂×C₃×C₄×C₅

—

C₂×C₃×C₄×C₅

C₂²⋊C₄

yes

characteristic

C₂×C₃×C₄×C₅

C₂²⋊C₄

yes

—

C₂×C₃×C₄×C₅

C₂²×C₄

yes

characteristic

C₂×C₃×C₄×C₅

C₂×D₄

yes

—

C₃⋊#168

—

C₃×C₄⋊Q₈

Dic₅

yes

—

C₃×#125

C₂×D₅

yes

—

C₃×C₂²⋊Q₈

—

C₃×C₄×Q₈

—

C₃×#117

Dic₅

yes

—

C₂×C₃×C₄²

C₂×D₅

yes

characteristic

C₂×#208

—

#803

—

C₄×C₃⋊C₈

—

C₂×C₅×Q₈

—

C₂²×C₄×C₅

C₂×Dic₃

yes

characteristic

C₂²×C₄×C₅

#67

yes

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

C₂×Dic₃

yes

—

C₅×C₂²⋊C₄

—

C₄²×C₅

—

C₄²×C₅

C₂×Dic₃

yes

characteristic

C₄²×C₅

#67

yes

—

C₄²×C₅

C₂²×D₃

yes

—

—

D₁₅

yes

characteristic

C₄×M₁₆

—

#330

—

C₂²×C₃×C₅

—

C₂²×C₃×C₅

—

C₂²×C₃×C₅

C₂×C₂²⋊C₄

yes

characteristic

C₃×C₄×C₅

—

C₃×C₄×C₅

—

C₃×C₄×C₅

—

C₃×C₄×C₅

#101

yes

—

C₃×C₄×C₅

#104

yes

—

C₂×C₃×Q₈

—

C₂²×C₃×C₄

C₂×Dic₅

yes

characteristic

C₂²×C₃×C₄

C₂²⋊D₅

yes

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

C₂×Dic₅

yes

—

C₃×C₂²⋊C₄

—

C₃×C₄²

—

C₃×C₄²

C₂×Dic₅

yes

characteristic

C₃×C₄²

C₂²⋊D₅

yes

—

C₃×C₄²

C₂²×D₅

yes

characteristic

#208

—

C₂×C₃⋊C₈

—

100

C₂³×C₅

#217

yes

characteristic

101

C₅×Q₈

—

102

C₂×C₄×C₅

—

103

C₂×C₄×C₅

—

104

C₂×C₄×C₅

—

105

C₂×C₄×C₅

#217

yes

—

106

C₂×C₄×C₅

#217

yes

characteristic

107

C₂×C₄×C₅

C₂²×Dic₃

yes

characteristic

108

C₂×C₄×C₅

C₂×#67

yes

—

109

#168

—

110

C₂×M₁₆

—

111

C₄⋊Q₈

Dic₁₅

yes

—

112

#125

C₂×D₁₅

yes

—

113

C₂²⋊Q₈

—

114

C₄×Q₈

—

115

#117

Dic₁₅

yes

—

116

C₂×C₄²

C₂×D₁₅

yes

characteristic

117

#97

—

118

C₄×C₈

—

119

C₂×C₃×C₅

—

120

C₂×C₃×C₅

C₂×#101

yes

characteristic

121

C₂×C₃×C₅

C₂×#104

yes

—

122

C₂³×C₃

C₂²⋊Dic₅

yes

characteristic

123

C₃×Q₈

—

124

C₂×C₃×C₄

—

125

C₂×C₃×C₄

—

126

C₂×C₃×C₄

—

127

C₂×C₃×C₄

C₂²⋊Dic₅

yes

—

128

C₂×C₃×C₄

C₂²⋊Dic₅

yes

characteristic

129

C₂×C₃×C₄

C₂²×Dic₅

yes

characteristic

130

C₂×C₃×C₄

C₂×C₂²⋊D₅

yes

—

131

C₃⋊C₈

—

132

C₂²×C₅

—

133

C₂²×C₅

—

134

C₂²×C₅

C₂×#217

yes

characteristic

135

C₄×C₅

—

136

C₄×C₅

—

137

C₄×C₅

—

138

C₄×C₅

#836

yes

—

139

C₄×C₅

#837

yes

—

140

C₂×Q₈

—

141

C₂²×C₄

C₂×Dic₁₅

yes

characteristic

142

C₂²×C₄

#1220

yes

—

143

M₁₆

—

144

C₂×C₈

—

145

C₄⋊C₄

—

146

C₄⋊C₄

—

147

C₄⋊C₄

C₂×Dic₁₅

yes

—

148

C₂²⋊C₄

—

149

C₄²

—

150

C₄²

C₂×Dic₁₅

yes

characteristic

151

C₄²

#1220

yes

—

152

C₄²

C₂²×D₁₅

yes

characteristic

153

C₃×C₅

#1526

yes

characteristic, complemented, Hall

154

C₂²×C₃

—

155

C₂²×C₃

—

156

C₂²×C₃

C₂×C₂²⋊Dic₅

yes

characteristic

157

C₃×C₄

—

158

C₃×C₄

—

159

C₃×C₄

—

160

C₃×C₄

#3968

yes

—

161

C₃×C₄

#3969

yes

—

162

C₂×C₅

—

163

C₂×C₅

C₂×#836

yes

characteristic

164

C₂×C₅

C₂×#837

yes

—

165

C₂³

#6719

yes

characteristic

166

Q₈

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

#6719

yes

—

171

C₂×C₄

#6719

yes

characteristic

172

C₂×C₄

C₂²×Dic₁₅

yes

characteristic

173

C₂×C₄

C₂×#1220

yes

—

174

C₈

—

175

C₂×C₃

—

176

C₂×C₃

C₂×#3968

yes

characteristic

177

C₂×C₃

C₂×#3969

yes

—

178

C₅

#71821

yes

characteristic, complemented, Sylow

179

C₂²

—

180

C₂²

—

181

C₂²

C₂×#6719

yes

characteristic

182

C₄

—

183

C₄

—

184

C₄

—

185

C₄

#91120

yes

—

186

C₄

#91121

yes

—

187

C₃

#10602614

yes

characteristic, complemented, Sylow

188

C₂

—

189

C₂

C₂×#91120

yes

characteristic

190

C₂

C₂×#91121

yes

—

	subgroup	G/H
center	C₂×C₄	#6719
commutator	C₂×C₃×C₄×C₅	C₂²×C₄
Frattini	C₄²	C₂²×D₁₅
Fitting	C₃×C₅×#533	C₂
socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
prime radical	C₂³×C₃×C₅	C₂²⋊C₄
cosocle	C₃×C₄²×C₅	C₂³
hypercenter	#533	D₁₅
hypocenter	C₃×C₅	#1526
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂²×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #533

subgroup

G/H

center

C₂×C₄

#6719

commutator

C₂×C₃×C₄×C₅

C₂²×C₄

Frattini

C₄²

C₂²×D₁₅

Fitting

C₃×C₅×#533

C₂

socle

C₂²×C₃×C₅

C₂×C₂²⋊C₄

prime radical

C₂³×C₃×C₅

C₂²⋊C₄

cosocle

C₃×C₄²×C₅

C₂³

D₁₅

C₃×C₅

G ▷ C₂×C₃×C₄×C₅ ▷ C₂²×C₃×C₅ ▷ C₃×C₅

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #533

N	H ≅ G/N	type
3. #11746364	C₂	semidirect
153. C₃×C₅	#1526	semidirect
178. C₅	#71821	semidirect
187. C₃	#10602614	semidirect

H ≅ G/N

type

3. #11746364

C₂

semidirect

153. C₃×C₅

#1526

semidirect

178. C₅

#71821

semidirect

187. C₃

#10602614

semidirect

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

subgroup	index
C₃×C₅×#533	2
C₄×#6710	2
#11746364	2
#10602614	3
#71821	5

subgroup

index

C₃×C₅×#533

Subgroups — #14284012

Conjugacy classes of subgroups

Complements

Maximal subgroups