Group #14285455 — subgroups

Subgroups — #14285455

Order 1920.

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

normal subgroups only

This group has 420 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	#11751025	C₂	1	2	yes	maximal
3	#11746368	C₂	1	4	yes	maximal, complemented
4	#10602627	—	3	1	no	maximal, Hall
5	C₃×C₅×C₄⋊Q₈	C₂²	1	2	yes	—
6	C₃×C₅×#125	C₄	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	C₄⋊(C₃⋊#168)	—	2	2	no	—
13	C₄⋊(C₃⋊#168)	C₂²	1	4	yes	—
14	#71834	—	5	1	no	maximal, Hall
15	C₅×#533	D₃	1	1	yes	characteristic
16	#65213	—	3	2	no	—
17	#64767	—	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	4	no	—
22	C₃×C₅×C₄⋊C₄	—	2	2	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	—	4	4	no	—
30	C₂×C₃⋊#168	—	2	4	no	—
31	C₃×#533	D₅	1	1	yes	characteristic
32	#4924	—	5	2	no	—
33	#4547	—	5	4	no	—
34	C₅×C₄⋊Q₈	C₂×D₃	1	2	yes	—
35	C₅×#125	Dic₃	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	C₄⋊#168	—	3	4	no	—
42	C₄⋊#168	—	6	2	no	—
43	#1561	—	15	1	no	Sylow
44	C₂³×C₃×C₅	C₂²⋊C₄	1	1	yes	characteristic
45	C₃×C₅×Q₈	—	4	2	no	—
46	C₃×C₅×Q₈	—	4	2	no	—
47	C₂×C₃×C₄×C₅	—	2	2	no	—
48	C₂×C₃×C₄×C₅	—	2	2	no	—
49	C₂×C₃×C₄×C₅	—	2	2	no	—
50	C₂×C₃×C₄×C₅	—	2	4	no	—
51	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
52	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
53	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
55	C₃⋊#168	—	4	4	no	—
56	C₃×C₄⋊Q₈	C₂×D₅	1	2	yes	—
57	C₃×#125	Dic₅	1	2	yes	—
58	C₃×C₂²⋊Q₈	—	2	2	no	—
59	C₃×C₄×Q₈	—	2	2	no	—
60	C₃×#117	Dic₅	1	2	yes	—
61	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
62	C₂×#208	—	10	2	no	—
63	C₄⋊(C₃⋊C₈)	—	5	4	no	—
64	C₄⋊(C₃⋊C₈)	—	10	2	no	—
65	C₂×C₅×Q₈	—	2	2	no	—
66	C₂²×C₄×C₅	C₂×Dic₃	1	1	yes	characteristic
67	C₂²×C₄×C₅	#67	1	2	yes	—
68	C₅×C₄⋊C₄	—	2	4	no	—
69	C₅×C₄⋊C₄	—	2	2	no	—
70	C₅×C₄⋊C₄	C₂×Dic₃	1	4	yes	—
71	C₅×C₂²⋊C₄	—	2	2	no	—
72	C₄²×C₅	—	2	2	no	—
73	C₄²×C₅	C₂×Dic₃	1	1	yes	characteristic
74	C₄²×C₅	#67	1	2	yes	—
75	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
76	#681	—	12	4	no	—
77	C₂×#168	—	6	4	no	—
78	#533	D₁₅	1	1	yes	characteristic
79	#423	—	15	2	no	—
80	#332	—	15	4	no	—
81	C₂²×C₃×C₅	—	2	1	no	—
82	C₂²×C₃×C₅	—	2	2	no	—
83	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
84	C₃×C₄×C₅	—	2	1	no	—
85	C₃×C₄×C₅	—	2	2	no	—
86	C₃×C₄×C₅	—	4	4	no	—
87	C₃×C₄×C₅	#101	1	2	yes	—
88	C₃×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
89	C₃×C₄×C₅	#137	1	2	yes	—
90	C₂×C₃×Q₈	—	2	2	no	—
91	C₂²×C₃×C₄	C₂×Dic₅	1	1	yes	characteristic
92	C₂²×C₃×C₄	C₂²⋊D₅	1	2	yes	—
93	C₃×C₄⋊C₄	—	2	4	no	—
94	C₃×C₄⋊C₄	—	2	2	no	—
95	C₃×C₄⋊C₄	C₂×Dic₅	1	4	yes	—
96	C₃×C₂²⋊C₄	—	2	2	no	—
97	C₃×C₄²	—	2	2	no	—
98	C₃×C₄²	C₂×Dic₅	1	1	yes	characteristic
99	C₃×C₄²	C₂²⋊D₅	1	2	yes	—
100	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
101	#208	—	20	4	no	—
102	C₂×C₃⋊C₈	—	10	4	no	—
103	C₂³×C₅	#217	1	1	yes	characteristic
104	C₅×Q₈	—	4	2	no	—
105	C₅×Q₈	—	4	2	no	—
106	C₂×C₄×C₅	—	2	2	no	—
107	C₂×C₄×C₅	—	2	2	no	—
108	C₂×C₄×C₅	—	2	2	no	—
109	C₂×C₄×C₅	—	2	4	no	—
110	C₂×C₄×C₅	#217	1	2	yes	—
111	C₂×C₄×C₅	#217	1	1	yes	characteristic
112	C₂×C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
113	C₂×C₄×C₅	C₂×#67	1	2	yes	—
114	#168	—	12	4	no	—
115	C₂×M₁₆	—	30	2	no	—
116	C₄⋊Q₈	C₂×D₁₅	1	2	yes	—
117	#125	Dic₁₅	1	2	yes	—
118	C₂²⋊Q₈	—	2	2	no	—
119	C₄×Q₈	—	2	2	no	—
120	#117	Dic₁₅	1	2	yes	—
121	C₂×C₄²	C₂×D₁₅	1	1	yes	characteristic
122	C₄⋊C₈	—	15	4	no	—
123	C₄⋊C₈	—	30	2	no	—
124	C₂×C₃×C₅	—	4	1	no	—
125	C₂×C₃×C₅	C₂×#101	1	1	yes	characteristic
126	C₂×C₃×C₅	#417	1	2	yes	—
127	C₂³×C₃	C₂²⋊Dic₅	1	1	yes	characteristic
128	C₃×Q₈	—	4	2	no	—
129	C₃×Q₈	—	4	2	no	—
130	C₂×C₃×C₄	—	2	2	no	—
131	C₂×C₃×C₄	—	2	2	no	—
132	C₂×C₃×C₄	—	2	2	no	—
133	C₂×C₃×C₄	—	2	4	no	—
134	C₂×C₃×C₄	C₂²⋊Dic₅	1	2	yes	—
135	C₂×C₃×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
136	C₂×C₃×C₄	C₂²×Dic₅	1	1	yes	characteristic
137	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	2	yes	—
138	C₃⋊C₈	—	20	4	no	—
139	C₂²×C₅	—	2	1	no	—
140	C₂²×C₅	—	2	2	no	—
141	C₂²×C₅	C₂×#217	1	1	yes	characteristic
142	C₄×C₅	—	2	1	no	—
143	C₄×C₅	—	2	2	no	—
144	C₄×C₅	—	4	4	no	—
145	C₄×C₅	#836	1	2	yes	—
146	C₄×C₅	#949	1	2	yes	—
147	C₄×C₅	#951	1	2	yes	—
148	C₂×Q₈	—	2	2	no	—
149	C₂²×C₄	C₂×Dic₁₅	1	1	yes	characteristic
150	C₂²×C₄	#1220	1	2	yes	—
151	M₁₆	—	60	4	no	—
152	C₂×C₈	—	30	4	no	—
153	C₄⋊C₄	—	2	4	no	—
154	C₄⋊C₄	—	2	2	no	—
155	C₄⋊C₄	C₂×Dic₁₅	1	4	yes	—
156	C₂²⋊C₄	—	2	2	no	—
157	C₄²	—	2	2	no	—
158	C₄²	C₂×Dic₁₅	1	1	yes	characteristic
159	C₄²	#1220	1	2	yes	—
160	C₄²	C₂²×D₁₅	1	1	yes	characteristic
161	C₃×C₅	#1561	1	1	yes	characteristic, complemented, Hall
162	C₂²×C₃	—	2	1	no	—
163	C₂²×C₃	—	2	2	no	—
164	C₂²×C₃	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
165	C₃×C₄	—	2	1	no	—
166	C₃×C₄	—	2	2	no	—
167	C₃×C₄	—	4	4	no	—
168	C₃×C₄	#3968	1	2	yes	—
169	C₃×C₄	#4095	1	2	yes	—
170	C₃×C₄	#4097	1	2	yes	—
171	C₂×C₅	—	4	1	no	—
172	C₂×C₅	C₂×#836	1	1	yes	characteristic
173	C₂×C₅	#5232	1	2	yes	—
174	C₂³	#6719	1	1	yes	characteristic
175	Q₈	—	4	2	no	—
176	Q₈	—	4	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	2	4	no	—
181	C₂×C₄	#6719	1	2	yes	—
182	C₂×C₄	#6719	1	1	yes	characteristic
183	C₂×C₄	C₂²×Dic₁₅	1	1	yes	characteristic
184	C₂×C₄	C₂×#1220	1	2	yes	—
185	C₈	—	60	4	no	—
186	C₂×C₃	—	4	1	no	—
187	C₂×C₃	C₂×#3968	1	1	yes	characteristic
188	C₂×C₃	#65521	1	2	yes	—
189	C₅	#71834	1	1	yes	characteristic, complemented, Sylow
190	C₂²	—	2	1	no	—
191	C₂²	—	2	2	no	—
192	C₂²	C₂×#6719	1	1	yes	characteristic
193	C₄	—	2	1	no	—
194	C₄	—	2	2	no	—
195	C₄	—	4	4	no	—
196	C₄	#91120	1	2	yes	—
197	C₄	#91838	1	2	yes	—
198	C₄	#91840	1	2	yes	—
199	C₃	#10602627	1	1	yes	characteristic, complemented, Sylow
200	C₂	—	4	1	no	—
201	C₂	C₂×#91120	1	1	yes	characteristic
202	C₂	#11751333	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×C₅×#533

C₂

yes

maximal, characteristic

#11751025

C₂

yes

maximal

#11746368

C₂

yes

maximal, complemented

#10602627

—

maximal, Hall

C₃×C₅×C₄⋊Q₈

C₂²

yes

—

C₃×C₅×#125

C₄

yes

—

C₃×C₅×C₂²⋊Q₈

—

C₃×C₄×C₅×Q₈

—

C₃×C₅×#117

C₄

yes

—

C₂×C₃×C₄²×C₅

C₂²

yes

—

—

C₂²

yes

—

#71834

—

maximal, Hall

C₅×#533

D₃

yes

characteristic

#65213

—

#64767

—

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

#4924

—

#4547

—

C₅×C₄⋊Q₈

C₂×D₃

yes

—

C₅×#125

Dic₃

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

—

C₃×C₅×Q₈

—

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

C₂²⋊C₄

yes

characteristic

C₂×C₃×C₄×C₅

C₂²×C₄

yes

characteristic

C₂×C₃×C₄×C₅

C₂×D₄

yes

—

C₃⋊#168

—

C₃×C₄⋊Q₈

C₂×D₅

yes

—

C₃×#125

Dic₅

yes

—

C₃×C₂²⋊Q₈

—

C₃×C₄×Q₈

—

C₃×#117

Dic₅

yes

—

C₂×C₃×C₄²

C₂×D₅

yes

characteristic

C₂×#208

—

C₄⋊(C₃⋊C₈)

—

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

#423

—

#332

—

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

AGL(1,ℤ/8)

yes

—

C₃×C₄×C₅

#137

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

—

100

C₃×C₄²

C₂²×D₅

yes

characteristic

101

#208

—

102

C₂×C₃⋊C₈

—

103

C₂³×C₅

#217

yes

characteristic

104

C₅×Q₈

—

105

C₅×Q₈

—

106

C₂×C₄×C₅

—

107

C₂×C₄×C₅

—

108

C₂×C₄×C₅

—

109

C₂×C₄×C₅

—

110

C₂×C₄×C₅

#217

yes

—

111

C₂×C₄×C₅

#217

yes

characteristic

112

C₂×C₄×C₅

C₂²×Dic₃

yes

characteristic

113

C₂×C₄×C₅

C₂×#67

yes

—

114

#168

—

115

C₂×M₁₆

—

116

C₄⋊Q₈

C₂×D₁₅

yes

—

117

#125

Dic₁₅

yes

—

118

C₂²⋊Q₈

—

119

C₄×Q₈

—

120

#117

Dic₁₅

yes

—

121

C₂×C₄²

C₂×D₁₅

yes

characteristic

122

C₄⋊C₈

—

123

C₄⋊C₈

—

124

C₂×C₃×C₅

—

125

C₂×C₃×C₅

C₂×#101

yes

characteristic

126

C₂×C₃×C₅

#417

yes

—

127

C₂³×C₃

C₂²⋊Dic₅

yes

characteristic

128

C₃×Q₈

—

129

C₃×Q₈

—

130

C₂×C₃×C₄

—

131

C₂×C₃×C₄

—

132

C₂×C₃×C₄

—

133

C₂×C₃×C₄

—

134

C₂×C₃×C₄

C₂²⋊Dic₅

yes

—

135

C₂×C₃×C₄

C₂²⋊Dic₅

yes

characteristic

136

C₂×C₃×C₄

C₂²×Dic₅

yes

characteristic

137

C₂×C₃×C₄

C₂×C₂²⋊D₅

yes

—

138

C₃⋊C₈

—

139

C₂²×C₅

—

140

C₂²×C₅

—

141

C₂²×C₅

C₂×#217

yes

characteristic

142

C₄×C₅

—

143

C₄×C₅

—

144

C₄×C₅

—

145

C₄×C₅

#836

yes

—

146

C₄×C₅

#949

yes

—

147

C₄×C₅

#951

yes

—

148

C₂×Q₈

—

149

C₂²×C₄

C₂×Dic₁₅

yes

characteristic

150

C₂²×C₄

#1220

yes

—

151

M₁₆

—

152

C₂×C₈

—

153

C₄⋊C₄

—

154

C₄⋊C₄

—

155

C₄⋊C₄

C₂×Dic₁₅

yes

—

156

C₂²⋊C₄

—

157

C₄²

—

158

C₄²

C₂×Dic₁₅

yes

characteristic

159

C₄²

#1220

yes

—

160

C₄²

C₂²×D₁₅

yes

characteristic

161

C₃×C₅

#1561

yes

characteristic, complemented, Hall

162

C₂²×C₃

—

163

C₂²×C₃

—

164

C₂²×C₃

C₂×C₂²⋊Dic₅

yes

characteristic

165

C₃×C₄

—

166

C₃×C₄

—

167

C₃×C₄

—

168

C₃×C₄

#3968

yes

—

169

C₃×C₄

#4095

yes

—

170

C₃×C₄

#4097

yes

—

171

C₂×C₅

—

172

C₂×C₅

C₂×#836

yes

characteristic

173

C₂×C₅

#5232

yes

—

174

C₂³

#6719

yes

characteristic

175

Q₈

—

176

Q₈

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

—

181

C₂×C₄

#6719

yes

—

182

C₂×C₄

#6719

yes

characteristic

183

C₂×C₄

C₂²×Dic₁₅

yes

characteristic

184

C₂×C₄

C₂×#1220

yes

—

185

C₈

—

186

C₂×C₃

—

187

C₂×C₃

C₂×#3968

yes

characteristic

188

C₂×C₃

#65521

yes

—

189

C₅

#71834

yes

characteristic, complemented, Sylow

190

C₂²

—

191

C₂²

—

192

C₂²

C₂×#6719

yes

characteristic

193

C₄

—

194

C₄

—

195

C₄

—

196

C₄

#91120

yes

—

197

C₄

#91838

yes

—

198

C₄

#91840

yes

—

199

C₃

#10602627

yes

characteristic, complemented, Sylow

200

C₂

—

201

C₂

C₂×#91120

yes

characteristic

202

C₂

#11751333

yes

—

	subgroup	G/H
center	C₂²	C₂×#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	#1561
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂²×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #533

subgroup

G/H

center

C₂²

C₂×#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 ◁ C₂² ◁ C₂×C₄² ◁ #533

N	H ≅ G/N	type
3. #11746368	C₂	semidirect
161. C₃×C₅	#1561	semidirect
189. C₅	#71834	semidirect
199. C₃	#10602627	semidirect

H ≅ G/N

type

3. #11746368

C₂

semidirect

161. C₃×C₅

#1561

semidirect

189. C₅

#71834

semidirect

199. C₃

#10602627

semidirect

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

subgroup	index
C₃×C₅×#533	2
#11751025	2
#11746368	2
#10602627	3
#71834	5

subgroup

index

C₃×C₅×#533

Subgroups — #14285455

Conjugacy classes of subgroups

Complements

Maximal subgroups