Group #14308979 — subgroups

Subgroups — #14308979

Order 1920.

932 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 932 conjugacy classes of subgroups — too many to draw. Only its 129 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×#543	C₂	1	1	yes	maximal, characteristic, complemented
2	#11751056	C₂	1	3	yes	maximal
3	#11746347	C₂	1	3	yes	maximal, complemented
4	#10602690	—	3	1	no	maximal, Hall
5	C₂⁴×D₁₅	C₂²	1	1	yes	characteristic
6	C₃×C₅×C₂²⋊Q₈	—	2	6	no	—
7	C₂×C₃×C₅×C₂²⋊C₄	C₂²	1	3	yes	—
8	C₂×#6719	C₂²	1	3	yes	—
9	C₂×#6714	—	2	6	no	—
10	#71897	—	5	1	no	maximal, Hall
11	C₅×#543	D₃	1	1	yes	characteristic, complemented
12	#65244	—	3	3	no	—
13	#64746	—	3	3	no	—
14	C₂⁴×C₃×C₅	C₂³	1	1	yes	characteristic
15	C₂³×D₁₅	—	2	3	no	—
16	C₂³×D₁₅	—	2	3	no	—
17	C₂³×D₁₅	—	4	4	no	—
18	C₂³×D₁₅	Q₈	1	2	yes	complemented
19	C₂×C₃×C₅×Q₈	—	2	2	no	—
20	C₂²×C₃×C₄×C₅	D₄	1	6	yes	—
21	C₂²×Dic₁₅	—	2	3	no	—
22	C₃×C₅×C₄⋊C₄	—	2	6	no	—
23	C₃×C₅×C₂²⋊C₄	—	2	6	no	—
24	#6719	—	2	6	no	—
25	#6714	—	4	12	no	—
26	C₃×#543	D₅	1	1	yes	characteristic, complemented
27	#4955	—	5	3	no	—
28	#4526	—	5	3	no	—
29	C₂⁴×D₅	—	3	1	no	—
30	C₅×C₂²⋊Q₈	—	2	6	no	—
31	C₂×C₅×C₂²⋊C₄	C₂×D₃	1	3	yes	—
32	C₂×C₂²⋊Dic₅	—	3	3	no	—
33	C₂×#685	—	6	6	no	—
34	#2029	—	15	1	no	Sylow
35	C₂³×C₃×C₅	—	4	2	no	—
36	C₂³×C₃×C₅	C₂×D₄	1	3	yes	—
37	C₂³×C₃×C₅	C₂×Q₈	1	1	yes	characteristic
38	C₂³×C₃×C₅	#41	1	3	yes	—
39	C₂²×D₁₅	—	2	12	no	—
40	C₂²×D₁₅	—	2	6	no	—
41	C₂²×D₁₅	—	4	4	no	—
42	C₂²×D₁₅	—	4	12	no	—
43	C₂²×D₁₅	—	4	6	no	—
44	C₂²×D₁₅	—	4	1	no	—
45	C₂²×D₁₅	—	4	3	no	—
46	C₃×C₅×Q₈	—	8	2	no	—
47	C₂×C₃×C₄×C₅	—	2	6	no	—
48	C₂×C₃×C₄×C₅	—	4	6	no	—
49	C₂×Dic₁₅	—	4	6	no	—
50	C₂×Dic₁₅	—	4	3	no	—
51	C₂⁴×D₃	—	5	1	no	—
52	C₃×C₂²⋊Q₈	—	2	6	no	—
53	C₂×C₃×C₂²⋊C₄	C₂×D₅	1	3	yes	—
54	C₂×#217	—	5	3	no	—
55	C₂×(C₂×D₃)⋊C₄	—	10	6	no	—
56	C₂⁴×C₅	C₂²×D₃	1	1	yes	characteristic
57	C₂³×D₅	—	3	2	no	—
58	C₂³×D₅	—	6	3	no	—
59	C₂³×D₅	—	6	3	no	—
60	C₂³×D₅	—	12	4	no	—
61	C₂×C₅×Q₈	—	2	2	no	—
62	C₂²×C₄×C₅	#67	1	6	yes	—
63	C₂²×Dic₅	—	6	3	no	—
64	C₅×C₄⋊C₄	—	2	6	no	—
65	C₅×C₂²⋊C₄	—	2	6	no	—
66	C₂²⋊Dic₅	—	6	6	no	—
67	#685	—	12	12	no	—
68	#543	D₁₅	1	1	yes	characteristic, complemented
69	#379	—	15	3	no	—
70	#342	—	15	3	no	—
71	C₂²×C₃×C₅	—	2	6	no	—
72	C₂²×C₃×C₅	—	4	1	no	—
73	C₂²×C₃×C₅	—	4	3	no	—
74	C₂²×C₃×C₅	#120	1	1	yes	characteristic
75	C₂²×C₃×C₅	C₂²⋊Q₈	1	3	yes	—
76	C₂²×C₃×C₅	#124	1	3	yes	—
77	C₂×D₁₅	—	2	12	no	—
78	C₂×D₁₅	—	4	12	no	—
79	C₂×D₁₅	—	4	6	no	—
80	C₂×D₁₅	—	4	2	no	—
81	C₂×D₁₅	—	4	4	no	—
82	C₃×C₄×C₅	—	4	6	no	—
83	Dic₁₅	—	8	3	no	—
84	C₂⁴×C₃	C₂²×D₅	1	1	yes	characteristic
85	C₂³×D₃	—	5	2	no	—
86	C₂³×D₃	—	10	3	no	—
87	C₂³×D₃	—	10	3	no	—
88	C₂³×D₃	—	20	4	no	—
89	C₂×C₃×Q₈	—	2	2	no	—
90	C₂²×C₃×C₄	C₂²⋊D₅	1	6	yes	—
91	C₂²×Dic₃	—	10	3	no	—
92	C₃×C₄⋊C₄	—	2	6	no	—
93	C₃×C₂²⋊C₄	—	2	6	no	—
94	#217	—	10	6	no	—
95	(C₂×D₃)⋊C₄	—	20	12	no	—
96	C₂³×C₅	—	4	2	no	—
97	C₂³×C₅	D₃×Q₈	1	1	yes	characteristic
98	C₂³×C₅	#239	1	3	yes	—
99	C₂³×C₅	C₂×#67	1	3	yes	—
100	C₂²×D₅	—	6	12	no	—
101	C₂²×D₅	—	6	6	no	—
102	C₂²×D₅	—	12	4	no	—
103	C₂²×D₅	—	12	12	no	—
104	C₂²×D₅	—	12	6	no	—
105	C₂²×D₅	—	12	3	no	—
106	C₂²×D₅	—	12	1	no	—
107	C₅×Q₈	—	8	2	no	—
108	C₂×C₄×C₅	—	2	6	no	—
109	C₂×C₄×C₅	—	4	6	no	—
110	C₂×Dic₅	—	12	6	no	—
111	C₂×Dic₅	—	12	3	no	—
112	C₂⁵	—	15	1	no	—
113	C₂²⋊Q₈	—	2	6	no	—
114	C₂×C₂²⋊C₄	—	15	3	no	—
115	C₂×C₂²⋊C₄	—	30	6	no	—
116	C₂×C₂²⋊C₄	C₂×D₁₅	1	3	yes	—
117	C₂×C₃×C₅	—	2	3	no	—
118	C₂×C₃×C₅	—	2	3	no	—
119	C₂×C₃×C₅	#393	1	1	yes	characteristic
120	C₂×C₃×C₅	B(4,𝔽₂)	1	2	yes	—
121	D₁₅	—	4	4	no	—
122	C₂³×C₃	—	4	2	no	—
123	C₂³×C₃	Q₈×D₅	1	1	yes	characteristic
124	C₂³×C₃	#713	1	3	yes	—
125	C₂³×C₃	C₂×C₂²⋊D₅	1	3	yes	—
126	C₂²×D₃	—	10	12	no	—
127	C₂²×D₃	—	10	6	no	—
128	C₂²×D₃	—	20	4	no	—
129	C₂²×D₃	—	20	12	no	—
130	C₂²×D₃	—	20	6	no	—
131	C₂²×D₃	—	20	1	no	—
132	C₂²×D₃	—	20	3	no	—
133	C₃×Q₈	—	8	2	no	—
134	C₂×C₃×C₄	—	2	6	no	—
135	C₂×C₃×C₄	—	4	6	no	—
136	C₂×Dic₃	—	20	6	no	—
137	C₂×Dic₃	—	20	3	no	—
138	C₂²×C₅	—	2	6	no	—
139	C₂²×C₅	—	4	1	no	—
140	C₂²×C₅	—	4	3	no	—
141	C₂²×C₅	#946	1	3	yes	—
142	C₂²×C₅	#947	1	3	yes	—
143	C₂²×C₅	#953	1	1	yes	characteristic
144	C₂×D₅	—	6	12	no	—
145	C₂×D₅	—	12	12	no	—
146	C₂×D₅	—	12	6	no	—
147	C₂×D₅	—	12	2	no	—
148	C₂×D₅	—	12	4	no	—
149	C₄×C₅	—	4	6	no	—
150	Dic₅	—	24	3	no	—
151	C₂⁴	—	15	2	no	—
152	C₂⁴	—	30	3	no	—
153	C₂⁴	—	30	3	no	—
154	C₂⁴	—	60	4	no	—
155	C₂⁴	C₂²×D₁₅	1	1	yes	characteristic
156	C₂×Q₈	—	2	2	no	—
157	C₂²×C₄	—	30	3	no	—
158	C₂²×C₄	#1220	1	6	yes	—
159	C₄⋊C₄	—	2	6	no	—
160	C₂²⋊C₄	—	2	6	no	—
161	C₂²⋊C₄	—	30	6	no	—
162	C₂²⋊C₄	—	60	12	no	—
163	C₃×C₅	#2029	1	1	yes	characteristic, complemented, Hall
164	C₂²×C₃	—	2	6	no	—
165	C₂²×C₃	—	4	1	no	—
166	C₂²×C₃	—	4	3	no	—
167	C₂²×C₃	#4092	1	3	yes	—
168	C₂²×C₃	#4093	1	3	yes	—
169	C₂²×C₃	#4099	1	1	yes	characteristic
170	C₂×D₃	—	10	12	no	—
171	C₂×D₃	—	20	12	no	—
172	C₂×D₃	—	20	6	no	—
173	C₂×D₃	—	20	2	no	—
174	C₂×D₃	—	20	4	no	—
175	C₃×C₄	—	4	6	no	—
176	Dic₃	—	40	3	no	—
177	C₂×C₅	—	2	3	no	—
178	C₂×C₅	—	2	3	no	—
179	C₂×C₅	#5231	1	1	yes	characteristic
180	C₂×C₅	#5244	1	2	yes	—
181	D₅	—	12	4	no	—
182	C₂³	—	4	2	no	—
183	C₂³	—	30	12	no	—
184	C₂³	—	30	6	no	—
185	C₂³	—	60	6	no	—
186	C₂³	—	60	4	no	—
187	C₂³	—	60	12	no	—
188	C₂³	—	60	3	no	—
189	C₂³	—	60	1	no	—
190	C₂³	Q₈×D₁₅	1	1	yes	characteristic
191	C₂³	#6821	1	3	yes	—
192	C₂³	C₂×#1220	1	3	yes	—
193	Q₈	—	8	2	no	—
194	C₂×C₄	—	2	6	no	—
195	C₂×C₄	—	4	6	no	—
196	C₂×C₄	—	60	6	no	—
197	C₂×C₄	—	60	3	no	—
198	C₂×C₃	—	2	3	no	—
199	C₂×C₃	—	2	3	no	—
200	C₂×C₃	#65520	1	1	yes	characteristic
201	C₂×C₃	#65533	1	2	yes	—
202	D₃	—	20	4	no	—
203	C₅	#71897	1	1	yes	characteristic, complemented, Sylow
204	C₂²	—	2	6	no	—
205	C₂²	—	4	1	no	—
206	C₂²	—	4	3	no	—
207	C₂²	—	30	12	no	—
208	C₂²	—	60	12	no	—
209	C₂²	—	60	6	no	—
210	C₂²	—	60	2	no	—
211	C₂²	—	60	4	no	—
212	C₂²	#91835	1	3	yes	—
213	C₂²	#91836	1	3	yes	—
214	C₂²	#91842	1	1	yes	characteristic
215	C₄	—	4	6	no	—
216	C₄	—	120	3	no	—
217	C₃	#10602690	1	1	yes	characteristic, complemented, Sylow
218	C₂	—	2	3	no	—
219	C₂	—	2	3	no	—
220	C₂	—	60	4	no	—
221	C₂	#11751332	1	1	yes	characteristic
222	C₂	#11751345	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×C₅×#543

C₂

yes

maximal, characteristic, complemented

#11751056

C₂

yes

maximal

#11746347

C₂

yes

maximal, complemented

#10602690

—

maximal, Hall

C₂⁴×D₁₅

C₂²

yes

characteristic

C₃×C₅×C₂²⋊Q₈

—

C₂×C₃×C₅×C₂²⋊C₄

C₂²

yes

—

C₂×#6719

C₂²

yes

—

—

—

D₃

yes

characteristic, complemented

#65244

—

#64746

—

C₂⁴×C₃×C₅

C₂³

yes

characteristic

C₂³×D₁₅

—

C₂³×D₁₅

—

C₂³×D₁₅

—

C₂³×D₁₅

Q₈

yes

complemented

C₂×C₃×C₅×Q₈

—

C₂²×C₃×C₄×C₅

D₄

yes

—

C₂²×Dic₁₅

—

C₃×C₅×C₄⋊C₄

—

C₃×C₅×C₂²⋊C₄

—

#6719

—

#6714

—

C₃×#543

D₅

yes

characteristic, complemented

#4955

—

#4526

—

C₂⁴×D₅

—

C₅×C₂²⋊Q₈

—

C₂×C₅×C₂²⋊C₄

C₂×D₃

yes

—

C₂×C₂²⋊Dic₅

—

C₂×#685

—

#2029

—

Sylow

C₂³×C₃×C₅

—

C₂³×C₃×C₅

C₂×D₄

yes

—

C₂³×C₃×C₅

C₂×Q₈

yes

characteristic

C₂³×C₃×C₅

#41

yes

—

C₂²×D₁₅

—

C₂²×D₁₅

—

C₂²×D₁₅

—

C₂²×D₁₅

—

C₂²×D₁₅

—

C₂²×D₁₅

—

C₂²×D₁₅

—

C₃×C₅×Q₈

—

C₂×C₃×C₄×C₅

—

C₂×C₃×C₄×C₅

—

C₂×Dic₁₅

—

C₂×Dic₁₅

—

C₂⁴×D₃

—

C₃×C₂²⋊Q₈

—

C₂×C₃×C₂²⋊C₄

C₂×D₅

yes

—

C₂×#217

—

C₂×(C₂×D₃)⋊C₄

—

C₂⁴×C₅

C₂²×D₃

yes

characteristic

C₂³×D₅

—

C₂³×D₅

—

C₂³×D₅

—

C₂³×D₅

—

C₂×C₅×Q₈

—

C₂²×C₄×C₅

#67

yes

—

C₂²×Dic₅

—

C₅×C₄⋊C₄

—

C₅×C₂²⋊C₄

—

C₂²⋊Dic₅

—

#685

—

#543

D₁₅

yes

characteristic, complemented

#379

—

#342

—

C₂²×C₃×C₅

—

C₂²×C₃×C₅

—

C₂²×C₃×C₅

—

C₂²×C₃×C₅

#120

yes

characteristic

C₂²×C₃×C₅

C₂²⋊Q₈

yes

—

C₂²×C₃×C₅

#124

yes

—

C₂×D₁₅

—

C₂×D₁₅

—

C₂×D₁₅

—

C₂×D₁₅

—

C₂×D₁₅

—

C₃×C₄×C₅

—

Dic₁₅

—

C₂⁴×C₃

C₂²×D₅

yes

characteristic

C₂³×D₃

—

C₂³×D₃

—

C₂³×D₃

—

C₂³×D₃

—

C₂×C₃×Q₈

—

C₂²×C₃×C₄

C₂²⋊D₅

yes

—

C₂²×Dic₃

—

C₃×C₄⋊C₄

—

C₃×C₂²⋊C₄

—

#217

—

(C₂×D₃)⋊C₄

—

C₂³×C₅

—

C₂³×C₅

D₃×Q₈

yes

characteristic

C₂³×C₅

#239

yes

—

C₂³×C₅

C₂×#67

yes

—

100

C₂²×D₅

—

101

C₂²×D₅

—

102

C₂²×D₅

—

103

C₂²×D₅

—

104

C₂²×D₅

—

105

C₂²×D₅

—

106

C₂²×D₅

—

107

C₅×Q₈

—

108

C₂×C₄×C₅

—

109

C₂×C₄×C₅

—

110

C₂×Dic₅

—

111

C₂×Dic₅

—

112

C₂⁵

—

113

C₂²⋊Q₈

—

114

C₂×C₂²⋊C₄

—

115

C₂×C₂²⋊C₄

—

116

C₂×C₂²⋊C₄

C₂×D₁₅

yes

—

117

C₂×C₃×C₅

—

118

C₂×C₃×C₅

—

119

C₂×C₃×C₅

#393

yes

characteristic

120

C₂×C₃×C₅

B(4,𝔽₂)

yes

—

121

D₁₅

—

122

C₂³×C₃

—

123

C₂³×C₃

Q₈×D₅

yes

characteristic

124

C₂³×C₃

#713

yes

—

125

C₂³×C₃

C₂×C₂²⋊D₅

yes

—

126

C₂²×D₃

—

127

C₂²×D₃

—

128

C₂²×D₃

—

129

C₂²×D₃

—

130

C₂²×D₃

—

131

C₂²×D₃

—

132

C₂²×D₃

—

133

C₃×Q₈

—

134

C₂×C₃×C₄

—

135

C₂×C₃×C₄

—

136

C₂×Dic₃

—

137

C₂×Dic₃

—

138

C₂²×C₅

—

139

C₂²×C₅

—

140

C₂²×C₅

—

141

C₂²×C₅

#946

yes

—

142

C₂²×C₅

#947

yes

—

143

C₂²×C₅

#953

yes

characteristic

144

C₂×D₅

—

145

C₂×D₅

—

146

C₂×D₅

—

147

C₂×D₅

—

148

C₂×D₅

—

149

C₄×C₅

—

150

Dic₅

—

151

C₂⁴

—

152

C₂⁴

—

153

C₂⁴

—

154

C₂⁴

—

155

C₂⁴

C₂²×D₁₅

yes

characteristic

156

C₂×Q₈

—

157

C₂²×C₄

—

158

C₂²×C₄

#1220

yes

—

159

C₄⋊C₄

—

160

C₂²⋊C₄

—

161

C₂²⋊C₄

—

162

C₂²⋊C₄

—

163

C₃×C₅

#2029

yes

characteristic, complemented, Hall

164

C₂²×C₃

—

165

C₂²×C₃

—

166

C₂²×C₃

—

167

C₂²×C₃

#4092

yes

—

168

C₂²×C₃

#4093

yes

—

169

C₂²×C₃

#4099

yes

characteristic

170

C₂×D₃

—

171

C₂×D₃

—

172

C₂×D₃

—

173

C₂×D₃

—

174

C₂×D₃

—

175

C₃×C₄

—

176

Dic₃

—

177

C₂×C₅

—

178

C₂×C₅

—

179

C₂×C₅

#5231

yes

characteristic

180

C₂×C₅

#5244

yes

—

181

D₅

—

182

C₂³

—

183

C₂³

—

184

C₂³

—

185

C₂³

—

186

C₂³

—

187

C₂³

—

188

C₂³

—

189

C₂³

—

190

C₂³

Q₈×D₁₅

yes

characteristic

191

C₂³

#6821

yes

—

192

C₂³

C₂×#1220

yes

—

193

Q₈

—

194

C₂×C₄

—

195

C₂×C₄

—

196

C₂×C₄

—

197

C₂×C₄

—

198

C₂×C₃

—

199

C₂×C₃

—

200

C₂×C₃

#65520

yes

characteristic

201

C₂×C₃

#65533

yes

—

202

D₃

—

203

C₅

#71897

yes

characteristic, complemented, Sylow

204

C₂²

—

205

C₂²

—

206

C₂²

—

207

C₂²

—

208

C₂²

—

209

C₂²

—

210

C₂²

—

211

C₂²

—

212

C₂²

#91835

yes

—

213

C₂²

#91836

yes

—

214

C₂²

#91842

yes

characteristic

215

C₄

—

216

C₄

—

120

—

217

C₃

#10602690

yes

characteristic, complemented, Sylow

218

C₂

—

219

C₂

—

220

C₂

—

221

C₂

#11751332

yes

characteristic

222

C₂

#11751345

yes

—

	subgroup	G/H
center	C₂²	#91842
commutator, cosocle	C₂⁴×C₃×C₅	C₂³
Frattini	C₂⁴	C₂²×D₁₅
Fitting	C₃×C₅×#543	C₂
socle	C₂²×C₃×C₅	#120
prime radical	C₂⁴×D₁₅	C₂²
hypercenter	#543	D₁₅
hypocenter	C₃×C₅	#2029
LCS	G ▷ C₂⁴×C₃×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ #543

subgroup

G/H

center

C₂²

#91842

commutator, cosocle

C₂⁴×C₃×C₅

C₂³

Frattini

C₂⁴

C₂²×D₁₅

Fitting

C₃×C₅×#543

C₂

socle

C₂²×C₃×C₅

#120

prime radical

C₂⁴×D₁₅

C₂²

D₁₅

C₃×C₅

G ▷ C₂⁴×C₃×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅

UCS

C₁ ◁ C₂² ◁ C₂⁴ ◁ #543

N	H ≅ G/N	type
1. C₃×C₅×#543	C₂	semidirect
3. #11746347	C₂	semidirect
11. C₅×#543	D₃	semidirect
18. C₂³×D₁₅	Q₈	semidirect
26. C₃×#543	D₅	semidirect
68. #543	D₁₅	semidirect
163. C₃×C₅	#2029	semidirect
203. C₅	#71897	semidirect
217. C₃	#10602690	semidirect

H ≅ G/N

type

1. C₃×C₅×#543

C₂

semidirect

3. #11746347

C₂

semidirect

11. C₅×#543

D₃

semidirect

18. C₂³×D₁₅

Q₈

semidirect

26. C₃×#543

D₅

semidirect

68. #543

D₁₅

semidirect

163. C₃×C₅

#2029

semidirect

203. C₅

#71897

semidirect

217. C₃

#10602690

semidirect

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

subgroup	index
C₃×C₅×#543	2
#11751056	2
#11746347	2
#10602690	3
#71897	5

subgroup

index

C₃×C₅×#543

Subgroups — #14308979

Conjugacy classes of subgroups

Complements

Maximal subgroups