Group #30601523 — subgroups

Subgroups — #30601523

Order 15616.

378 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 120 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	#21461917	C₂	1	2	yes	maximal
2	C₂×#16840625	C₂	1	1	yes	maximal, characteristic
3	C₂×#92420⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#21460655	C₂	1	2	yes	maximal
5	#21445707	C₂	1	1	yes	maximal, characteristic
6	C₂×#14519844	—	2	2	no	—
7	C₂×C₄⋊#92420	C₂²	1	2	yes	—
8	C₂×C₄×#92420	C₂²	1	1	yes	characteristic
9	C₂×C₄×#92420	C₂²	1	1	yes	characteristic
10	#16840628	—	2	2	no	—
11	#16840625	—	2	2	no	—
12	#92420⋊C₈	—	2	2	no	—
13	C₂×#14519708	C₄	1	2	yes	—
14	C₂×C₈×Dic₆₁	C₂²	1	1	yes	characteristic
15	C₄⋊C₄×Dic₆₁	C₂²	1	2	yes	—
16	#16839525	C₄	1	1	yes	characteristic
17	C₆₁×#340	C₄	1	1	yes	characteristic
18	C₂²×#92420	—	2	1	no	—
19	C₂²×#92420	—	2	2	no	—
20	C₂²×#92420	D₄	1	1	yes	characteristic
21	C₂²×#92420	D₄	1	1	yes	characteristic
22	#14519844	—	4	4	no	—
23	C₄⋊#92420	—	2	2	no	—
24	C₄⋊#92420	—	2	2	no	—
25	C₄×#92420	—	2	1	no	—
26	C₄×#92420	—	2	1	no	—
27	C₄×#92420	—	2	1	no	—
28	C₄×#92420	Q₈	1	2	yes	—
29	C₂²×#92418	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄⋊Dic₆₁	C₂×C₄	1	2	yes	—
31	C₂×C₄×Dic₆₁	—	2	2	no	—
32	C₂×C₄×Dic₆₁	C₂³	1	1	yes	characteristic
33	#14519708	—	2	4	no	—
34	C₈×Dic₆₁	D₄	1	2	yes	—
35	C₈×Dic₆₁	Q₈	1	2	yes	—
36	#14519684	—	2	2	no	—
37	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	2	yes	—
39	C₂²×C₄×C₆₁	—	2	2	no	—
40	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
41	C₂²×Dic₆₁	—	2	2	no	—
42	C₂²×Dic₆₁	C₂×D₄	1	1	yes	characteristic
43	C₂²×Dic₆₁	#41	1	1	yes	characteristic
44	C₂×#92420	—	2	2	no	—
45	C₂×#92420	—	2	2	no	—
46	C₂×#92420	—	2	2	no	—
47	C₂×#92420	—	2	1	no	—
48	C₂×#92420	—	2	1	no	—
49	C₂×#92420	—	2	2	no	—
50	C₂×#92420	—	2	2	no	—
51	C₂×#92420	—	4	2	no	—
52	C₂×#92420	—	4	2	no	—
53	C₂×#92420	—	4	2	no	—
54	C₂×C₈×C₆₁	—	2	2	no	—
55	C₂×C₈×C₆₁	C₄⋊C₄	1	2	yes	—
56	C₄⋊C₄×C₆₁	—	2	2	no	—
57	C₄⋊C₄×C₆₁	—	2	2	no	—
58	C₄⋊Dic₆₁	—	2	2	no	—
59	C₄⋊Dic₆₁	M₁₆	1	4	yes	—
60	C₄×Dic₆₁	—	4	2	no	—
61	C₄×Dic₆₁	—	4	2	no	—
62	C₄×Dic₆₁	C₂×D₄	1	1	yes	characteristic
63	C₄×Dic₆₁	C₂×Q₈	1	1	yes	characteristic
64	C₄×Dic₆₁	C₂×Q₈	1	1	yes	characteristic
65	C₄×Dic₆₁	#41	1	1	yes	characteristic
66	C₂×#92418	—	2	2	no	—
67	C₂×#92418	C₄⋊C₄	1	2	yes	—
68	C₂³×C₆₁	C₄×D₄	1	1	yes	characteristic
69	C₂×C₄×C₆₁	—	4	2	no	—
70	C₂×C₄×C₆₁	—	4	2	no	—
71	C₂×C₄×C₆₁	—	4	2	no	—
72	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
73	C₂×C₄×C₆₁	C₄×Q₈	1	1	yes	characteristic
74	C₂×C₄×C₆₁	C₂×M₁₆	1	2	yes	—
75	C₂×C₄×C₆₁	#131	1	2	yes	—
76	C₂×Dic₆₁	—	2	1	no	—
77	C₂×Dic₆₁	—	2	1	no	—
78	C₂×Dic₆₁	—	2	4	no	—
79	C₂×Dic₆₁	—	4	2	no	—
80	C₂×Dic₆₁	—	4	2	no	—
81	C₂×Dic₆₁	#121	1	1	yes	characteristic
82	C₂×Dic₆₁	C₂²⋊Q₈	1	1	yes	characteristic
83	C₂×Dic₆₁	#125	1	1	yes	characteristic
84	C₂×Dic₆₁	C₄⋊Q₈	1	1	yes	characteristic
85	C₂×Dic₆₁	AGL(1,ℤ/8)	1	2	yes	—
86	C₂×Dic₆₁	#137	1	2	yes	—
87	#92420	—	2	4	no	—
88	#92420	—	4	1	no	—
89	#92420	—	4	1	no	—
90	#92420	—	8	2	no	—
91	C₈×C₆₁	—	2	2	no	—
92	#92418	—	2	2	no	—
93	#9600	—	61	1	no	maximal, Sylow
94	C₂²×C₆₁	#389	1	1	yes	characteristic
95	C₂²×C₆₁	#435	1	2	yes	—
96	C₂²×C₆₁	#440	1	2	yes	—
97	C₂²×C₆₁	#443	1	1	yes	characteristic
98	C₂²×C₆₁	#443	1	1	yes	characteristic
99	C₄×C₆₁	—	8	2	no	—
100	C₄×C₆₁	#423	1	1	yes	characteristic
101	C₄×C₆₁	#424	1	1	yes	characteristic
102	C₄×C₆₁	#446	1	2	yes	—
103	Dic₆₁	—	2	1	no	—
104	Dic₆₁	—	2	1	no	—
105	Dic₆₁	—	4	4	no	—
106	Dic₆₁	#469	1	1	yes	characteristic
107	Dic₆₁	#470	1	1	yes	characteristic
108	Dic₆₁	#501	1	2	yes	—
109	#1939	—	61	2	no	—
110	C₄×#106	—	61	1	no	—
111	C₂×#335	—	61	1	no	—
112	C₂×#322	—	61	1	no	—
113	#1279	—	61	2	no	—
114	C₂×C₆₁	#1585	1	2	yes	—
115	C₂×C₆₁	#1588	1	2	yes	—
116	C₂×C₆₁	#1940	1	1	yes	characteristic
117	C₂×C₆₁	#1944	1	1	yes	characteristic
118	C₂×C₆₁	#1945	1	1	yes	characteristic
119	C₂×#106	—	61	2	no	—
120	C₂×C₄⋊C₈	—	61	2	no	—
121	C₂×C₄⋊C₈	—	122	2	no	—
122	C₂×C₄×C₈	—	61	1	no	—
123	C₂×C₄×C₈	—	61	1	no	—
124	C₂×C₄×C₈	—	61	1	no	—
125	C₄×C₄⋊C₄	—	61	2	no	—
126	#340	—	61	1	no	—
127	#340	#6923	1	1	yes	characteristic
128	#336	—	122	2	no	—
129	#335	—	122	2	no	—
130	#322	—	122	2	no	—
131	C₆₁	#9600	1	1	yes	characteristic, complemented, Sylow
132	C₂²×C₈	—	61	1	no	—
133	C₂²×C₈	—	61	1	no	—
134	C₂²×C₈	—	61	1	no	—
135	C₂²×C₈	—	122	1	no	—
136	C₂²×C₈	—	122	2	no	—
137	C₂²×C₈	C₂×#6923	1	1	yes	characteristic
138	C₂×C₄⋊C₄	—	61	2	no	—
139	C₂×C₄⋊C₄	C₂×#6923	1	2	yes	—
140	C₂×C₄²	—	61	1	no	—
141	C₂×C₄²	—	122	2	no	—
142	#106	—	122	4	no	—
143	C₄⋊C₈	—	122	2	no	—
144	C₄⋊C₈	—	122	2	no	—
145	C₄⋊C₈	—	244	4	no	—
146	C₄×C₈	—	61	2	no	—
147	C₄×C₈	—	61	2	no	—
148	C₄×C₈	—	61	2	no	—
149	C₄×C₈	—	122	1	no	—
150	C₄×C₈	—	122	1	no	—
151	C₄×C₈	—	122	1	no	—
152	#95	—	122	2	no	—
153	C₂²×C₄	—	2	2	no	—
154	C₂²×C₄	—	61	1	no	—
155	C₂²×C₄	—	61	1	no	—
156	C₂²×C₄	—	122	2	no	—
157	C₂²×C₄	C₂²×#6923	1	1	yes	characteristic
158	C₂×C₈	—	2	2	no	—
159	C₂×C₈	—	61	2	no	—
160	C₂×C₈	—	122	2	no	—
161	C₂×C₈	—	122	2	no	—
162	C₂×C₈	—	122	2	no	—
163	C₂×C₈	—	122	2	no	—
164	C₂×C₈	—	122	1	no	—
165	C₂×C₈	—	122	1	no	—
166	C₂×C₈	—	122	2	no	—
167	C₂×C₈	—	122	2	no	—
168	C₂×C₈	—	244	2	no	—
169	C₂×C₈	—	244	2	no	—
170	C₂×C₈	—	244	2	no	—
171	C₂×C₈	C₄⋊#6923	1	2	yes	—
172	C₄⋊C₄	—	2	2	no	—
173	C₄⋊C₄	—	2	2	no	—
174	C₄⋊C₄	—	61	4	no	—
175	C₄⋊C₄	—	122	2	no	—
176	C₄²	—	61	1	no	—
177	C₄²	—	61	1	no	—
178	C₄²	—	61	1	no	—
179	C₄²	—	61	1	no	—
180	C₄²	—	244	2	no	—
181	C₄²	—	244	2	no	—
182	C₂³	D₄×#6923	1	1	yes	characteristic
183	C₂×C₄	—	4	2	no	—
184	C₂×C₄	—	4	2	no	—
185	C₂×C₄	—	4	2	no	—
186	C₂×C₄	—	61	1	no	—
187	C₂×C₄	—	61	1	no	—
188	C₂×C₄	—	61	2	no	—
189	C₂×C₄	—	61	2	no	—
190	C₂×C₄	—	61	1	no	—
191	C₂×C₄	—	61	1	no	—
192	C₂×C₄	—	122	1	no	—
193	C₂×C₄	—	122	1	no	—
194	C₂×C₄	—	122	4	no	—
195	C₂×C₄	—	244	2	no	—
196	C₂×C₄	—	244	2	no	—
197	C₂×C₄	C₂×C₄⋊#6923	1	1	yes	characteristic
198	C₂×C₄	Q₈×#6923	1	1	yes	characteristic
199	C₂×C₄	D₆₁⋊M₁₆	1	2	yes	—
200	C₂×C₄	#14519864	1	2	yes	—
201	C₈	—	2	2	no	—
202	C₈	—	122	4	no	—
203	C₈	—	122	2	no	—
204	C₈	—	488	2	no	—
205	C₈	—	244	1	no	—
206	C₈	—	244	1	no	—
207	C₂²	#16840713	1	1	yes	characteristic
208	C₂²	#16840794	1	2	yes	—
209	C₂²	#16840801	1	2	yes	—
210	C₂²	#16840804	1	1	yes	characteristic
211	C₂²	#16840806	1	1	yes	characteristic
212	C₄	—	8	2	no	—
213	C₄	—	61	1	no	—
214	C₄	—	61	1	no	—
215	C₄	—	61	2	no	—
216	C₄	—	122	1	no	—
217	C₄	—	122	1	no	—
218	C₄	—	244	4	no	—
219	C₄	#16840767	1	1	yes	characteristic
220	C₄	#16840768	1	1	yes	characteristic
221	C₄	#16840814	1	2	yes	—
222	C₂	#21461437	1	2	yes	—
223	C₂	#21461449	1	2	yes	—
224	C₂	#21461920	1	1	yes	characteristic
225	C₂	#21461925	1	1	yes	characteristic
226	C₂	#21461927	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21461917

C₂

yes

maximal

C₂×#16840625

C₂

yes

maximal, characteristic

C₂×#92420⋊C₈

C₂

yes

maximal, characteristic

#21460655

C₂

yes

maximal

#21445707

C₂

yes

maximal, characteristic

C₂×#14519844

—

C₂×C₄⋊#92420

C₂²

yes

—

C₂×C₄×#92420

C₂²

yes

characteristic

C₂×C₄×#92420

C₂²

yes

—

—

—

C₄

yes

—

C₂×C₈×Dic₆₁

C₂²

yes

characteristic

C₄⋊C₄×Dic₆₁

C₂²

yes

—

#16839525

C₄

yes

characteristic

C₆₁×#340

C₄

yes

characteristic

C₂²×#92420

—

C₂²×#92420

—

C₂²×#92420

D₄

yes

characteristic

C₂²×#92420

D₄

yes

—

—

—

—

—

—

Q₈

yes

—

C₂²×#92418

C₂×C₄

yes

characteristic

C₂×C₄⋊Dic₆₁

C₂×C₄

yes

—

C₂×C₄×Dic₆₁

—

C₂×C₄×Dic₆₁

C₂³

yes

characteristic

#14519708

—

C₈×Dic₆₁

D₄

yes

—

C₈×Dic₆₁

Q₈

yes

—

#14519684

—

C₂²×C₈×C₆₁

C₂×C₄

yes

characteristic

C₂×C₄⋊C₄×C₆₁

C₂×C₄

yes

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₂²×Dic₆₁

—

C₂²×Dic₆₁

C₂×D₄

yes

C₂²×Dic₆₁

yes

—

—

—

—

—

—

—

—

—

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

C₄⋊C₄

yes

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

M₁₆

yes

—

C₄×Dic₆₁

—

C₄×Dic₆₁

—

C₄×Dic₆₁

C₂×D₄

yes

characteristic

C₄×Dic₆₁

C₂×Q₈

yes

characteristic

C₄×Dic₆₁

C₂×Q₈

yes

C₄×Dic₆₁

yes

—

C₄⋊C₄

yes

—

C₂³×C₆₁

C₄×D₄

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₁

C₄×Q₈

yes

characteristic

C₂×C₄×C₆₁

C₂×M₁₆

yes

—

C₂×C₄×C₆₁

#131

yes

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

#121

yes

characteristic

C₂×Dic₆₁

C₂²⋊Q₈

yes

characteristic

C₂×Dic₆₁

#125

yes

characteristic

C₂×Dic₆₁

C₄⋊Q₈

yes

characteristic

C₂×Dic₆₁

AGL(1,ℤ/8)

yes

—

C₂×Dic₆₁

#137

yes

—

—

—

—

—

C₈×C₆₁

—

—

—

C₂²×C₆₁

yes

C₂²×C₆₁

yes

—

C₂²×C₆₁

#440

yes

—

C₂²×C₆₁

yes

C₂²×C₆₁

yes

C₄×C₆₁

—

100

C₄×C₆₁

#423

yes

characteristic

101

C₄×C₆₁

#424

yes

characteristic

102

C₄×C₆₁

#446

yes

—

103

Dic₆₁

—

104

Dic₆₁

—

105

Dic₆₁

—

106

Dic₆₁

#469

yes

characteristic

107

Dic₆₁

#470

yes

characteristic

108

Dic₆₁

#501

yes

—

109

#1939

—

110

C₄×#106

—

111

C₂×#335

—

112

C₂×#322

—

113

#1279

—

114

C₂×C₆₁

#1585

yes

—

115

C₂×C₆₁

#1588

yes

—

116

C₂×C₆₁

#1940

yes

characteristic

117

C₂×C₆₁

#1944

yes

characteristic

118

C₂×C₆₁

#1945

yes

characteristic

119

C₂×#106

—

120

C₂×C₄⋊C₈

—

121

C₂×C₄⋊C₈

—

122

—

122

C₂×C₄×C₈

—

123

C₂×C₄×C₈

—

124

C₂×C₄×C₈

—

125

C₄×C₄⋊C₄

—

126

#340

—

127

yes

128

—

122

—

129

#335

—

122

—

130

#322

—

122

—

131

C₆₁

#9600

yes

characteristic, complemented, Sylow

132

C₂²×C₈

—

133

C₂²×C₈

—

134

C₂²×C₈

—

135

C₂²×C₈

—

122

—

136

C₂²×C₈

—

122

—

137

C₂²×C₈

C₂×#6923

yes

characteristic

138

C₂×C₄⋊C₄

—

139

C₂×C₄⋊C₄

C₂×#6923

yes

—

140

C₂×C₄²

—

141

C₂×C₄²

—

122

—

142

#106

—

122

—

143

C₄⋊C₈

—

122

—

144

C₄⋊C₈

—

122

—

145

C₄⋊C₈

—

244

—

146

C₄×C₈

—

147

C₄×C₈

—

148

C₄×C₈

—

149

C₄×C₈

—

122

—

150

C₄×C₈

—

122

—

151

C₄×C₈

—

122

—

152

#95

—

122

—

153

C₂²×C₄

—

154

C₂²×C₄

—

155

C₂²×C₄

—

156

C₂²×C₄

—

122

—

157

C₂²×C₄

C₂²×#6923

yes

characteristic

158

C₂×C₈

—

159

C₂×C₈

—

160

C₂×C₈

—

122

—

161

C₂×C₈

—

122

—

162

C₂×C₈

—

122

—

163

C₂×C₈

—

122

—

164

C₂×C₈

—

122

—

165

C₂×C₈

—

122

—

166

C₂×C₈

—

122

—

167

C₂×C₈

—

122

—

168

C₂×C₈

—

244

—

169

C₂×C₈

—

244

—

170

C₂×C₈

—

244

—

171

C₂×C₈

C₄⋊#6923

yes

—

172

C₄⋊C₄

—

173

C₄⋊C₄

—

174

C₄⋊C₄

—

175

C₄⋊C₄

—

122

—

176

C₄²

—

177

C₄²

—

178

C₄²

—

179

C₄²

—

180

C₄²

—

244

—

181

C₄²

—

244

—

182

C₂³

D₄×#6923

yes

characteristic

183

C₂×C₄

—

184

C₂×C₄

—

185

C₂×C₄

—

186

C₂×C₄

—

187

C₂×C₄

—

188

C₂×C₄

—

189

C₂×C₄

—

190

C₂×C₄

—

191

C₂×C₄

—

192

C₂×C₄

—

122

—

193

C₂×C₄

—

122

—

194

C₂×C₄

—

122

—

195

C₂×C₄

—

244

—

196

C₂×C₄

—

244

—

197

C₂×C₄

C₂×C₄⋊#6923

yes

characteristic

198

C₂×C₄

Q₈×#6923

yes

characteristic

199

C₂×C₄

D₆₁⋊M₁₆

yes

—

200

C₂×C₄

#14519864

yes

—

201

C₈

—

202

C₈

—

122

—

203

C₈

—

122

—

204

C₈

—

488

—

205

C₈

—

244

—

206

C₈

—

244

—

207

C₂²

#16840713

yes

characteristic

208

C₂²

#16840794

yes

—

209

C₂²

#16840801

yes

—

210

C₂²

#16840804

yes

characteristic

211

C₂²

#16840806

yes

characteristic

212

C₄

—

213

C₄

—

214

C₄

—

215

C₄

—

216

C₄

—

122

—

217

C₄

—

122

—

218

C₄

—

244

—

219

C₄

#16840767

yes

characteristic

220

C₄

#16840768

yes

characteristic

221

C₄

#16840814

yes

—

222

C₂

#21461437

yes

—

223

C₂

#21461449

yes

—

224

C₂

#21461920

yes

characteristic

225

C₂

#21461925

yes

characteristic

226

C₂

#21461927

yes

characteristic

	subgroup	G/H
center	C₂³	D₄×#6923
commutator	C₂²×C₄×C₆₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×#6923
Fitting	C₆₁×#340	C₄
socle, prime radical	C₂³×C₆₁	C₄×D₄
cosocle	C₂×C₄×Dic₆₁	C₂³
hypercenter	#340	#6923
hypocenter	C₆₁	#9600
LCS	G ▷ C₂²×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ #340

subgroup

G/H

center

C₂³

D₄×#6923

commutator

C₂²×C₄×C₆₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×#6923

Fitting

C₆₁×#340

C₄

socle, prime radical

C₂³×C₆₁

C₄×D₄

cosocle

C₂×C₄×Dic₆₁

C₂³

C₆₁

G ▷ C₂²×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ #340

N	H ≅ G/N	type
131. C₆₁	#9600	semidirect

H ≅ G/N

type

131. C₆₁

#9600

semidirect

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

subgroup	index
#21461917	2
C₂×#16840625	2
C₂×#92420⋊C₈	2
#21460655	2
#21445707	2
#9600	61

subgroup

index

C₂×#92420⋊C₈

Subgroups — #30601523

Conjugacy classes of subgroups

Complements

Maximal subgroups