Group #30601514 — subgroups

Subgroups — #30601514

Order 15616.

378 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 124 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×#16840623	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	#16840623	—	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	2	yes	—
21	#14519844	—	4	4	no	—
22	C₄⋊#92420	—	2	2	no	—
23	C₄⋊#92420	—	2	2	no	—
24	C₄×#92420	—	2	1	no	—
25	C₄×#92420	—	2	1	no	—
26	C₄×#92420	—	2	1	no	—
27	C₄×#92420	Q₈	1	2	yes	—
28	C₂²×#92418	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄⋊Dic₆₁	C₂×C₄	1	2	yes	—
30	C₂×C₄×Dic₆₁	—	2	2	no	—
31	C₂×C₄×Dic₆₁	C₂³	1	1	yes	characteristic
32	#14519708	—	2	4	no	—
33	C₈×Dic₆₁	D₄	1	2	yes	—
34	C₈×Dic₆₁	Q₈	1	2	yes	—
35	#14519684	—	2	2	no	—
36	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	2	yes	—
38	C₂²×C₄×C₆₁	—	2	2	no	—
39	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
40	C₂²×Dic₆₁	—	2	2	no	—
41	C₂²×Dic₆₁	C₂×D₄	1	1	yes	characteristic
42	C₂²×Dic₆₁	#41	1	1	yes	characteristic
43	C₂×#92420	—	2	2	no	—
44	C₂×#92420	—	2	2	no	—
45	C₂×#92420	—	2	2	no	—
46	C₂×#92420	—	4	1	no	—
47	C₂×#92420	—	4	1	no	—
48	C₂×#92420	—	4	2	no	—
49	C₂×#92420	—	4	2	no	—
50	C₂×#92420	—	4	2	no	—
51	C₂×#92420	SD₁₆	1	4	yes	—
52	C₂×C₈×C₆₁	—	2	2	no	—
53	C₂×C₈×C₆₁	C₄⋊C₄	1	2	yes	—
54	C₄⋊C₄×C₆₁	—	2	2	no	—
55	C₄⋊C₄×C₆₁	—	2	2	no	—
56	C₄⋊Dic₆₁	—	2	2	no	—
57	C₄⋊Dic₆₁	M₁₆	1	4	yes	—
58	C₄×Dic₆₁	—	4	2	no	—
59	C₄×Dic₆₁	—	4	2	no	—
60	C₄×Dic₆₁	C₂×D₄	1	1	yes	characteristic
61	C₄×Dic₆₁	C₂×Q₈	1	1	yes	characteristic
62	C₄×Dic₆₁	C₂×Q₈	1	1	yes	characteristic
63	C₄×Dic₆₁	#41	1	1	yes	characteristic
64	C₂×#92418	—	2	2	no	—
65	C₂×#92418	C₄⋊C₄	1	2	yes	—
66	C₂³×C₆₁	C₄×D₄	1	1	yes	characteristic
67	C₂×C₄×C₆₁	—	4	2	no	—
68	C₂×C₄×C₆₁	—	4	2	no	—
69	C₂×C₄×C₆₁	—	4	2	no	—
70	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
71	C₂×C₄×C₆₁	C₄×Q₈	1	1	yes	characteristic
72	C₂×C₄×C₆₁	C₂×M₁₆	1	2	yes	—
73	C₂×C₄×C₆₁	#131	1	2	yes	—
74	C₂×Dic₆₁	—	2	1	no	—
75	C₂×Dic₆₁	—	2	1	no	—
76	C₂×Dic₆₁	—	2	4	no	—
77	C₂×Dic₆₁	—	4	2	no	—
78	C₂×Dic₆₁	—	4	2	no	—
79	C₂×Dic₆₁	#121	1	1	yes	characteristic
80	C₂×Dic₆₁	C₂²⋊Q₈	1	1	yes	characteristic
81	C₂×Dic₆₁	#125	1	1	yes	characteristic
82	C₂×Dic₆₁	C₄⋊Q₈	1	1	yes	characteristic
83	C₂×Dic₆₁	C₂×SD₁₆	1	2	yes	—
84	C₂×Dic₆₁	#135	1	2	yes	—
85	#92420	—	2	4	no	—
86	#92420	—	4	2	no	—
87	#92420	—	8	2	no	—
88	C₈×C₆₁	—	2	2	no	—
89	#92418	—	2	2	no	—
90	#9597	—	61	1	no	maximal, Sylow
91	C₂²×C₆₁	#389	1	1	yes	characteristic
92	C₂²×C₆₁	#435	1	2	yes	—
93	C₂²×C₆₁	C₄×SD₁₆	1	2	yes	—
94	C₂²×C₆₁	#444	1	1	yes	characteristic
95	C₂²×C₆₁	#444	1	1	yes	characteristic
96	C₄×C₆₁	—	8	2	no	—
97	C₄×C₆₁	#423	1	1	yes	characteristic
98	C₄×C₆₁	#424	1	1	yes	characteristic
99	C₄×C₆₁	#446	1	2	yes	—
100	Dic₆₁	—	2	1	no	—
101	Dic₆₁	—	2	1	no	—
102	Dic₆₁	—	4	4	no	—
103	Dic₆₁	#465	1	2	yes	—
104	Dic₆₁	#498	1	1	yes	characteristic
105	Dic₆₁	#499	1	1	yes	characteristic
106	#1939	—	61	2	no	—
107	C₄×#106	—	61	1	no	—
108	C₂×#334	—	61	1	no	—
109	C₂×#322	—	61	1	no	—
110	#1279	—	61	2	no	—
111	C₂×C₆₁	#1582	1	2	yes	—
112	C₂×C₆₁	#1583	1	2	yes	—
113	C₂×C₆₁	#1940	1	1	yes	characteristic
114	C₂×C₆₁	#1941	1	1	yes	characteristic
115	C₂×C₆₁	#1946	1	1	yes	characteristic
116	C₂×#106	—	61	2	no	—
117	C₂×C₄⋊C₈	—	61	2	no	—
118	C₂×C₄⋊C₈	—	122	2	no	—
119	C₂×C₄×C₈	—	61	1	no	—
120	C₂×C₄×C₈	—	61	1	no	—
121	C₂×C₄×C₈	—	61	1	no	—
122	C₄×C₄⋊C₄	—	61	2	no	—
123	#340	—	61	1	no	—
124	#340	#6923	1	1	yes	characteristic
125	#336	—	122	2	no	—
126	#334	—	122	2	no	—
127	#322	—	122	2	no	—
128	C₆₁	#9597	1	1	yes	characteristic, complemented, Sylow
129	C₂²×C₈	—	61	2	no	—
130	C₂²×C₈	—	61	1	no	—
131	C₂²×C₈	—	122	1	no	—
132	C₂²×C₈	—	122	2	no	—
133	C₂²×C₈	C₂×#6923	1	1	yes	characteristic
134	C₂×C₄⋊C₄	—	61	2	no	—
135	C₂×C₄⋊C₄	C₂×#6923	1	2	yes	—
136	C₂×C₄²	—	61	1	no	—
137	C₂×C₄²	—	122	2	no	—
138	#106	—	122	4	no	—
139	C₄⋊C₈	—	122	2	no	—
140	C₄⋊C₈	—	122	2	no	—
141	C₄⋊C₈	—	244	4	no	—
142	C₄×C₈	—	61	2	no	—
143	C₄×C₈	—	61	2	no	—
144	C₄×C₈	—	61	2	no	—
145	C₄×C₈	—	122	1	no	—
146	C₄×C₈	—	122	1	no	—
147	C₄×C₈	—	122	1	no	—
148	#95	—	122	2	no	—
149	C₂²×C₄	—	2	2	no	—
150	C₂²×C₄	—	61	1	no	—
151	C₂²×C₄	—	61	1	no	—
152	C₂²×C₄	—	122	2	no	—
153	C₂²×C₄	C₂²×#6923	1	1	yes	characteristic
154	C₂×C₈	—	2	2	no	—
155	C₂×C₈	—	61	4	no	—
156	C₂×C₈	—	61	2	no	—
157	C₂×C₈	—	122	2	no	—
158	C₂×C₈	—	122	2	no	—
159	C₂×C₈	—	122	2	no	—
160	C₂×C₈	—	122	2	no	—
161	C₂×C₈	—	244	2	no	—
162	C₂×C₈	—	244	2	no	—
163	C₂×C₈	—	244	2	no	—
164	C₂×C₈	—	244	1	no	—
165	C₂×C₈	—	244	1	no	—
166	C₂×C₈	C₄⋊#6923	1	2	yes	—
167	C₄⋊C₄	—	2	2	no	—
168	C₄⋊C₄	—	2	2	no	—
169	C₄⋊C₄	—	61	4	no	—
170	C₄⋊C₄	—	122	2	no	—
171	C₄²	—	61	1	no	—
172	C₄²	—	61	1	no	—
173	C₄²	—	61	1	no	—
174	C₄²	—	61	1	no	—
175	C₄²	—	244	2	no	—
176	C₄²	—	244	2	no	—
177	C₂³	D₄×#6923	1	1	yes	characteristic
178	C₂×C₄	—	4	2	no	—
179	C₂×C₄	—	4	2	no	—
180	C₂×C₄	—	4	2	no	—
181	C₂×C₄	—	61	1	no	—
182	C₂×C₄	—	61	1	no	—
183	C₂×C₄	—	61	2	no	—
184	C₂×C₄	—	61	2	no	—
185	C₂×C₄	—	61	1	no	—
186	C₂×C₄	—	61	1	no	—
187	C₂×C₄	—	122	1	no	—
188	C₂×C₄	—	122	1	no	—
189	C₂×C₄	—	122	4	no	—
190	C₂×C₄	—	244	2	no	—
191	C₂×C₄	—	244	2	no	—
192	C₂×C₄	C₂×C₄⋊#6923	1	1	yes	characteristic
193	C₂×C₄	Q₈×#6923	1	1	yes	characteristic
194	C₂×C₄	D₆₁⋊M₁₆	1	2	yes	—
195	C₂×C₄	#14519864	1	2	yes	—
196	C₈	—	2	2	no	—
197	C₈	—	122	4	no	—
198	C₈	—	122	2	no	—
199	C₈	—	488	2	no	—
200	C₈	—	244	2	no	—
201	C₂²	#16840713	1	1	yes	characteristic
202	C₂²	#16840794	1	2	yes	—
203	C₂²	SD₁₆×#6923	1	2	yes	—
204	C₂²	#16840808	1	1	yes	characteristic
205	C₂²	#16840809	1	1	yes	characteristic
206	C₄	—	8	2	no	—
207	C₄	—	61	2	no	—
208	C₄	—	61	1	no	—
209	C₄	—	61	1	no	—
210	C₄	—	122	1	no	—
211	C₄	—	122	1	no	—
212	C₄	—	244	4	no	—
213	C₄	#16840767	1	1	yes	characteristic
214	C₄	#16840768	1	1	yes	characteristic
215	C₄	#16840814	1	2	yes	—
216	C₂	#21461425	1	2	yes	—
217	C₂	#21461429	1	2	yes	—
218	C₂	#21461920	1	1	yes	characteristic
219	C₂	#21461922	1	1	yes	characteristic
220	C₂	#21461928	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21461917

C₂

yes

maximal

C₂×#16840623

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

—

—

—

—

—

—

—

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

—

—

—

—

—

—

—

—

SD₁₆

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

C₂×SD₁₆

yes

—

C₂×Dic₆₁

#135

yes

—

#92420

—

#92420

—

#92420

—

C₈×C₆₁

—

—

—

C₂²×C₆₁

yes

C₂²×C₆₁

yes

—

C₂²×C₆₁

C₄×SD₁₆

yes

—

C₂²×C₆₁

yes

C₂²×C₆₁

yes

C₄×C₆₁

—

C₄×C₆₁

yes

C₄×C₆₁

yes

C₄×C₆₁

yes

—

100

Dic₆₁

—

101

Dic₆₁

—

102

Dic₆₁

—

103

Dic₆₁

#465

yes

—

104

Dic₆₁

#498

yes

characteristic

105

Dic₆₁

#499

yes

characteristic

106

#1939

—

107

C₄×#106

—

108

C₂×#334

—

109

C₂×#322

—

110

#1279

—

111

C₂×C₆₁

#1582

yes

—

112

C₂×C₆₁

#1583

yes

—

113

C₂×C₆₁

#1940

yes

characteristic

114

C₂×C₆₁

#1941

yes

characteristic

115

C₂×C₆₁

#1946

yes

characteristic

116

C₂×#106

—

117

C₂×C₄⋊C₈

—

118

C₂×C₄⋊C₈

—

122

—

119

C₂×C₄×C₈

—

120

C₂×C₄×C₈

—

121

C₂×C₄×C₈

—

122

C₄×C₄⋊C₄

—

123

#340

—

124

yes

125

—

122

—

126

#334

—

122

—

127

#322

—

122

—

128

C₆₁

#9597

yes

characteristic, complemented, Sylow

129

C₂²×C₈

—

130

C₂²×C₈

—

131

C₂²×C₈

—

122

—

132

C₂²×C₈

—

122

—

133

C₂²×C₈

C₂×#6923

yes

characteristic

134

C₂×C₄⋊C₄

—

135

C₂×C₄⋊C₄

C₂×#6923

yes

—

136

C₂×C₄²

—

137

C₂×C₄²

—

122

—

138

#106

—

122

—

139

C₄⋊C₈

—

122

—

140

C₄⋊C₈

—

122

—

141

C₄⋊C₈

—

244

—

142

C₄×C₈

—

143

C₄×C₈

—

144

C₄×C₈

—

145

C₄×C₈

—

122

—

146

C₄×C₈

—

122

—

147

C₄×C₈

—

122

—

148

#95

—

122

—

149

C₂²×C₄

—

150

C₂²×C₄

—

151

C₂²×C₄

—

152

C₂²×C₄

—

122

—

153

C₂²×C₄

C₂²×#6923

yes

characteristic

154

C₂×C₈

—

155

C₂×C₈

—

156

C₂×C₈

—

157

C₂×C₈

—

122

—

158

C₂×C₈

—

122

—

159

C₂×C₈

—

122

—

160

C₂×C₈

—

122

—

161

C₂×C₈

—

244

—

162

C₂×C₈

—

244

—

163

C₂×C₈

—

244

—

164

C₂×C₈

—

244

—

165

C₂×C₈

—

244

—

166

C₂×C₈

C₄⋊#6923

yes

—

167

C₄⋊C₄

—

168

C₄⋊C₄

—

169

C₄⋊C₄

—

170

C₄⋊C₄

—

122

—

171

C₄²

—

172

C₄²

—

173

C₄²

—

174

C₄²

—

175

C₄²

—

244

—

176

C₄²

—

244

—

177

C₂³

D₄×#6923

yes

characteristic

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

—

181

C₂×C₄

—

182

C₂×C₄

—

183

C₂×C₄

—

184

C₂×C₄

—

185

C₂×C₄

—

186

C₂×C₄

—

187

C₂×C₄

—

122

—

188

C₂×C₄

—

122

—

189

C₂×C₄

—

122

—

190

C₂×C₄

—

244

—

191

C₂×C₄

—

244

—

192

C₂×C₄

C₂×C₄⋊#6923

yes

characteristic

193

C₂×C₄

Q₈×#6923

yes

characteristic

194

C₂×C₄

D₆₁⋊M₁₆

yes

—

195

C₂×C₄

#14519864

yes

—

196

C₈

—

197

C₈

—

122

—

198

C₈

—

122

—

199

C₈

—

488

—

200

C₈

—

244

—

201

C₂²

#16840713

yes

characteristic

202

C₂²

#16840794

yes

—

203

C₂²

SD₁₆×#6923

yes

—

204

C₂²

#16840808

yes

characteristic

205

C₂²

#16840809

yes

characteristic

206

C₄

—

207

C₄

—

208

C₄

—

209

C₄

—

210

C₄

—

122

—

211

C₄

—

122

—

212

C₄

—

244

—

213

C₄

#16840767

yes

characteristic

214

C₄

#16840768

yes

characteristic

215

C₄

#16840814

yes

—

216

C₂

#21461425

yes

—

217

C₂

#21461429

yes

—

218

C₂

#21461920

yes

characteristic

219

C₂

#21461922

yes

characteristic

220

C₂

#21461928

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	#9597
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
128. C₆₁	#9597	semidirect

H ≅ G/N

type

128. C₆₁

#9597

semidirect

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

subgroup	index
#21461917	2
C₂×#16840623	2
C₂×#92420⋊C₈	2
#21460655	2
#21445707	2
#9597	61

subgroup

index

C₂×#92420⋊C₈

Subgroups — #30601514

Conjugacy classes of subgroups

Complements

Maximal subgroups