Group #30601519 — subgroups

Subgroups — #30601519

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	#21461916	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	#21445711	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#92420	—	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×#14519710	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	C₄⋊#92420	—	2	2	no	—
22	C₄⋊#92420	—	2	2	no	—
23	C₄⋊#92420	—	4	4	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	#14519710	—	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	4	no	—
44	C₂×#92420	—	2	2	no	—
45	C₂×#92420	—	2	2	no	—
46	C₂×#92420	—	2	2	no	—
47	C₂×#92420	—	2	2	no	—
48	C₂×#92420	—	4	2	no	—
49	C₂×#92420	—	4	2	no	—
50	C₂×#92420	—	4	2	no	—
51	C₂×C₈×C₆₁	—	2	2	no	—
52	C₂×C₈×C₆₁	C₄⋊C₄	1	2	yes	—
53	C₄⋊C₄×C₆₁	—	2	2	no	—
54	C₄⋊C₄×C₆₁	M₁₆	1	4	yes	—
55	C₄⋊Dic₆₁	—	2	2	no	—
56	C₄⋊Dic₆₁	—	2	2	no	—
57	C₄×Dic₆₁	—	4	2	no	—
58	C₄×Dic₆₁	—	4	2	no	—
59	C₄×Dic₆₁	C₂×D₄	1	1	yes	characteristic
60	C₄×Dic₆₁	C₂×Q₈	1	1	yes	characteristic
61	C₄×Dic₆₁	C₂×Q₈	1	1	yes	characteristic
62	C₄×Dic₆₁	#41	1	1	yes	characteristic
63	C₂×#92418	—	2	2	no	—
64	C₂×#92418	C₄⋊C₄	1	2	yes	—
65	C₂³×C₆₁	C₄×D₄	1	1	yes	characteristic
66	C₂×C₄×C₆₁	—	2	4	no	—
67	C₂×C₄×C₆₁	—	4	2	no	—
68	C₂×C₄×C₆₁	—	4	2	no	—
69	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
70	C₂×C₄×C₆₁	C₄×Q₈	1	1	yes	characteristic
71	C₂×C₄×C₆₁	C₂×M₁₆	1	2	yes	—
72	C₂×C₄×C₆₁	#131	1	2	yes	—
73	C₂×Dic₆₁	—	2	2	no	—
74	C₂×Dic₆₁	—	4	2	no	—
75	C₂×Dic₆₁	—	4	2	no	—
76	C₂×Dic₆₁	—	4	2	no	—
77	C₂×Dic₆₁	#121	1	1	yes	characteristic
78	C₂×Dic₆₁	C₂²⋊Q₈	1	1	yes	characteristic
79	C₂×Dic₆₁	#125	1	1	yes	characteristic
80	C₂×Dic₆₁	C₄⋊Q₈	1	1	yes	characteristic
81	C₂×Dic₆₁	AGL(1,ℤ/8)	1	2	yes	—
82	C₂×Dic₆₁	#137	1	2	yes	—
83	#92420	—	2	4	no	—
84	#92420	—	4	2	no	—
85	#92420	—	8	2	no	—
86	C₈×C₆₁	—	2	2	no	—
87	#92418	—	2	2	no	—
88	#9599	—	61	1	no	maximal, Sylow
89	C₂²×C₆₁	#389	1	1	yes	characteristic
90	C₂²×C₆₁	#435	1	2	yes	—
91	C₂²×C₆₁	#441	1	1	yes	characteristic
92	C₂²×C₆₁	#442	1	1	yes	characteristic
93	C₂²×C₆₁	#443	1	2	yes	—
94	C₄×C₆₁	—	4	4	no	—
95	C₄×C₆₁	#423	1	1	yes	characteristic
96	C₄×C₆₁	#424	1	1	yes	characteristic
97	C₄×C₆₁	#446	1	2	yes	—
98	Dic₆₁	—	2	1	no	—
99	Dic₆₁	—	2	1	no	—
100	Dic₆₁	—	8	2	no	—
101	Dic₆₁	#468	1	2	yes	—
102	Dic₆₁	#501	1	2	yes	—
103	#1939	—	61	2	no	—
104	C₄×#107	—	61	1	no	—
105	C₂×#334	—	61	1	no	—
106	C₂×#322	—	61	1	no	—
107	#1279	—	61	2	no	—
108	C₂×C₆₁	#1586	1	2	yes	—
109	C₂×C₆₁	#1587	1	2	yes	—
110	C₂×C₆₁	#1940	1	1	yes	characteristic
111	C₂×C₆₁	#1943	1	1	yes	characteristic
112	C₂×C₆₁	#1945	1	1	yes	characteristic
113	C₂×#107	—	61	2	no	—
114	C₂×C₄⋊C₈	—	61	2	no	—
115	C₂×C₄⋊C₈	—	122	2	no	—
116	C₂×C₄×C₈	—	61	1	no	—
117	C₂×C₄×C₈	—	61	1	no	—
118	C₂×C₄×C₈	—	61	1	no	—
119	C₄×C₄⋊C₄	—	61	2	no	—
120	#340	—	61	1	no	—
121	#340	#6923	1	1	yes	characteristic
122	#336	—	122	2	no	—
123	#334	—	122	2	no	—
124	#322	—	122	2	no	—
125	C₆₁	#9599	1	1	yes	characteristic, complemented, Sylow
126	C₂²×C₈	—	61	1	no	—
127	C₂²×C₈	—	61	2	no	—
128	C₂²×C₈	—	122	1	no	—
129	C₂²×C₈	—	122	2	no	—
130	C₂²×C₈	C₂×#6923	1	1	yes	characteristic
131	C₂×C₄⋊C₄	—	61	2	no	—
132	C₂×C₄⋊C₄	C₂×#6923	1	2	yes	—
133	C₂×C₄²	—	61	1	no	—
134	C₂×C₄²	—	122	2	no	—
135	#107	—	122	4	no	—
136	C₄⋊C₈	—	122	2	no	—
137	C₄⋊C₈	—	122	2	no	—
138	C₄⋊C₈	—	244	4	no	—
139	C₄×C₈	—	61	2	no	—
140	C₄×C₈	—	61	2	no	—
141	C₄×C₈	—	61	2	no	—
142	C₄×C₈	—	122	1	no	—
143	C₄×C₈	—	122	1	no	—
144	C₄×C₈	—	122	1	no	—
145	#95	—	122	2	no	—
146	C₂²×C₄	—	2	2	no	—
147	C₂²×C₄	—	61	1	no	—
148	C₂²×C₄	—	61	1	no	—
149	C₂²×C₄	—	122	2	no	—
150	C₂²×C₄	C₂²×#6923	1	1	yes	characteristic
151	C₂×C₈	—	2	2	no	—
152	C₂×C₈	—	61	2	no	—
153	C₂×C₈	—	122	2	no	—
154	C₂×C₈	—	122	4	no	—
155	C₂×C₈	—	122	2	no	—
156	C₂×C₈	—	122	2	no	—
157	C₂×C₈	—	122	2	no	—
158	C₂×C₈	—	122	2	no	—
159	C₂×C₈	—	244	2	no	—
160	C₂×C₈	—	244	2	no	—
161	C₂×C₈	—	244	2	no	—
162	C₂×C₈	C₄⋊#6923	1	2	yes	—
163	C₄⋊C₄	—	2	2	no	—
164	C₄⋊C₄	—	122	2	no	—
165	C₄⋊C₄	—	122	2	no	—
166	C₄⋊C₄	#11758213	1	4	yes	—
167	C₄²	—	61	1	no	—
168	C₄²	—	61	1	no	—
169	C₄²	—	61	1	no	—
170	C₄²	—	61	1	no	—
171	C₄²	—	244	2	no	—
172	C₄²	—	244	2	no	—
173	C₂³	D₄×#6923	1	1	yes	characteristic
174	C₂×C₄	—	2	4	no	—
175	C₂×C₄	—	4	2	no	—
176	C₂×C₄	—	4	2	no	—
177	C₂×C₄	—	61	1	no	—
178	C₂×C₄	—	61	1	no	—
179	C₂×C₄	—	61	2	no	—
180	C₂×C₄	—	61	2	no	—
181	C₂×C₄	—	61	1	no	—
182	C₂×C₄	—	61	1	no	—
183	C₂×C₄	—	122	2	no	—
184	C₂×C₄	—	244	2	no	—
185	C₂×C₄	—	244	2	no	—
186	C₂×C₄	—	244	2	no	—
187	C₂×C₄	C₂×C₄⋊#6923	1	1	yes	characteristic
188	C₂×C₄	Q₈×#6923	1	1	yes	characteristic
189	C₂×C₄	C₂×#11758213	1	2	yes	—
190	C₂×C₄	#14519864	1	2	yes	—
191	C₈	—	2	2	no	—
192	C₈	—	122	4	no	—
193	C₈	—	122	2	no	—
194	C₈	—	488	2	no	—
195	C₈	—	244	2	no	—
196	C₂²	#16840713	1	1	yes	characteristic
197	C₂²	#16840791	1	2	yes	—
198	C₂²	#16840802	1	1	yes	characteristic
199	C₂²	#16840803	1	1	yes	characteristic
200	C₂²	#16840805	1	2	yes	—
201	C₄	—	4	4	no	—
202	C₄	—	61	2	no	—
203	C₄	—	61	2	no	—
204	C₄	—	122	1	no	—
205	C₄	—	122	1	no	—
206	C₄	—	488	2	no	—
207	C₄	#16840766	1	1	yes	characteristic
208	C₄	#16840768	1	1	yes	characteristic
209	C₄	#16840812	1	2	yes	—
210	C₂	#21461438	1	2	yes	—
211	C₂	#21461442	1	2	yes	—
212	C₂	#21461919	1	1	yes	characteristic
213	C₂	#21461924	1	1	yes	characteristic
214	C₂	#21461926	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21461916

C₂

yes

maximal

C₂×#16840623

C₂

yes

maximal, characteristic

C₂×#92420⋊C₈

C₂

yes

maximal, characteristic

#21460655

C₂

yes

maximal

#21445711

C₂

yes

maximal, characteristic

C₂×C₄⋊#92420

—

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

#14519710

—

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

M₁₆

yes

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

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

#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

—

#92420

—

#92420

—

#92420

—

C₈×C₆₁

—

—

—

C₂²×C₆₁

yes

C₂²×C₆₁

yes

—

C₂²×C₆₁

yes

C₂²×C₆₁

yes

C₂²×C₆₁

yes

—

C₄×C₆₁

—

C₄×C₆₁

yes

C₄×C₆₁

yes

C₄×C₆₁

yes

—

Dic₆₁

—

Dic₆₁

—

100

Dic₆₁

—

101

Dic₆₁

#468

yes

—

102

Dic₆₁

#501

yes

—

103

#1939

—

104

C₄×#107

—

105

C₂×#334

—

106

C₂×#322

—

107

#1279

—

108

C₂×C₆₁

#1586

yes

—

109

C₂×C₆₁

#1587

yes

—

110

C₂×C₆₁

#1940

yes

characteristic

111

C₂×C₆₁

#1943

yes

characteristic

112

C₂×C₆₁

#1945

yes

characteristic

113

C₂×#107

—

114

C₂×C₄⋊C₈

—

115

C₂×C₄⋊C₈

—

122

—

116

C₂×C₄×C₈

—

117

C₂×C₄×C₈

—

118

C₂×C₄×C₈

—

119

C₄×C₄⋊C₄

—

120

#340

—

121

yes

122

—

122

—

123

#334

—

122

—

124

#322

—

122

—

125

C₆₁

#9599

yes

characteristic, complemented, Sylow

126

C₂²×C₈

—

127

C₂²×C₈

—

128

C₂²×C₈

—

122

—

129

C₂²×C₈

—

122

—

130

C₂²×C₈

C₂×#6923

yes

characteristic

131

C₂×C₄⋊C₄

—

132

C₂×C₄⋊C₄

C₂×#6923

yes

—

133

C₂×C₄²

—

134

C₂×C₄²

—

122

—

135

#107

—

122

—

136

C₄⋊C₈

—

122

—

137

C₄⋊C₈

—

122

—

138

C₄⋊C₈

—

244

—

139

C₄×C₈

—

140

C₄×C₈

—

141

C₄×C₈

—

142

C₄×C₈

—

122

—

143

C₄×C₈

—

122

—

144

C₄×C₈

—

122

—

145

#95

—

122

—

146

C₂²×C₄

—

147

C₂²×C₄

—

148

C₂²×C₄

—

149

C₂²×C₄

—

122

—

150

C₂²×C₄

C₂²×#6923

yes

characteristic

151

C₂×C₈

—

152

C₂×C₈

—

153

C₂×C₈

—

122

—

154

C₂×C₈

—

122

—

155

C₂×C₈

—

122

—

156

C₂×C₈

—

122

—

157

C₂×C₈

—

122

—

158

C₂×C₈

—

122

—

159

C₂×C₈

—

244

—

160

C₂×C₈

—

244

—

161

C₂×C₈

—

244

—

162

C₂×C₈

C₄⋊#6923

yes

—

163

C₄⋊C₄

—

164

C₄⋊C₄

—

122

—

165

C₄⋊C₄

—

122

—

166

C₄⋊C₄

#11758213

yes

—

167

C₄²

—

168

C₄²

—

169

C₄²

—

170

C₄²

—

171

C₄²

—

244

—

172

C₄²

—

244

—

173

C₂³

D₄×#6923

yes

characteristic

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

—

181

C₂×C₄

—

182

C₂×C₄

—

183

C₂×C₄

—

122

—

184

C₂×C₄

—

244

—

185

C₂×C₄

—

244

—

186

C₂×C₄

—

244

—

187

C₂×C₄

C₂×C₄⋊#6923

yes

characteristic

188

C₂×C₄

Q₈×#6923

yes

characteristic

189

C₂×C₄

C₂×#11758213

yes

—

190

C₂×C₄

#14519864

yes

—

191

C₈

—

192

C₈

—

122

—

193

C₈

—

122

—

194

C₈

—

488

—

195

C₈

—

244

—

196

C₂²

#16840713

yes

characteristic

197

C₂²

#16840791

yes

—

198

C₂²

#16840802

yes

characteristic

199

C₂²

#16840803

yes

characteristic

200

C₂²

#16840805

yes

—

201

C₄

—

202

C₄

—

203

C₄

—

204

C₄

—

122

—

205

C₄

—

122

—

206

C₄

—

488

—

207

C₄

#16840766

yes

characteristic

208

C₄

#16840768

yes

characteristic

209

C₄

#16840812

yes

—

210

C₂

#21461438

yes

—

211

C₂

#21461442

yes

—

212

C₂

#21461919

yes

characteristic

213

C₂

#21461924

yes

characteristic

214

C₂

#21461926

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	#9599
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
125. C₆₁	#9599	semidirect

H ≅ G/N

type

125. C₆₁

#9599

semidirect

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

subgroup	index
#21461916	2
C₂×#16840623	2
C₂×#92420⋊C₈	2
#21460655	2
#21445711	2
#9599	61

subgroup

index

C₂×#92420⋊C₈

Subgroups — #30601519

Conjugacy classes of subgroups

Complements

Maximal subgroups