Group #30602144 — subgroups

Subgroups — #30602144

Order 15616.

494 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 152 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	#21461734	C₂	1	1	yes	maximal, characteristic
2	C₄²×#92420	C₂	1	1	yes	maximal, characteristic
3	#21460655	C₂	1	4	yes	maximal
4	#125×Dic₆₁	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊#92420	C₂²	1	2	yes	—
6	C₂×C₄×#92420	—	2	2	no	—
7	C₂×C₄×#92420	C₂²	1	2	yes	—
8	#16840628	—	2	2	no	—
9	#16839661	C₄	1	2	yes	—
10	#16839648	—	2	2	no	—
11	#16839628	C₄	1	1	yes	characteristic
12	C₄⋊C₄×Dic₆₁	—	2	2	no	—
13	C₄⋊C₄×Dic₆₁	C₂²	1	2	yes	—
14	C₄²×Dic₆₁	C₂²	1	1	yes	characteristic
15	C₂×#125×C₆₁	C₄	1	1	yes	characteristic
16	C₂²×#92420	—	2	2	no	—
17	C₂²×#92420	—	2	2	no	—
18	C₄⋊#92420	—	2	4	no	—
19	C₄×#92420	—	2	8	no	—
20	C₄×#92420	—	2	2	no	—
21	C₄×#92420	Q₈	1	4	yes	—
22	C₂×C₄⋊Dic₆₁	—	2	2	no	—
23	C₂×C₄⋊Dic₆₁	C₂×C₄	1	2	yes	—
24	C₂×C₄×Dic₆₁	—	2	2	no	—
25	C₂×C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄×Dic₆₁	D₄	1	2	yes	—
27	C₂×C₄×Dic₆₁	D₄	1	2	yes	—
28	C₂×C₄×Dic₆₁	C₂³	1	1	yes	characteristic
29	#14519684	—	2	2	no	—
30	#14519684	—	2	2	no	—
31	#14519684	—	2	2	no	—
32	#125×C₆₁	—	2	2	no	—
33	#125×C₆₁	C₈	1	4	yes	complemented
34	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
35	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	2	yes	—
36	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₄×C₆₁	—	2	2	no	—
38	C₂²×C₄×C₆₁	C₂²⋊C₄	1	2	yes	—
39	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
40	C₂²×Dic₆₁	—	2	2	no	—
41	C₂²×Dic₆₁	C₂²⋊C₄	1	2	yes	—
42	C₂²×Dic₆₁	C₂×D₄	1	1	yes	characteristic
43	C₂²×Dic₆₁	C₂×D₄	1	1	yes	characteristic
44	C₂×#92420	—	2	8	no	—
45	C₂×#92420	—	2	4	no	—
46	C₂×#92420	—	4	2	no	—
47	C₂×#92420	—	4	4	no	—
48	C₄⋊C₄×C₆₁	—	2	8	no	—
49	C₄⋊C₄×C₆₁	—	2	2	no	—
50	C₄⋊C₄×C₆₁	C₂×C₈	1	4	yes	—
51	C₄²×C₆₁	C₂×C₈	1	2	yes	—
52	C₄²×C₆₁	M₁₆	1	2	yes	—
53	C₄⋊Dic₆₁	—	2	4	no	—
54	C₄⋊Dic₆₁	—	4	4	no	—
55	C₄×Dic₆₁	—	2	2	no	—
56	C₄×Dic₆₁	—	2	4	no	—
57	C₄×Dic₆₁	—	2	4	no	—
58	C₄×Dic₆₁	—	4	4	no	—
59	C₄×Dic₆₁	C₂×Q₈	1	2	yes	—
60	C₄×Dic₆₁	#41	1	2	yes	—
61	C₂³×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
62	C₂×C₄×C₆₁	—	2	2	no	—
63	C₂×C₄×C₆₁	—	2	2	no	—
64	C₂×C₄×C₆₁	—	2	4	no	—
65	C₂×C₄×C₆₁	—	4	2	no	—
66	C₂×C₄×C₆₁	—	4	2	no	—
67	C₂×C₄×C₆₁	C₂²⋊C₈	1	4	yes	—
68	C₂×C₄×C₆₁	C₄×Q₈	1	2	yes	—
69	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
70	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
71	C₂×C₄×C₆₁	#131	1	2	yes	—
72	C₂×Dic₆₁	—	2	2	no	—
73	C₂×Dic₆₁	—	2	2	no	—
74	C₂×Dic₆₁	—	2	2	no	—
75	C₂×Dic₆₁	—	4	2	no	—
76	C₂×Dic₆₁	—	4	2	no	—
77	C₂×Dic₆₁	—	4	2	no	—
78	C₂×Dic₆₁	#104	1	4	yes	—
79	C₂×Dic₆₁	C₂²⋊Q₈	1	2	yes	—
80	C₂×Dic₆₁	#124	1	1	yes	characteristic
81	C₂×Dic₆₁	C₄⋊Q₈	1	1	yes	characteristic
82	C₂×Dic₆₁	#135	1	4	yes	—
83	#92420	—	2	8	no	—
84	#92420	—	8	2	no	—
85	#9871	—	61	1	no	maximal, Sylow
86	C₂²×C₆₁	#391	1	1	yes	characteristic
87	C₂²×C₆₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
88	C₂²×C₆₁	#408	1	1	yes	characteristic
89	C₂²×C₆₁	#416	1	2	yes	—
90	C₂²×C₆₁	C₂×#104	1	2	yes	—
91	C₄×C₆₁	—	2	4	no	—
92	C₄×C₆₁	—	4	4	no	—
93	C₄×C₆₁	C₈×Q₈	1	2	yes	—
94	C₄×C₆₁	#446	1	2	yes	—
95	Dic₆₁	—	2	2	no	—
96	Dic₆₁	—	2	4	no	—
97	Dic₆₁	—	8	2	no	—
98	Dic₆₁	#488	1	2	yes	—
99	Dic₆₁	#499	1	2	yes	—
100	C₄×#125	—	61	1	no	—
101	#1831	—	61	1	no	—
102	C₄²×C₈	—	61	1	no	—
103	#1279	—	61	4	no	—
104	C₂×C₆₁	#1476	1	2	yes	—
105	C₂×C₆₁	#1494	1	2	yes	—
106	C₂×C₆₁	#1984	1	1	yes	characteristic
107	C₂×C₆₁	#1991	1	1	yes	characteristic
108	C₂×C₆₁	#1994	1	1	yes	characteristic
109	C₂×#125	#6923	1	1	yes	characteristic
110	C₂×C₄⋊C₈	—	61	2	no	—
111	C₂×C₄×C₈	—	61	2	no	—
112	C₂×C₄×C₈	—	122	2	no	—
113	#389	—	61	2	no	—
114	#387	—	122	2	no	—
115	#382	—	61	1	no	—
116	C₄×C₄⋊C₄	—	61	2	no	—
117	C₄×C₄⋊C₄	—	122	2	no	—
118	C₄³	—	61	1	no	—
119	#336	—	122	2	no	—
120	C₆₁	#9871	1	1	yes	characteristic, complemented, Sylow
121	C₂²×C₈	—	122	2	no	—
122	C₂²×C₈	—	122	2	no	—
123	#125	—	2	2	no	—
124	#125	#92420	1	4	yes	complemented
125	C₂×C₄⋊C₄	—	2	2	no	—
126	C₂×C₄⋊C₄	—	61	2	no	—
127	C₂×C₄⋊C₄	—	122	2	no	—
128	C₂×C₄⋊C₄	C₂×#6923	1	2	yes	—
129	C₂×C₄²	—	61	1	no	—
130	C₂×C₄²	—	61	2	no	—
131	C₂×C₄²	—	61	2	no	—
132	C₂×C₄²	—	61	1	no	—
133	C₂×C₄²	—	122	2	no	—
134	C₂×C₄²	C₂×#6923	1	1	yes	characteristic
135	C₄⋊C₈	—	122	4	no	—
136	C₄×C₈	—	61	4	no	—
137	C₄×C₈	—	122	2	no	—
138	C₄×C₈	—	122	8	no	—
139	#95	—	122	2	no	—
140	#95	—	122	2	no	—
141	#95	—	122	2	no	—
142	C₂²×C₄	—	2	2	no	—
143	C₂²×C₄	—	61	1	no	—
144	C₂²×C₄	—	61	1	no	—
145	C₂²×C₄	—	61	2	no	—
146	C₂²×C₄	—	122	2	no	—
147	C₂²×C₄	C₂²⋊#6923	1	2	yes	—
148	C₂²×C₄	C₂²×#6923	1	1	yes	characteristic
149	C₂×C₈	—	122	4	no	—
150	C₂×C₈	—	122	8	no	—
151	C₂×C₈	—	244	2	no	—
152	C₂×C₈	—	244	4	no	—
153	C₄⋊C₄	—	2	8	no	—
154	C₄⋊C₄	—	2	2	no	—
155	C₄⋊C₄	—	122	4	no	—
156	C₄⋊C₄	—	244	4	no	—
157	C₄⋊C₄	C₂×#92420	1	4	yes	—
158	C₄²	—	61	2	no	—
159	C₄²	—	61	2	no	—
160	C₄²	—	122	4	no	—
161	C₄²	—	122	4	no	—
162	C₄²	—	122	2	no	—
163	C₄²	—	244	4	no	—
164	C₄²	C₂×#92420	1	2	yes	—
165	C₄²	#11758217	1	2	yes	—
166	C₂³	C₂×C₂²⋊#6923	1	1	yes	characteristic
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	2	4	no	—
170	C₂×C₄	—	4	2	no	—
171	C₂×C₄	—	4	2	no	—
172	C₂×C₄	—	61	1	no	—
173	C₂×C₄	—	61	1	no	—
174	C₂×C₄	—	61	4	no	—
175	C₂×C₄	—	61	2	no	—
176	C₂×C₄	—	61	4	no	—
177	C₂×C₄	—	122	2	no	—
178	C₂×C₄	—	122	2	no	—
179	C₂×C₄	—	122	2	no	—
180	C₂×C₄	—	244	2	no	—
181	C₂×C₄	—	244	2	no	—
182	C₂×C₄	—	244	2	no	—
183	C₂×C₄	C₂²⋊#92420	1	4	yes	—
184	C₂×C₄	Q₈×#6923	1	2	yes	—
185	C₂×C₄	C₂²×#92420	1	1	yes	characteristic
186	C₂×C₄	C₂×#11758217	1	1	yes	characteristic
187	C₂×C₄	#14519865	1	2	yes	—
188	C₈	—	122	8	no	—
189	C₈	—	488	2	no	—
190	C₂²	(C₂×#6923)⋊Q₈	1	1	yes	characteristic
191	C₂²	C₂×C₂²⋊#92420	1	1	yes	characteristic
192	C₂²	#16840737	1	1	yes	characteristic
193	C₂²	#16840752	1	2	yes	—
194	C₂²	D₆₁⋊#104	1	2	yes	—
195	C₄	—	2	4	no	—
196	C₄	—	4	4	no	—
197	C₄	—	61	2	no	—
198	C₄	—	61	2	no	—
199	C₄	—	122	4	no	—
200	C₄	—	122	2	no	—
201	C₄	—	488	2	no	—
202	C₄	Q₈×#92420	1	2	yes	—
203	C₄	#16840813	1	2	yes	—
204	C₂	#21461110	1	2	yes	—
205	C₂	#21461173	1	2	yes	—
206	C₂	#21461988	1	1	yes	characteristic
207	C₂	#21462001	1	1	yes	characteristic
208	C₂	#21462005	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21461734

C₂

yes

maximal, characteristic

C₄²×#92420

C₂

yes

maximal, characteristic

#21460655

C₂

yes

maximal

#125×Dic₆₁

C₂

yes

maximal, characteristic

C₂×C₄⋊#92420

C₂²

yes

—

C₂×C₄×#92420

—

C₂×C₄×#92420

C₂²

yes

—

#16840628

—

#16839661

C₄

yes

—

#16839648

—

#16839628

C₄

yes

characteristic

C₄⋊C₄×Dic₆₁

—

C₄⋊C₄×Dic₆₁

C₂²

yes

—

C₄²×Dic₆₁

C₂²

yes

characteristic

C₂×#125×C₆₁

C₄

yes

characteristic

C₂²×#92420

—

C₂²×#92420

—

—

—

—

Q₈

yes

—

C₂×C₄⋊Dic₆₁

—

C₂×C₄⋊Dic₆₁

C₂×C₄

yes

—

C₂×C₄×Dic₆₁

—

C₂×C₄×Dic₆₁

C₂×C₄

yes

characteristic

C₂×C₄×Dic₆₁

D₄

yes

—

C₂×C₄×Dic₆₁

D₄

yes

—

C₂×C₄×Dic₆₁

C₂³

yes

—

—

—

#125×C₆₁

—

#125×C₆₁

C₈

yes

complemented

C₂×C₄⋊C₄×C₆₁

—

C₂×C₄⋊C₄×C₆₁

C₂×C₄

yes

—

C₂×C₄²×C₆₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

C₂²⋊C₄

yes

—

C₂²×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₂²×Dic₆₁

—

C₂²×Dic₆₁

C₂²⋊C₄

yes

—

C₂²×Dic₆₁

C₂×D₄

yes

characteristic

C₂²×Dic₆₁

C₂×D₄

yes

—

—

—

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

C₂×C₈

yes

—

C₄²×C₆₁

C₂×C₈

yes

—

C₄²×C₆₁

M₁₆

yes

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄×Dic₆₁

—

C₄×Dic₆₁

—

C₄×Dic₆₁

—

C₄×Dic₆₁

—

C₄×Dic₆₁

C₂×Q₈

yes

—

C₄×Dic₆₁

#41

yes

—

C₂³×C₆₁

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂²⋊C₈

yes

—

C₂×C₄×C₆₁

C₄×Q₈

yes

—

C₂×C₄×C₆₁

C₂²×C₈

yes

characteristic

C₂×C₄×C₆₁

C₂×M₁₆

yes

characteristic

C₂×C₄×C₆₁

#131

yes

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

#104

yes

—

C₂×Dic₆₁

C₂²⋊Q₈

yes

—

C₂×Dic₆₁

#124

yes

characteristic

C₂×Dic₆₁

C₄⋊Q₈

yes

characteristic

C₂×Dic₆₁

#135

yes

—

—

—

—

C₂²×C₆₁

yes

C₂²×C₆₁

C₂×C₂²⋊C₈

yes

C₂²×C₆₁

yes

C₂²×C₆₁

yes

—

C₂²×C₆₁

C₂×#104

yes

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

C₈×Q₈

yes

—

C₄×C₆₁

#446

yes

—

Dic₆₁

—

Dic₆₁

—

Dic₆₁

—

Dic₆₁

#488

yes

—

Dic₆₁

#499

yes

—

100

C₄×#125

—

101

#1831

—

102

C₄²×C₈

—

103

#1279

—

104

C₂×C₆₁

#1476

yes

—

105

C₂×C₆₁

#1494

yes

—

106

C₂×C₆₁

#1984

yes

characteristic

107

C₂×C₆₁

#1991

yes

characteristic

108

C₂×C₆₁

yes

109

yes

110

C₂×C₄⋊C₈

—

111

C₂×C₄×C₈

—

112

C₂×C₄×C₈

—

122

—

113

#389

—

114

#387

—

122

—

115

#382

—

116

C₄×C₄⋊C₄

—

117

C₄×C₄⋊C₄

—

122

—

118

C₄³

—

119

#336

—

122

—

120

C₆₁

#9871

yes

characteristic, complemented, Sylow

121

C₂²×C₈

—

122

—

122

C₂²×C₈

—

122

—

123

#125

—

124

#125

#92420

yes

complemented

125

C₂×C₄⋊C₄

—

126

C₂×C₄⋊C₄

—

127

C₂×C₄⋊C₄

—

122

—

128

C₂×C₄⋊C₄

C₂×#6923

yes

—

129

C₂×C₄²

—

130

C₂×C₄²

—

131

C₂×C₄²

—

132

C₂×C₄²

—

133

C₂×C₄²

—

122

—

134

C₂×C₄²

C₂×#6923

yes

characteristic

135

C₄⋊C₈

—

122

—

136

C₄×C₈

—

137

C₄×C₈

—

122

—

138

C₄×C₈

—

122

—

139

#95

—

122

—

140

#95

—

122

—

141

#95

—

122

—

142

C₂²×C₄

—

143

C₂²×C₄

—

144

C₂²×C₄

—

145

C₂²×C₄

—

146

C₂²×C₄

—

122

—

147

C₂²×C₄

C₂²⋊#6923

yes

—

148

C₂²×C₄

C₂²×#6923

yes

characteristic

149

C₂×C₈

—

122

—

150

C₂×C₈

—

122

—

151

C₂×C₈

—

244

—

152

C₂×C₈

—

244

—

153

C₄⋊C₄

—

154

C₄⋊C₄

—

155

C₄⋊C₄

—

122

—

156

C₄⋊C₄

—

244

—

157

C₄⋊C₄

C₂×#92420

yes

—

158

C₄²

—

159

C₄²

—

160

C₄²

—

122

—

161

C₄²

—

122

—

162

C₄²

—

122

—

163

C₄²

—

244

—

164

C₄²

C₂×#92420

yes

—

165

C₄²

#11758217

yes

—

166

C₂³

C₂×C₂²⋊#6923

yes

characteristic

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

122

—

178

C₂×C₄

—

122

—

179

C₂×C₄

—

122

—

180

C₂×C₄

—

244

—

181

C₂×C₄

—

244

—

182

C₂×C₄

—

244

—

183

C₂×C₄

C₂²⋊#92420

yes

—

184

C₂×C₄

Q₈×#6923

yes

—

185

C₂×C₄

C₂²×#92420

yes

characteristic

186

C₂×C₄

C₂×#11758217

yes

characteristic

187

C₂×C₄

#14519865

yes

—

188

C₈

—

122

—

189

C₈

—

488

—

190

C₂²

(C₂×#6923)⋊Q₈

yes

characteristic

191

C₂²

C₂×C₂²⋊#92420

yes

characteristic

192

C₂²

#16840737

yes

characteristic

193

C₂²

#16840752

yes

—

194

C₂²

D₆₁⋊#104

yes

—

195

C₄

—

196

C₄

—

197

C₄

—

198

C₄

—

199

C₄

—

122

—

200

C₄

—

122

—

201

C₄

—

488

—

202

C₄

Q₈×#92420

yes

—

203

C₄

#16840813

yes

—

204

C₂

#21461110

yes

—

205

C₂

#21461173

yes

—

206

C₂

#21461988

yes

characteristic

207

C₂

#21462001

yes

characteristic

208

C₂

#21462005

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊#6923
commutator	C₂×C₄×C₆₁	C₂²×C₈
Frattini	C₂²×C₄	C₂²×#6923
Fitting	C₂×#125×C₆₁	C₄
socle, prime radical	C₂³×C₆₁	C₂×C₂²⋊C₄
cosocle	C₂×C₄×Dic₆₁	C₂³
hypercenter	C₂×#125	#6923
hypocenter	C₆₁	#9871
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#125

subgroup

G/H

center

C₂³

C₂×C₂²⋊#6923

commutator

C₂×C₄×C₆₁

C₂²×C₈

Frattini

C₂²×C₄

C₂²×#6923

Fitting

C₂×#125×C₆₁

C₄

socle, prime radical

C₂³×C₆₁

C₂×C₂²⋊C₄

cosocle

C₂×C₄×Dic₆₁

C₂³

C₆₁

G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#125

N	H ≅ G/N	type
33. #125×C₆₁	C₈	semidirect
120. C₆₁	#9871	semidirect
124. #125	#92420	semidirect

H ≅ G/N

type

33. #125×C₆₁

C₈

semidirect

120. C₆₁

#9871

semidirect

124. #125

#92420

semidirect

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

subgroup	index
#21461734	2
C₄²×#92420	2
#21460655	2
#125×Dic₆₁	2
#9871	61

subgroup

index

#21461734

C₄²×#92420

#21460655

#125×Dic₆₁

#9871

Subgroups — #30602144

Conjugacy classes of subgroups

Complements

Maximal subgroups