Group #30612178 — subgroups

Subgroups — #30612178

Order 15616.

702 conjugacy classes of subgroups; only the 272 normal subgroups are shown.

normal subgroups only

This group has 702 conjugacy classes of subgroups — too many to draw. Only its 272 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	#21462465	C₂	1	2	yes	maximal
2	#21461387	C₂	1	6	yes	maximal, complemented
3	#21461374	C₂	1	3	yes	maximal, complemented
4	D₆₁⋊#334	C₂	1	3	yes	maximal, complemented
5	C₈×#11758226	C₂	1	1	yes	maximal, characteristic
6	C₂×#14519865	—	2	2	no	—
7	#16840769	—	2	6	no	—
8	#16840766	C₂²	1	6	yes	—
9	D₆₁⋊(C₄⋊C₈)	C₂²	1	6	yes	—
10	#16840624	C₂²	1	6	yes	—
11	#16840623	C₂²	1	1	yes	characteristic
12	#16840623	C₂²	1	3	yes	—
13	#16840623	C₂²	1	6	yes	—
14	C₄×#11758226	C₂²	1	1	yes	characteristic
15	Q₈×#92418	C₄	1	1	yes	characteristic
16	C₈×D₂₄₄	C₄	1	3	yes	—
17	#16839827	C₄	1	3	yes	—
18	#16839814	C₂²	1	3	yes	—
19	C₄×C₈×D₆₁	C₂²	1	3	yes	—
20	C₈×Q₈×C₆₁	C₄	1	1	yes	characteristic
21	#14519865	—	4	8	no	—
22	C₂×#11758213	—	2	6	no	—
23	C₂×D₆₁⋊C₈	—	2	6	no	—
24	#14519844	—	2	6	no	—
25	C₄⋊#92420	—	2	6	no	—
26	C₄⋊#92420	C₂³	1	2	yes	—
27	C₄⋊#92420	C₂³	1	6	yes	—
28	C₂×#11758226	D₄	1	1	yes	characteristic
29	C₂×C₈×D₆₁	C₂×C₄	1	3	yes	—
30	C₂×C₈×D₆₁	C₂×C₄	1	3	yes	—
31	Q₈×Dic₆₁	Q₈	1	1	yes	characteristic
32	C₄×D₂₄₄	C₂×C₄	1	3	yes	—
33	#14519739	Q₈	1	3	yes	—
34	#14519735	D₄	1	3	yes	—
35	C₄²×D₆₁	C₂³	1	3	yes	—
36	C₄⋊#92418	C₂×C₄	1	3	yes	—
37	#14519689	C₂×C₄	1	6	yes	—
38	C₄×#92418	C₂×C₄	1	3	yes	—
39	C₈×Dic₆₁	C₂³	1	1	yes	characteristic
40	C₈×Dic₆₁	C₂³	1	3	yes	—
41	C₄×Q₈×C₆₁	C₂×C₄	1	1	yes	characteristic
42	C₄⋊C₈×C₆₁	C₂×C₄	1	3	yes	—
43	C₄×C₈×C₆₁	C₂×C₄	1	3	yes	—
44	C₂×Q₈×C₆₁	C₄⋊C₄	1	1	yes	characteristic
45	#11758226	SD₁₆	1	8	yes	—
46	C₂×D₂₄₄	C₄⋊C₄	1	3	yes	—
47	C₂×C₄×D₆₁	C₂²×C₄	1	3	yes	—
48	C₂×C₄×D₆₁	C₂×D₄	1	3	yes	—
49	C₂×C₄×D₆₁	C₂×Q₈	1	3	yes	—
50	C₂×#92420	—	2	6	no	—
51	C₂×#92420	—	2	2	no	—
52	#11758213	—	4	12	no	—
53	D₆₁⋊C₈	—	4	12	no	—
54	C₂×C₈×C₆₁	C₂²×C₄	1	3	yes	—
55	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
56	C₄⋊C₄×C₆₁	C₄⋊C₄	1	3	yes	—
57	C₄²×C₆₁	C₂²×C₄	1	3	yes	—
58	#11758198	C₄⋊C₄	1	6	yes	—
59	C₄⋊Dic₆₁	C₄⋊C₄	1	3	yes	—
60	C₄×Dic₆₁	C₂×D₄	1	3	yes	—
61	C₄×Dic₆₁	C₂×Q₈	1	3	yes	—
62	C₄×Dic₆₁	C₂⁴	1	1	yes	characteristic
63	C₂×#92418	C₂²×C₄	1	1	yes	characteristic
64	C₂×#92418	C₂²×C₄	1	3	yes	—
65	C₈×D₆₁	—	2	6	no	—
66	C₈×D₆₁	—	4	3	no	—
67	C₂²×D₆₁	C₂×C₄⋊C₄	1	3	yes	—
68	Q₈×C₆₁	#106	1	4	yes	—
69	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	3	yes	—
70	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	3	yes	—
71	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
72	C₂×Dic₆₁	C₂×C₄⋊C₄	1	3	yes	—
73	C₂×Dic₆₁	C₂²×D₄	1	1	yes	characteristic
74	C₂×Dic₆₁	C₂²×Q₈	1	1	yes	characteristic
75	D₂₄₄	#106	1	12	yes	—
76	C₄×D₆₁	—	2	3	no	—
77	C₄×D₆₁	—	2	3	no	—
78	C₄×D₆₁	—	4	3	no	—
79	C₄×D₆₁	C₂×SD₁₆	1	12	yes	—
80	#92420	—	4	2	no	—
81	#92420	—	4	6	no	—
82	C₈×C₆₁	—	2	3	no	—
83	C₈×C₆₁	#131	1	2	yes	complemented
84	#92418	—	2	3	no	—
85	#92418	—	2	1	no	—
86	#17826	—	61	1	no	maximal, Sylow
87	C₂²×C₆₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
88	C₂×D₆₁	—	2	3	no	—
89	C₂×D₆₁	—	2	3	no	—
90	C₂×D₆₁	C₂×#106	1	6	yes	—
91	C₄×C₆₁	—	2	3	no	—
92	C₄×C₆₁	C₂×#106	1	6	yes	—
93	C₄×C₆₁	C₂×#131	1	1	yes	characteristic
94	C₄×C₆₁	#568	1	1	yes	characteristic
95	Dic₆₁	—	2	3	no	—
96	Dic₆₁	—	2	1	no	—
97	Dic₆₁	C₂²×SD₁₆	1	2	yes	—
98	C₈×#41	—	61	1	no	—
99	#2906	—	61	2	no	—
100	#1570	—	61	6	no	—
101	#1566	—	61	3	no	—
102	C₂×#334	—	61	3	no	—
103	C₂×C₆₁	#2906	1	1	yes	characteristic
104	C₂×C₆₁	C₂²×#106	1	1	yes	characteristic
105	C₂×C₆₁	#2916	1	1	yes	characteristic
106	D₆₁	—	2	6	no	—
107	C₂×#131	—	122	2	no	—
108	C₄×#41	—	61	1	no	—
109	C₈×Q₈	—	61	1	no	—
110	C₈×Q₈	#6923	1	1	yes	characteristic
111	C₈×D₄	—	61	3	no	—
112	C₈×D₄	—	61	3	no	—
113	#431	—	61	3	no	—
114	#424	—	122	6	no	—
115	#423	—	61	6	no	—
116	C₂×C₄⋊C₈	—	61	6	no	—
117	C₂×C₄×C₈	—	61	3	no	—
118	#334	—	61	1	no	—
119	#334	—	61	3	no	—
120	#334	—	61	6	no	—
121	#334	—	61	6	no	—
122	C₆₁	#17826	1	1	yes	characteristic, complemented, Sylow
123	C₂×#41	—	61	1	no	—
124	#131	—	244	8	no	—
125	C₂×M₁₆	—	122	6	no	—
126	C₂²×C₈	—	61	3	no	—
127	C₂²×C₈	—	61	3	no	—
128	C₂²×C₈	—	122	6	no	—
129	C₄×Q₈	—	61	1	no	—
130	C₄×Q₈	C₂×#6923	1	1	yes	characteristic
131	C₄×D₄	—	61	3	no	—
132	C₄×D₄	—	61	3	no	—
133	#117	—	61	3	no	—
134	C₂×C₄²	—	61	3	no	—
135	C₄⋊C₈	—	61	2	no	—
136	C₄⋊C₈	—	61	6	no	—
137	C₄⋊C₈	—	61	3	no	—
138	C₄⋊C₈	—	122	6	no	—
139	C₄⋊C₈	—	122	6	no	—
140	C₄⋊C₈	C₂×#6923	1	3	yes	—
141	C₂²⋊C₈	—	61	6	no	—
142	C₄×C₈	—	61	1	no	—
143	C₄×C₈	—	61	3	no	—
144	C₄×C₈	—	61	3	no	—
145	C₄×C₈	C₂×#6923	1	3	yes	—
146	#41	—	61	8	no	—
147	C₂×Q₈	C₄⋊#6923	1	1	yes	characteristic
148	C₂×D₄	—	61	3	no	—
149	C₂²×C₄	—	61	3	no	—
150	C₂²×C₄	—	61	3	no	—
151	C₂²×C₄	—	61	3	no	—
152	M₁₆	—	244	12	no	—
153	C₂×C₈	—	61	1	no	—
154	C₂×C₈	—	61	3	no	—
155	C₂×C₈	—	122	6	no	—
156	C₂×C₈	—	122	2	no	—
157	C₂×C₈	—	122	6	no	—
158	C₂×C₈	—	244	3	no	—
159	C₂×C₈	—	244	12	no	—
160	C₂×C₈	C₂²×#6923	1	3	yes	—
161	C₂×C₈	C₂²×#6923	1	1	yes	characteristic
162	C₄⋊C₄	—	61	3	no	—
163	C₄⋊C₄	C₄⋊#6923	1	3	yes	—
164	C₂²⋊C₄	—	61	6	no	—
165	C₄²	—	61	3	no	—
166	C₄²	—	61	3	no	—
167	C₄²	—	61	1	no	—
168	C₄²	C₂²×#6923	1	3	yes	—
169	C₂³	—	61	3	no	—
170	Q₈	#14519845	1	4	yes	—
171	D₄	—	61	12	no	—
172	C₂×C₄	—	61	1	no	—
173	C₂×C₄	—	61	12	no	—
174	C₂×C₄	—	61	3	no	—
175	C₂×C₄	—	61	1	no	—
176	C₂×C₄	—	122	3	no	—
177	C₂×C₄	—	122	3	no	—
178	C₂×C₄	—	244	3	no	—
179	C₂×C₄	C₂×C₄⋊#6923	1	3	yes	—
180	C₂×C₄	C₂×C₄⋊#6923	1	3	yes	—
181	C₂×C₄	C₂³×#6923	1	1	yes	characteristic
182	C₈	—	2	3	no	—
183	C₈	—	122	3	no	—
184	C₈	—	122	1	no	—
185	C₈	—	244	2	no	—
186	C₈	—	244	6	no	—
187	C₈	#14519865	1	2	yes	complemented
188	C₂²	—	61	6	no	—
189	C₂²	—	122	3	no	—
190	C₂²	—	122	3	no	—
191	C₂²	C₂²×C₄⋊#6923	1	1	yes	characteristic
192	C₄	—	2	3	no	—
193	C₄	—	61	2	no	—
194	C₄	—	122	3	no	—
195	C₄	—	122	1	no	—
196	C₄	C₂×#14519845	1	6	yes	—
197	C₄	C₂×#14519865	1	1	yes	characteristic
198	C₄	#16840841	1	1	yes	characteristic
199	C₂	—	122	6	no	—
200	C₂	#21462465	1	1	yes	characteristic
201	C₂	C₂²×#14519845	1	1	yes	characteristic
202	C₂	#21462480	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21462465

C₂

yes

maximal

#21461387

C₂

yes

maximal, complemented

#21461374

C₂

yes

maximal, complemented

D₆₁⋊#334

C₂

yes

maximal, complemented

C₈×#11758226

C₂

yes

maximal, characteristic

C₂×#14519865

—

#16840769

—

#16840766

C₂²

yes

—

D₆₁⋊(C₄⋊C₈)

C₂²

yes

—

#16840624

C₂²

yes

—

#16840623

C₂²

yes

characteristic

#16840623

C₂²

yes

—

#16840623

C₂²

yes

—

C₄×#11758226

C₂²

yes

characteristic

Q₈×#92418

C₄

yes

characteristic

C₈×D₂₄₄

C₄

yes

—

#16839827

C₄

yes

—

#16839814

C₂²

yes

—

C₄×C₈×D₆₁

C₂²

yes

—

C₈×Q₈×C₆₁

C₄

yes

characteristic

#14519865

—

C₂×#11758213

—

C₂×D₆₁⋊C₈

—

#14519844

—

C₄⋊#92420

—

C₄⋊#92420

C₂³

yes

—

C₄⋊#92420

C₂³

yes

—

C₂×#11758226

D₄

yes

characteristic

C₂×C₈×D₆₁

C₂×C₄

yes

—

C₂×C₈×D₆₁

C₂×C₄

yes

—

Q₈×Dic₆₁

Q₈

yes

characteristic

C₄×D₂₄₄

C₂×C₄

yes

—

#14519739

Q₈

yes

—

#14519735

D₄

yes

—

C₄²×D₆₁

C₂³

yes

—

C₄⋊#92418

C₂×C₄

yes

—

#14519689

C₂×C₄

yes

—

C₄×#92418

C₂×C₄

yes

—

C₈×Dic₆₁

C₂³

yes

characteristic

C₈×Dic₆₁

C₂³

yes

—

C₄×Q₈×C₆₁

C₂×C₄

yes

characteristic

C₄⋊C₈×C₆₁

C₂×C₄

yes

—

C₄×C₈×C₆₁

C₂×C₄

yes

—

C₂×Q₈×C₆₁

C₄⋊C₄

yes

characteristic

#11758226

SD₁₆

yes

—

C₂×D₂₄₄

C₄⋊C₄

yes

—

C₂×C₄×D₆₁

C₂²×C₄

yes

—

C₂×C₄×D₆₁

C₂×D₄

yes

—

C₂×C₄×D₆₁

C₂×Q₈

yes

—

C₂×#92420

—

C₂×#92420

—

#11758213

—

D₆₁⋊C₈

—

C₂×C₈×C₆₁

C₂²×C₄

yes

—

C₂×C₈×C₆₁

C₂²×C₄

yes

characteristic

C₄⋊C₄×C₆₁

C₄⋊C₄

yes

—

C₄²×C₆₁

C₂²×C₄

yes

—

#11758198

C₄⋊C₄

yes

—

C₄⋊Dic₆₁

C₄⋊C₄

yes

—

C₄×Dic₆₁

C₂×D₄

yes

—

C₄×Dic₆₁

C₂×Q₈

yes

—

C₄×Dic₆₁

C₂⁴

yes

characteristic

C₂×#92418

C₂²×C₄

yes

characteristic

C₂×#92418

C₂²×C₄

yes

—

C₈×D₆₁

—

C₈×D₆₁

—

C₂²×D₆₁

C₂×C₄⋊C₄

yes

—

Q₈×C₆₁

#106

yes

—

C₂×C₄×C₆₁

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₆₁

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₆₁

C₂³×C₄

yes

characteristic

C₂×Dic₆₁

C₂×C₄⋊C₄

yes

—

C₂×Dic₆₁

C₂²×D₄

yes

characteristic

C₂×Dic₆₁

C₂²×Q₈

yes

characteristic

D₂₄₄

#106

yes

—

C₄×D₆₁

—

C₄×D₆₁

—

C₄×D₆₁

—

C₄×D₆₁

C₂×SD₁₆

yes

—

#92420

—

#92420

—

C₈×C₆₁

—

C₈×C₆₁

yes

—

—

—

C₂²×C₆₁

C₂²×C₄⋊C₄

yes

characteristic

C₂×D₆₁

—

C₂×D₆₁

—

C₂×D₆₁

C₂×#106

yes

—

C₄×C₆₁

—

C₄×C₆₁

C₂×#106

yes

—

C₄×C₆₁

yes

C₄×C₆₁

yes

Dic₆₁

—

Dic₆₁

—

Dic₆₁

C₂²×SD₁₆

yes

—

C₈×#41

—

#2906

—

100

#1570

—

101

#1566

—

102

C₂×#334

—

103

C₂×C₆₁

#2906

yes

characteristic

104

C₂×C₆₁

C₂²×#106

yes

characteristic

105

C₂×C₆₁

#2916

yes

characteristic

106

D₆₁

—

107

C₂×#131

—

122

—

108

C₄×#41

—

109

C₈×Q₈

—

110

C₈×Q₈

#6923

yes

characteristic

111

C₈×D₄

—

112

C₈×D₄

—

113

#431

—

114

#424

—

122

—

115

#423

—

116

C₂×C₄⋊C₈

—

117

C₂×C₄×C₈

—

118

#334

—

119

#334

—

120

#334

—

121

#334

—

122

C₆₁

#17826

yes

characteristic, complemented, Sylow

123

C₂×#41

—

124

#131

—

244

—

125

C₂×M₁₆

—

122

—

126

C₂²×C₈

—

127

C₂²×C₈

—

128

C₂²×C₈

—

122

—

129

C₄×Q₈

—

130

C₄×Q₈

C₂×#6923

yes

characteristic

131

C₄×D₄

—

132

C₄×D₄

—

133

#117

—

134

C₂×C₄²

—

135

C₄⋊C₈

—

136

C₄⋊C₈

—

137

C₄⋊C₈

—

138

C₄⋊C₈

—

122

—

139

C₄⋊C₈

—

122

—

140

C₄⋊C₈

C₂×#6923

yes

—

141

C₂²⋊C₈

—

142

C₄×C₈

—

143

C₄×C₈

—

144

C₄×C₈

—

145

C₄×C₈

C₂×#6923

yes

—

146

#41

—

147

C₂×Q₈

C₄⋊#6923

yes

characteristic

148

C₂×D₄

—

149

C₂²×C₄

—

150

C₂²×C₄

—

151

C₂²×C₄

—

152

M₁₆

—

244

—

153

C₂×C₈

—

154

C₂×C₈

—

155

C₂×C₈

—

122

—

156

C₂×C₈

—

122

—

157

C₂×C₈

—

122

—

158

C₂×C₈

—

244

—

159

C₂×C₈

—

244

—

160

C₂×C₈

C₂²×#6923

yes

—

161

C₂×C₈

C₂²×#6923

yes

characteristic

162

C₄⋊C₄

—

163

C₄⋊C₄

C₄⋊#6923

yes

—

164

C₂²⋊C₄

—

165

C₄²

—

166

C₄²

—

167

C₄²

—

168

C₄²

C₂²×#6923

yes

—

169

C₂³

—

170

Q₈

#14519845

yes

—

171

D₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

—

122

—

177

C₂×C₄

—

122

—

178

C₂×C₄

—

244

—

179

C₂×C₄

C₂×C₄⋊#6923

yes

—

180

C₂×C₄

C₂×C₄⋊#6923

yes

—

181

C₂×C₄

C₂³×#6923

yes

characteristic

182

C₈

—

183

C₈

—

122

—

184

C₈

—

122

—

185

C₈

—

244

—

186

C₈

—

244

—

187

C₈

#14519865

yes

complemented

188

C₂²

—

189

C₂²

—

122

—

190

C₂²

—

122

—

191

C₂²

C₂²×C₄⋊#6923

yes

characteristic

192

C₄

—

193

C₄

—

194

C₄

—

122

—

195

C₄

—

122

—

196

C₄

C₂×#14519845

yes

—

197

C₄

C₂×#14519865

yes

characteristic

198

C₄

#16840841

yes

characteristic

199

C₂

—

122

—

200

C₂

#21462465

yes

characteristic

201

C₂

C₂²×#14519845

yes

characteristic

202

C₂

#21462480

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×C₄⋊#6923
commutator	C₂×C₄×C₆₁	C₂³×C₄
Frattini	C₂×C₄	C₂³×#6923
Fitting	C₈×Q₈×C₆₁	C₄
socle	C₂²×C₆₁	C₂²×C₄⋊C₄
prime radical	C₂×#11758226	D₄
cosocle	C₄×Dic₆₁	C₂⁴
hypercenter	C₈×Q₈	#6923
hypocenter	C₆₁	#17826
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈

subgroup

G/H

center

C₂²

C₂²×C₄⋊#6923

commutator

C₂×C₄×C₆₁

C₂³×C₄

Frattini

C₂×C₄

C₂³×#6923

Fitting

C₈×Q₈×C₆₁

C₄

socle

C₂²×C₆₁

C₂²×C₄⋊C₄

prime radical

C₂×#11758226

D₄

cosocle

C₄×Dic₆₁

C₂⁴

C₈×Q₈

C₆₁

G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈

N	H ≅ G/N	type
2. #21461387	C₂	semidirect
3. #21461374	C₂	semidirect
4. D₆₁⋊#334	C₂	semidirect
83. C₈×C₆₁	#131	semidirect
122. C₆₁	#17826	semidirect
187. C₈	#14519865	semidirect

H ≅ G/N

type

2. #21461387

C₂

semidirect

3. #21461374

C₂

semidirect

4. D₆₁⋊#334

C₂

semidirect

83. C₈×C₆₁

#131

semidirect

122. C₆₁

#17826

semidirect

187. C₈

#14519865

semidirect

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

subgroup	index
#21462465	2
#21461387	2
#21461374	2
D₆₁⋊#334	2
C₈×#11758226	2
#17826	61

subgroup

index

D₆₁⋊#334

Subgroups — #30612178

Conjugacy classes of subgroups

Complements

Maximal subgroups