Group #30612230 — subgroups

Subgroups — #30612230

Order 15616.

612 conjugacy classes of subgroups; only the 232 normal subgroups are shown.

normal subgroups only

This group has 612 conjugacy classes of subgroups — too many to draw. Only its 232 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	#21462753	C₂	1	2	yes	maximal
2	#21461397	C₂	1	6	yes	maximal, complemented
3	#21461375	C₂	1	3	yes	maximal, complemented
4	#21461369	C₂	1	3	yes	maximal, complemented
5	C₈×#11758226	C₂	1	1	yes	maximal, characteristic
6	Q₈×#92420	—	2	2	no	—
7	#16840796	—	2	6	no	—
8	#16840781	C₂²	1	6	yes	—
9	#16840766	C₂²	1	6	yes	—
10	#16840625	C₂²	1	6	yes	—
11	#16840624	C₂²	1	1	yes	characteristic
12	#16840624	C₂²	1	3	yes	—
13	#16840624	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	C₂×#11758213	—	2	6	no	—
22	C₄⋊#92420	—	2	6	no	—
23	C₄⋊#92420	C₂³	1	6	yes	—
24	C₄⋊#92420	C₂³	1	2	yes	—
25	#14519834	—	2	6	no	—
26	C₄×#92420	—	2	6	no	—
27	C₂×#11758226	Q₈	1	1	yes	characteristic
28	C₂×C₈×D₆₁	C₂×C₄	1	3	yes	—
29	C₂×C₈×D₆₁	C₂×C₄	1	3	yes	—
30	Q₈×Dic₆₁	D₄	1	1	yes	characteristic
31	C₄×D₂₄₄	C₂×C₄	1	3	yes	—
32	#14519739	D₄	1	3	yes	—
33	#14519735	Q₈	1	3	yes	—
34	C₄²×D₆₁	C₂³	1	3	yes	—
35	C₄⋊#92418	C₂×C₄	1	3	yes	—
36	#14519689	C₂×C₄	1	6	yes	—
37	C₄×#92418	C₂×C₄	1	3	yes	—
38	C₈×Dic₆₁	C₂³	1	1	yes	characteristic
39	C₈×Dic₆₁	C₂³	1	3	yes	—
40	C₄×Q₈×C₆₁	C₂×C₄	1	1	yes	characteristic
41	C₄⋊C₈×C₆₁	C₂×C₄	1	3	yes	—
42	C₄×C₈×C₆₁	C₂×C₄	1	3	yes	—
43	C₂×Q₈×C₆₁	C₄⋊C₄	1	1	yes	characteristic
44	#11758226	—	2	4	no	—
45	C₂×D₂₄₄	C₄⋊C₄	1	3	yes	—
46	C₂×C₄×D₆₁	C₂²×C₄	1	3	yes	—
47	C₂×C₄×D₆₁	C₂×D₄	1	3	yes	—
48	C₂×C₄×D₆₁	C₂×Q₈	1	3	yes	—
49	C₂×#92420	—	2	2	no	—
50	C₂×#92420	—	2	6	no	—
51	#11758213	—	8	6	no	—
52	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₈×C₆₁	C₂²×C₄	1	3	yes	—
54	C₄⋊C₄×C₆₁	C₄⋊C₄	1	3	yes	—
55	C₄²×C₆₁	C₂²×C₄	1	3	yes	—
56	#11758198	C₄⋊C₄	1	6	yes	—
57	C₄⋊Dic₆₁	C₄⋊C₄	1	3	yes	—
58	C₄×Dic₆₁	C₂×D₄	1	3	yes	—
59	C₄×Dic₆₁	C₂×Q₈	1	3	yes	—
60	C₄×Dic₆₁	C₂⁴	1	1	yes	characteristic
61	C₂×#92418	C₂²×C₄	1	3	yes	—
62	C₂×#92418	C₂²×C₄	1	1	yes	characteristic
63	C₈×D₆₁	—	2	6	no	—
64	C₈×D₆₁	—	4	3	no	—
65	C₂²×D₆₁	C₂×C₄⋊C₄	1	3	yes	—
66	Q₈×C₆₁	#108	1	4	yes	—
67	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	3	yes	—
68	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	3	yes	—
69	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
70	C₂×Dic₆₁	C₂×C₄⋊C₄	1	3	yes	—
71	C₂×Dic₆₁	C₂²×D₄	1	1	yes	characteristic
72	C₂×Dic₆₁	C₂²×Q₈	1	1	yes	characteristic
73	D₂₄₄	—	2	6	no	—
74	C₄×D₆₁	—	2	3	no	—
75	C₄×D₆₁	—	2	3	no	—
76	C₄×D₆₁	—	2	6	no	—
77	C₄×D₆₁	—	2	6	no	—
78	#92420	—	4	6	no	—
79	#92420	—	4	2	no	—
80	C₈×C₆₁	—	2	3	no	—
81	C₈×C₆₁	—	2	1	no	—
82	#92418	—	2	3	no	—
83	#92418	#131	1	2	yes	—
84	#17835	—	61	1	no	maximal, Sylow
85	C₂²×C₆₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
86	C₂×D₆₁	—	2	3	no	—
87	C₂×D₆₁	—	2	3	no	—
88	C₂×D₆₁	—	2	3	no	—
89	C₄×C₆₁	—	2	3	no	—
90	C₄×C₆₁	C₂×#108	1	6	yes	—
91	C₄×C₆₁	C₂×#131	1	1	yes	characteristic
92	C₄×C₆₁	#568	1	1	yes	characteristic
93	Dic₆₁	—	2	3	no	—
94	Dic₆₁	—	2	1	no	—
95	Dic₆₁	#577	1	2	yes	—
96	#2991	—	61	2	no	—
97	C₈×#41	—	61	1	no	—
98	#1574	—	61	6	no	—
99	#1566	—	61	3	no	—
100	#1564	—	61	3	no	—
101	C₂×C₆₁	#2906	1	1	yes	characteristic
102	C₂×C₆₁	#2912	1	1	yes	characteristic
103	C₂×C₆₁	C₂²×#108	1	1	yes	characteristic
104	D₆₁	—	4	3	no	—
105	C₄×#41	—	61	1	no	—
106	C₈×Q₈	—	61	1	no	—
107	C₈×Q₈	—	122	2	no	—
108	C₈×Q₈	#6923	1	1	yes	characteristic
109	#436	—	122	6	no	—
110	C₈×D₄	—	61	3	no	—
111	C₈×D₄	—	61	3	no	—
112	#431	—	61	3	no	—
113	#431	—	61	6	no	—
114	#423	—	61	6	no	—
115	C₂×C₄×C₈	—	61	3	no	—
116	#335	—	61	6	no	—
117	#334	—	61	1	no	—
118	#334	—	61	3	no	—
119	#334	—	61	6	no	—
120	C₆₁	#17835	1	1	yes	characteristic, complemented, Sylow
121	C₂×#41	—	61	1	no	—
122	C₂×M₁₆	—	122	6	no	—
123	C₂²×C₈	—	61	3	no	—
124	C₂²×C₈	—	61	3	no	—
125	C₄×Q₈	—	61	1	no	—
126	C₄×Q₈	C₂×#6923	1	1	yes	characteristic
127	C₄×D₄	—	61	3	no	—
128	C₄×D₄	—	61	3	no	—
129	#117	—	61	3	no	—
130	C₂×C₄²	—	61	3	no	—
131	C₄⋊C₈	—	61	6	no	—
132	C₄⋊C₈	—	61	2	no	—
133	C₄⋊C₈	—	61	3	no	—
134	C₄⋊C₈	—	122	6	no	—
135	C₄⋊C₈	C₂×#6923	1	3	yes	—
136	C₂²⋊C₈	—	61	6	no	—
137	C₂²⋊C₈	—	122	6	no	—
138	C₄×C₈	—	61	1	no	—
139	C₄×C₈	—	61	3	no	—
140	C₄×C₈	—	61	3	no	—
141	C₄×C₈	—	122	6	no	—
142	C₄×C₈	C₂×#6923	1	3	yes	—
143	#41	—	122	4	no	—
144	C₂×Q₈	C₄⋊#6923	1	1	yes	characteristic
145	C₂×D₄	—	61	3	no	—
146	C₂²×C₄	—	61	3	no	—
147	C₂²×C₄	—	61	3	no	—
148	C₂²×C₄	—	61	3	no	—
149	M₁₆	—	488	6	no	—
150	C₂×C₈	—	61	1	no	—
151	C₂×C₈	—	61	3	no	—
152	C₂×C₈	—	122	6	no	—
153	C₂×C₈	—	122	2	no	—
154	C₂×C₈	—	122	6	no	—
155	C₂×C₈	—	244	3	no	—
156	C₂×C₈	C₂²×#6923	1	1	yes	characteristic
157	C₂×C₈	C₂²×#6923	1	3	yes	—
158	C₄⋊C₄	—	61	3	no	—
159	C₄⋊C₄	C₄⋊#6923	1	3	yes	—
160	C₂²⋊C₄	—	61	6	no	—
161	C₄²	—	61	3	no	—
162	C₄²	—	61	3	no	—
163	C₄²	—	61	1	no	—
164	C₄²	C₂²×#6923	1	3	yes	—
165	C₂³	—	61	3	no	—
166	Q₈	#14519848	1	4	yes	—
167	D₄	—	122	6	no	—
168	C₂×C₄	—	61	1	no	—
169	C₂×C₄	—	61	3	no	—
170	C₂×C₄	—	61	1	no	—
171	C₂×C₄	—	122	6	no	—
172	C₂×C₄	—	122	6	no	—
173	C₂×C₄	—	122	3	no	—
174	C₂×C₄	—	122	3	no	—
175	C₂×C₄	C₂×C₄⋊#6923	1	3	yes	—
176	C₂×C₄	C₂×C₄⋊#6923	1	3	yes	—
177	C₂×C₄	C₂³×#6923	1	1	yes	characteristic
178	C₈	—	2	3	no	—
179	C₈	—	2	1	no	—
180	C₈	—	61	2	no	—
181	C₈	—	122	3	no	—
182	C₈	—	244	2	no	—
183	C₈	—	244	6	no	—
184	C₂²	—	122	3	no	—
185	C₂²	—	122	3	no	—
186	C₂²	—	122	3	no	—
187	C₂²	C₂²×C₄⋊#6923	1	1	yes	characteristic
188	C₄	—	2	3	no	—
189	C₄	—	61	2	no	—
190	C₄	—	122	3	no	—
191	C₄	—	122	1	no	—
192	C₄	C₂×#14519848	1	6	yes	—
193	C₄	D₆₁⋊#131	1	1	yes	characteristic
194	C₄	#16840842	1	1	yes	characteristic
195	C₂	—	244	3	no	—
196	C₂	#21462467	1	1	yes	characteristic
197	C₂	#21462473	1	1	yes	characteristic
198	C₂	C₂²×#14519848	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21462753

C₂

yes

maximal

#21461397

C₂

yes

maximal, complemented

#21461375

C₂

yes

maximal, complemented

#21461369

C₂

yes

maximal, complemented

C₈×#11758226

C₂

yes

maximal, characteristic

Q₈×#92420

—

#16840796

—

#16840781

C₂²

yes

—

#16840766

C₂²

yes

—

#16840625

C₂²

yes

—

#16840624

C₂²

yes

characteristic

#16840624

C₂²

yes

—

#16840624

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

—

—

C₂³

yes

—

C₄⋊#92420

C₂³

yes

—

#14519834

—

C₄×#92420

—

C₂×#11758226

Q₈

yes

characteristic

C₂×C₈×D₆₁

C₂×C₄

yes

—

C₂×C₈×D₆₁

C₂×C₄

yes

—

Q₈×Dic₆₁

D₄

yes

characteristic

C₄×D₂₄₄

C₂×C₄

yes

—

#14519739

D₄

yes

—

#14519735

Q₈

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

—

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

—

C₂×C₈×C₆₁

C₂²×C₄

yes

characteristic

C₂×C₈×C₆₁

C₂²×C₄

yes

—

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

—

C₂×#92418

C₂²×C₄

yes

characteristic

C₈×D₆₁

—

C₈×D₆₁

—

C₂²×D₆₁

C₂×C₄⋊C₄

yes

—

Q₈×C₆₁

#108

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

—

C₄×D₆₁

—

C₄×D₆₁

—

C₄×D₆₁

—

C₄×D₆₁

—

#92420

—

#92420

—

C₈×C₆₁

—

C₈×C₆₁

—

#92418

—

#92418

#131

yes

—

#17835

—

maximal, Sylow

C₂²×C₆₁

C₂²×C₄⋊C₄

yes

characteristic

C₂×D₆₁

—

C₂×D₆₁

—

C₂×D₆₁

—

C₄×C₆₁

—

C₄×C₆₁

C₂×#108

yes

—

C₄×C₆₁

yes

C₄×C₆₁

yes

Dic₆₁

—

Dic₆₁

—

Dic₆₁

#577

yes

—

#2991

—

C₈×#41

—

#1574

—

#1566

—

100

#1564

—

101

C₂×C₆₁

#2906

yes

characteristic

102

C₂×C₆₁

#2912

yes

characteristic

103

C₂×C₆₁

C₂²×#108

yes

characteristic

104

D₆₁

—

105

C₄×#41

—

106

C₈×Q₈

—

107

C₈×Q₈

—

122

—

108

C₈×Q₈

#6923

yes

characteristic

109

#436

—

122

—

110

C₈×D₄

—

111

C₈×D₄

—

112

#431

—

113

#431

—

114

#423

—

115

C₂×C₄×C₈

—

116

#335

—

117

#334

—

118

#334

—

119

#334

—

120

C₆₁

#17835

yes

characteristic, complemented, Sylow

121

C₂×#41

—

122

C₂×M₁₆

—

122

—

123

C₂²×C₈

—

124

C₂²×C₈

—

125

C₄×Q₈

—

126

C₄×Q₈

C₂×#6923

yes

characteristic

127

C₄×D₄

—

128

C₄×D₄

—

129

#117

—

130

C₂×C₄²

—

131

C₄⋊C₈

—

132

C₄⋊C₈

—

133

C₄⋊C₈

—

134

C₄⋊C₈

—

122

—

135

C₄⋊C₈

C₂×#6923

yes

—

136

C₂²⋊C₈

—

137

C₂²⋊C₈

—

122

—

138

C₄×C₈

—

139

C₄×C₈

—

140

C₄×C₈

—

141

C₄×C₈

—

122

—

142

C₄×C₈

C₂×#6923

yes

—

143

#41

—

122

—

144

C₂×Q₈

C₄⋊#6923

yes

characteristic

145

C₂×D₄

—

146

C₂²×C₄

—

147

C₂²×C₄

—

148

C₂²×C₄

—

149

M₁₆

—

488

—

150

C₂×C₈

—

151

C₂×C₈

—

152

C₂×C₈

—

122

—

153

C₂×C₈

—

122

—

154

C₂×C₈

—

122

—

155

C₂×C₈

—

244

—

156

C₂×C₈

C₂²×#6923

yes

characteristic

157

C₂×C₈

C₂²×#6923

yes

—

158

C₄⋊C₄

—

159

C₄⋊C₄

C₄⋊#6923

yes

—

160

C₂²⋊C₄

—

161

C₄²

—

162

C₄²

—

163

C₄²

—

164

C₄²

C₂²×#6923

yes

—

165

C₂³

—

166

Q₈

#14519848

yes

—

167

D₄

—

122

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

—

122

—

172

C₂×C₄

—

122

—

173

C₂×C₄

—

122

—

174

C₂×C₄

—

122

—

175

C₂×C₄

C₂×C₄⋊#6923

yes

—

176

C₂×C₄

C₂×C₄⋊#6923

yes

—

177

C₂×C₄

C₂³×#6923

yes

characteristic

178

C₈

—

179

C₈

—

180

C₈

—

181

C₈

—

122

—

182

C₈

—

244

—

183

C₈

—

244

—

184

C₂²

—

122

—

185

C₂²

—

122

—

186

C₂²

—

122

—

187

C₂²

C₂²×C₄⋊#6923

yes

characteristic

188

C₄

—

189

C₄

—

190

C₄

—

122

—

191

C₄

—

122

—

192

C₄

C₂×#14519848

yes

—

193

C₄

D₆₁⋊#131

yes

characteristic

194

C₄

#16840842

yes

characteristic

195

C₂

—

244

—

196

C₂

#21462467

yes

characteristic

197

C₂

#21462473

yes

characteristic

198

C₂

C₂²×#14519848

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	Q₈
cosocle	C₄×Dic₆₁	C₂⁴
hypercenter	C₈×Q₈	#6923
hypocenter	C₆₁	#17835
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

Q₈

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. #21461397	C₂	semidirect
3. #21461375	C₂	semidirect
4. #21461369	C₂	semidirect
120. C₆₁	#17835	semidirect

H ≅ G/N

type

2. #21461397

C₂

semidirect

3. #21461375

C₂

semidirect

4. #21461369

C₂

semidirect

120. C₆₁

#17835

semidirect

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

subgroup	index
#21462753	2
#21461397	2
#21461375	2
#21461369	2
C₈×#11758226	2
#17835	61

subgroup

index

Subgroups — #30612230

Conjugacy classes of subgroups

Complements

Maximal subgroups