Group #30606718 — subgroups

Subgroups — #30606718

Order 15616.

326 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 128 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	#21462031	C₂	1	2	yes	maximal
2	#21462020	C₂	1	1	yes	maximal, characteristic
3	#21461621	C₂	1	1	yes	maximal, characteristic
4	#21460875	C₂	1	2	yes	maximal
5	#364×D₆₁	C₂	1	1	yes	maximal, characteristic
6	C₂×D₆₁⋊C₁₆	C₂²	1	1	yes	characteristic
7	C₂×D₆₁⋊C₁₆	C₂²	1	1	yes	characteristic
8	D₆₁⋊#108	C₄	1	1	yes	characteristic
9	#16840772	C₄	1	1	yes	characteristic
10	#16840725	C₂²	1	2	yes	—
11	#16840651	C₄	1	2	yes	complemented
12	#16840650	C₄	1	2	yes	complemented
13	#16840649	—	2	2	no	—
14	C₁₆×#6923	—	2	2	no	—
15	#16840633	C₄	1	2	yes	—
16	M₃₂×D₆₁	C₂²	1	2	yes	—
17	C₄×C₈×D₆₁	C₂²	1	1	yes	characteristic
18	#16839583	C₄	1	1	yes	characteristic, complemented
19	#16839582	C₄	1	2	yes	complemented
20	C₆₁×#364	C₄	1	1	yes	characteristic, complemented
21	D₆₁⋊M₁₆	C₂×C₄	1	2	yes	—
22	#14519856	C₂×C₄	1	2	yes	—
23	D₆₁⋊C₁₆	—	2	1	no	—
24	D₆₁⋊C₁₆	—	2	1	no	—
25	D₆₁⋊C₁₆	D₄	1	2	yes	—
26	D₆₁⋊C₁₆	Q₈	1	2	yes	—
27	C₂×#11758187	C₂×C₄	1	2	yes	—
28	C₂×#11758187	C₂×C₄	1	2	yes	—
29	#14519848	C₈	1	2	yes	—
30	#14519847	C₈	1	2	yes	—
31	#14519846	C₈	1	2	yes	—
32	#14519845	C₈	1	2	yes	—
33	#14519831	—	2	2	no	—
34	C₈×#6923	—	2	2	no	—
35	C₂×C₈×D₆₁	C₂×C₄	1	1	yes	characteristic
36	C₂×C₈×D₆₁	C₂³	1	1	yes	characteristic
37	C₄²×D₆₁	C₂×C₄	1	1	yes	characteristic
38	#14519717	C₂×C₄	1	2	yes	—
39	#14519716	—	2	2	no	—
40	C₁₆×D₆₁	—	2	2	no	—
41	C₄×#92418	C₂×C₄	1	1	yes	characteristic
42	C₈×Dic₆₁	C₂×C₄	1	1	yes	characteristic
43	C₈×Dic₆₁	C₂×C₄	1	1	yes	characteristic
44	M₃₂×C₆₁	C₂×C₄	1	2	yes	—
45	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄×D₆₁	—	2	1	no	—
47	C₂×C₄×D₆₁	C₂²×C₄	1	1	yes	characteristic
48	C₂²⋊#6923	—	2	2	no	—
49	#11758217	—	2	2	no	—
50	C₄⋊#6923	C₂×C₈	1	4	yes	—
51	C₄×#6923	—	2	2	no	—
52	#11758213	C₂×C₈	1	4	yes	—
53	D₆₁⋊C₈	—	2	2	no	—
54	C₂×C₈×C₆₁	C₄²	1	1	yes	characteristic
55	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
56	C₄²×C₆₁	C₄²	1	1	yes	characteristic
57	C₄×Dic₆₁	—	2	1	no	—
58	C₄×Dic₆₁	C₄²	1	1	yes	characteristic
59	C₂×#92418	C₄²	1	1	yes	characteristic
60	C₂×#92418	C₂²×C₄	1	1	yes	characteristic
61	C₈×D₆₁	C₂×C₈	1	1	yes	characteristic
62	C₈×D₆₁	C₂×C₈	1	1	yes	characteristic
63	C₈×D₆₁	C₂×C₈	1	1	yes	characteristic
64	C₈×D₆₁	C₂×C₈	1	1	yes	characteristic
65	C₈×D₆₁	C₂×D₄	1	1	yes	characteristic
66	C₈×D₆₁	C₂×Q₈	1	1	yes	characteristic
67	C₈×D₆₁	#41	1	1	yes	characteristic
68	C₈×D₆₁	#41	1	1	yes	characteristic
69	#11758187	—	2	2	no	—
70	#11758187	C₄⋊C₄	1	4	yes	—
71	C₁₆×C₆₁	—	2	2	no	—
72	#11758185	—	2	2	no	—
73	C₂²×D₆₁	#131	1	1	yes	characteristic
74	C₂×#6923	—	2	4	no	—
75	C₂×C₄×C₆₁	—	2	1	no	—
76	C₂×C₄×C₆₁	C₂×C₄²	1	1	yes	characteristic
77	C₂×Dic₆₁	—	2	1	no	—
78	C₂×Dic₆₁	#131	1	1	yes	characteristic
79	C₄×D₆₁	—	4	2	no	—
80	C₄×D₆₁	C₄×D₄	1	1	yes	characteristic
81	C₄×D₆₁	C₄×Q₈	1	1	yes	characteristic
82	C₄×D₆₁	C₂²×C₈	1	1	yes	characteristic
83	C₄×D₆₁	C₂²×C₈	1	1	yes	characteristic
84	#92420	—	2	4	no	—
85	C₈×C₆₁	C₄×C₈	1	1	yes	characteristic
86	C₈×C₆₁	C₄×C₈	1	1	yes	characteristic
87	C₈×C₆₁	C₄×D₄	1	1	yes	characteristic
88	C₈×C₆₁	C₄×Q₈	1	1	yes	characteristic
89	#92418	C₄×C₈	1	1	yes	characteristic
90	#92418	C₄×C₈	1	1	yes	characteristic
91	#92418	C₂×C₄⋊C₄	1	1	yes	characteristic
92	#92418	#117	1	1	yes	characteristic
93	#12665	—	61	1	no	maximal, Sylow
94	C₂²×C₆₁	#405	1	1	yes	characteristic
95	C₂×D₆₁	—	2	2	no	—
96	C₂×D₆₁	C₈×D₄	1	1	yes	characteristic
97	C₂×D₆₁	C₈×Q₈	1	1	yes	characteristic
98	#6923	—	4	4	no	—
99	C₄×C₆₁	—	4	1	no	—
100	C₄×C₆₁	C₄×C₄⋊C₄	1	1	yes	characteristic
101	C₄×C₆₁	C₂×C₄×C₈	1	1	yes	characteristic
102	Dic₆₁	—	4	1	no	—
103	Dic₆₁	C₈×D₄	1	1	yes	characteristic
104	Dic₆₁	C₈×Q₈	1	1	yes	characteristic
105	C₂×#364	—	61	1	no	—
106	#2108	—	61	2	no	—
107	C₂×#346	—	61	1	no	—
108	#1772	—	61	1	no	—
109	#1379	—	61	2	no	—
110	C₂×C₆₁	—	2	1	no	—
111	C₂×C₆₁	C₈×C₄⋊C₄	1	1	yes	characteristic
112	D₆₁	—	2	1	no	—
113	D₆₁	#2171	1	2	yes	—
114	C₂×M₃₂	—	61	2	no	—
115	C₂²×C₁₆	—	61	1	no	—
116	C₂²×C₁₆	—	61	1	no	—
117	C₂×#108	—	61	1	no	—
118	#427	—	61	1	no	—
119	#405	—	61	2	no	—
120	C₂×C₄×C₈	—	61	1	no	—
121	#364	—	61	2	no	—
122	#364	—	61	1	no	—
123	#364	#6923	1	1	yes	characteristic, complemented
124	#346	—	61	2	no	—
125	#346	—	61	2	no	—
126	#346	—	122	2	no	—
127	C₄×C₁₆	—	122	2	no	—
128	#339	—	61	2	no	—
129	C₆₁	#12665	1	1	yes	characteristic, complemented, Sylow
130	C₂×M₁₆	—	61	2	no	—
131	C₂²×C₈	—	61	1	no	—
132	C₂²×C₈	—	61	1	no	—
133	#117	—	61	2	no	—
134	C₂×C₄²	—	61	1	no	—
135	M₃₂	—	61	2	no	—
136	M₃₂	—	122	2	no	—
137	M₃₂	C₂×#6923	1	2	yes	—
138	C₂×C₁₆	—	61	2	no	—
139	C₂×C₁₆	—	61	2	no	—
140	C₂×C₁₆	—	61	2	no	—
141	C₂×C₁₆	—	61	2	no	—
142	C₂×C₁₆	—	122	2	no	—
143	C₂×C₁₆	—	122	1	no	—
144	C₂×C₁₆	—	122	1	no	—
145	#108	—	61	2	no	—
146	#108	—	61	2	no	—
147	#107	—	61	2	no	—
148	#106	—	61	2	no	—
149	#97	—	122	2	no	—
150	C₄×C₈	—	61	1	no	—
151	C₄×C₈	—	61	1	no	—
152	C₄×C₈	—	61	1	no	—
153	C₄×C₈	—	122	2	no	—
154	C₄×C₈	C₂×#6923	1	1	yes	characteristic
155	C₂²×C₄	—	61	1	no	—
156	C₂²×C₄	—	122	1	no	—
157	M₁₆	—	61	4	no	—
158	M₁₆	—	122	2	no	—
159	C₂×C₈	—	61	1	no	—
160	C₂×C₈	—	61	1	no	—
161	C₂×C₈	—	61	1	no	—
162	C₂×C₈	—	61	1	no	—
163	C₂×C₈	—	61	1	no	—
164	C₂×C₈	—	61	1	no	—
165	C₂×C₈	—	61	1	no	—
166	C₂×C₈	—	61	1	no	—
167	C₂×C₈	—	61	1	no	—
168	C₂×C₈	—	61	1	no	—
169	C₂×C₈	—	122	2	no	—
170	C₂×C₈	C₄×#6923	1	1	yes	characteristic
171	C₂×C₈	C₂²×#6923	1	1	yes	characteristic
172	C₄⋊C₄	—	61	4	no	—
173	C₂²⋊C₄	—	122	2	no	—
174	C₄²	—	61	1	no	—
175	C₄²	—	122	1	no	—
176	C₄²	—	122	2	no	—
177	C₄²	C₄×#6923	1	1	yes	characteristic
178	C₁₆	—	2	2	no	—
179	C₁₆	—	61	4	no	—
180	C₁₆	—	122	2	no	—
181	C₁₆	—	122	2	no	—
182	C₂³	—	61	1	no	—
183	C₂×C₄	—	2	1	no	—
184	C₂×C₄	—	61	1	no	—
185	C₂×C₄	—	61	1	no	—
186	C₂×C₄	—	61	1	no	—
187	C₂×C₄	—	61	1	no	—
188	C₂×C₄	—	61	1	no	—
189	C₂×C₄	—	122	4	no	—
190	C₂×C₄	—	122	1	no	—
191	C₂×C₄	—	244	2	no	—
192	C₂×C₄	C₂×C₄×#6923	1	1	yes	characteristic
193	C₈	—	61	1	no	—
194	C₈	—	61	1	no	—
195	C₈	—	61	1	no	—
196	C₈	—	61	1	no	—
197	C₈	—	122	4	no	—
198	C₈	C₈×#6923	1	1	yes	characteristic
199	C₈	C₈×#6923	1	1	yes	characteristic
200	C₈	D₄×#6923	1	1	yes	characteristic
201	C₈	Q₈×#6923	1	1	yes	characteristic
202	C₂²	—	61	1	no	—
203	C₂²	—	61	1	no	—
204	C₂²	—	122	2	no	—
205	C₂²	#16840728	1	1	yes	characteristic
206	C₄	—	4	1	no	—
207	C₄	—	61	1	no	—
208	C₄	—	61	1	no	—
209	C₄	—	244	1	no	—
210	C₄	—	244	4	no	—
211	C₄	C₄⋊C₄×#6923	1	1	yes	characteristic
212	C₄	C₂×C₈×#6923	1	1	yes	characteristic
213	C₂	—	2	1	no	—
214	C₂	—	61	2	no	—
215	C₂	—	122	1	no	—
216	C₂	#21461612	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21462031

C₂

yes

maximal

#21462020

C₂

yes

maximal, characteristic

#21461621

C₂

yes

maximal, characteristic

#21460875

C₂

yes

maximal

#364×D₆₁

C₂

yes

maximal, characteristic

C₂×D₆₁⋊C₁₆

C₂²

yes

characteristic

C₂×D₆₁⋊C₁₆

C₂²

yes

characteristic

D₆₁⋊#108

C₄

yes

characteristic

#16840772

C₄

yes

characteristic

#16840725

C₂²

yes

—

#16840651

C₄

yes

complemented

#16840650

C₄

yes

complemented

#16840649

—

C₁₆×#6923

—

#16840633

C₄

yes

—

M₃₂×D₆₁

C₂²

yes

—

C₄×C₈×D₆₁

C₂²

yes

characteristic

#16839583

C₄

yes

characteristic, complemented

#16839582

C₄

yes

complemented

C₆₁×#364

C₄

yes

characteristic, complemented

D₆₁⋊M₁₆

C₂×C₄

yes

—

#14519856

C₂×C₄

yes

—

D₆₁⋊C₁₆

—

D₆₁⋊C₁₆

—

D₆₁⋊C₁₆

D₄

yes

—

D₆₁⋊C₁₆

Q₈

yes

—

C₂×#11758187

C₂×C₄

yes

—

C₂×#11758187

C₂×C₄

yes

—

#14519848

C₈

yes

—

#14519847

C₈

yes

—

#14519846

C₈

yes

—

#14519845

C₈

yes

—

#14519831

—

C₈×#6923

—

C₂×C₈×D₆₁

C₂×C₄

yes

characteristic

C₂×C₈×D₆₁

C₂³

yes

characteristic

C₄²×D₆₁

C₂×C₄

yes

characteristic

#14519717

C₂×C₄

yes

—

#14519716

—

C₁₆×D₆₁

—

C₄×#92418

C₂×C₄

yes

characteristic

C₈×Dic₆₁

C₂×C₄

yes

characteristic

C₈×Dic₆₁

C₂×C₄

yes

characteristic

M₃₂×C₆₁

C₂×C₄

yes

—

C₄×C₈×C₆₁

C₂×C₄

yes

characteristic

C₂×C₄×D₆₁

—

C₂×C₄×D₆₁

C₂²×C₄

yes

characteristic

C₂²⋊#6923

—

#11758217

—

C₄⋊#6923

C₂×C₈

yes

—

C₄×#6923

—

#11758213

C₂×C₈

yes

—

D₆₁⋊C₈

—

C₂×C₈×C₆₁

C₄²

yes

characteristic

C₂×C₈×C₆₁

C₂²×C₄

yes

characteristic

C₄²×C₆₁

C₄²

yes

characteristic

C₄×Dic₆₁

—

C₄×Dic₆₁

C₄²

yes

characteristic

C₂×#92418

C₄²

yes

characteristic

C₂×#92418

C₂²×C₄

yes

characteristic

C₈×D₆₁

C₂×C₈

yes

characteristic

C₈×D₆₁

C₂×C₈

yes

characteristic

C₈×D₆₁

C₂×C₈

yes

characteristic

C₈×D₆₁

C₂×C₈

yes

characteristic

C₈×D₆₁

C₂×D₄

yes

characteristic

C₈×D₆₁

C₂×Q₈

yes

C₈×D₆₁

yes

C₈×D₆₁

yes

—

C₄⋊C₄

yes

—

C₁₆×C₆₁

—

#11758185

—

C₂²×D₆₁

#131

yes

characteristic

C₂×#6923

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂×C₄²

yes

characteristic

C₂×Dic₆₁

—

C₂×Dic₆₁

#131

yes

characteristic

C₄×D₆₁

—

C₄×D₆₁

C₄×D₄

yes

characteristic

C₄×D₆₁

C₄×Q₈

yes

characteristic

C₄×D₆₁

C₂²×C₈

yes

characteristic

C₄×D₆₁

C₂²×C₈

yes

characteristic

#92420

—

C₈×C₆₁

C₄×C₈

yes

characteristic

C₈×C₆₁

C₄×C₈

yes

characteristic

C₈×C₆₁

C₄×D₄

yes

characteristic

C₈×C₆₁

C₄×Q₈

yes

characteristic

#92418

C₄×C₈

yes

characteristic

#92418

C₄×C₈

yes

characteristic

#92418

C₂×C₄⋊C₄

yes

yes

—

C₂²×C₆₁

yes

C₂×D₆₁

—

C₂×D₆₁

C₈×D₄

yes

characteristic

C₂×D₆₁

C₈×Q₈

yes

characteristic

#6923

—

C₄×C₆₁

—

100

C₄×C₆₁

C₄×C₄⋊C₄

yes

characteristic

101

C₄×C₆₁

C₂×C₄×C₈

yes

characteristic

102

Dic₆₁

—

103

Dic₆₁

C₈×D₄

yes

characteristic

104

Dic₆₁

C₈×Q₈

yes

characteristic

105

C₂×#364

—

106

#2108

—

107

C₂×#346

—

108

#1772

—

109

#1379

—

110

C₂×C₆₁

—

111

C₂×C₆₁

C₈×C₄⋊C₄

yes

characteristic

112

D₆₁

—

113

D₆₁

#2171

yes

—

114

C₂×M₃₂

—

115

C₂²×C₁₆

—

116

C₂²×C₁₆

—

117

C₂×#108

—

118

#427

—

119

#405

—

120

C₂×C₄×C₈

—

121

#364

—

122

#364

—

123

#364

#6923

yes

characteristic, complemented

124

#346

—

125

#346

—

126

#346

—

122

—

127

C₄×C₁₆

—

122

—

128

#339

—

129

C₆₁

#12665

yes

characteristic, complemented, Sylow

130

C₂×M₁₆

—

131

C₂²×C₈

—

132

C₂²×C₈

—

133

#117

—

134

C₂×C₄²

—

135

M₃₂

—

136

M₃₂

—

122

—

137

M₃₂

C₂×#6923

yes

—

138

C₂×C₁₆

—

139

C₂×C₁₆

—

140

C₂×C₁₆

—

141

C₂×C₁₆

—

142

C₂×C₁₆

—

122

—

143

C₂×C₁₆

—

122

—

144

C₂×C₁₆

—

122

—

145

#108

—

146

#108

—

147

#107

—

148

#106

—

149

#97

—

122

—

150

C₄×C₈

—

151

C₄×C₈

—

152

C₄×C₈

—

153

C₄×C₈

—

122

—

154

C₄×C₈

C₂×#6923

yes

characteristic

155

C₂²×C₄

—

156

C₂²×C₄

—

122

—

157

M₁₆

—

158

M₁₆

—

122

—

159

C₂×C₈

—

160

C₂×C₈

—

161

C₂×C₈

—

162

C₂×C₈

—

163

C₂×C₈

—

164

C₂×C₈

—

165

C₂×C₈

—

166

C₂×C₈

—

167

C₂×C₈

—

168

C₂×C₈

—

169

C₂×C₈

—

122

—

170

C₂×C₈

C₄×#6923

yes

characteristic

171

C₂×C₈

C₂²×#6923

yes

characteristic

172

C₄⋊C₄

—

173

C₂²⋊C₄

—

122

—

174

C₄²

—

175

C₄²

—

122

—

176

C₄²

—

122

—

177

C₄²

C₄×#6923

yes

characteristic

178

C₁₆

—

179

C₁₆

—

180

C₁₆

—

122

—

181

C₁₆

—

122

—

182

C₂³

—

183

C₂×C₄

—

184

C₂×C₄

—

185

C₂×C₄

—

186

C₂×C₄

—

187

C₂×C₄

—

188

C₂×C₄

—

189

C₂×C₄

—

122

—

190

C₂×C₄

—

122

—

191

C₂×C₄

—

244

—

192

C₂×C₄

C₂×C₄×#6923

yes

characteristic

193

C₈

—

194

C₈

—

195

C₈

—

196

C₈

—

197

C₈

—

122

—

198

C₈

C₈×#6923

yes

characteristic

199

C₈

C₈×#6923

yes

characteristic

200

C₈

D₄×#6923

yes

characteristic

201

C₈

Q₈×#6923

yes

characteristic

202

C₂²

—

203

C₂²

—

204

C₂²

—

122

—

205

C₂²

#16840728

yes

characteristic

206

C₄

—

207

C₄

—

208

C₄

—

209

C₄

—

244

—

210

C₄

—

244

—

211

C₄

C₄⋊C₄×#6923

yes

characteristic

212

C₄

C₂×C₈×#6923

yes

characteristic

213

C₂

—

214

C₂

—

215

C₂

—

122

—

216

C₂

#21461612

yes

characteristic

	subgroup	G/H
center	C₈	D₄×#6923
commutator	C₄×C₆₁	C₂×C₄×C₈
Frattini	C₂×C₈	C₂²×#6923
Fitting	C₆₁×#364	C₄
socle	C₂×C₆₁	C₈×C₄⋊C₄
prime radical	C₂²×D₆₁	#131
cosocle	C₂×C₈×D₆₁	C₂³
hypercenter	#364	#6923
hypocenter	C₆₁	#12665
LCS	G ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ #364

subgroup

G/H

center

C₈

D₄×#6923

commutator

C₄×C₆₁

C₂×C₄×C₈

Frattini

C₂×C₈

C₂²×#6923

Fitting

C₆₁×#364

C₄

socle

C₂×C₆₁

C₈×C₄⋊C₄

prime radical

C₂²×D₆₁

#131

cosocle

C₂×C₈×D₆₁

C₂³

C₆₁

G ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₈ ◁ C₂×C₈ ◁ #364

N	H ≅ G/N	type
11. #16840651	C₄	semidirect
12. #16840650	C₄	semidirect
18. #16839583	C₄	semidirect
19. #16839582	C₄	semidirect
20. C₆₁×#364	C₄	semidirect
123. #364	#6923	semidirect
129. C₆₁	#12665	semidirect

H ≅ G/N

type

11. #16840651

C₄

semidirect

12. #16840650

C₄

semidirect

18. #16839583

C₄

semidirect

19. #16839582

C₄

semidirect

20. C₆₁×#364

C₄

semidirect

123. #364

#6923

semidirect

129. C₆₁

#12665

semidirect

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

subgroup	index
#21462031	2
#21462020	2
#21461621	2
#21460875	2
#364×D₆₁	2
#12665	61

subgroup

index

#364×D₆₁

Subgroups — #30606718

Conjugacy classes of subgroups

Complements

Maximal subgroups