Group #30598062 — subgroups

Subgroups — #30598062

Order 15616.

398 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 132 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	C₄×#14519846	C₂	1	1	yes	maximal, characteristic
2	#21461616	C₂	1	2	yes	maximal
3	#21461013	C₂	1	1	yes	maximal, characteristic
4	#21460653	C₂	1	2	yes	maximal
5	#334×D₆₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#14519846	C₄	1	2	yes	—
7	C₂×#14519831	—	2	2	no	—
8	C₄×D₆₁⋊C₈	C₂²	1	1	yes	characteristic
9	C₄×D₆₁⋊C₈	C₂²	1	1	yes	characteristic
10	C₄×C₄⋊#6923	C₂²	1	2	yes	—
11	#16840636	C₄	1	2	yes	—
12	(C₂×#6923)⋊C₈	—	2	2	no	—
13	#16840603	C₄	1	2	yes	complemented
14	#16840603	C₄	1	2	yes	complemented
15	C₄⋊C₈×D₆₁	C₂²	1	2	yes	—
16	C₄×C₈×D₆₁	C₂²	1	1	yes	characteristic
17	#16839513	C₄	1	1	yes	characteristic, complemented
18	#16839512	C₄	1	2	yes	complemented
19	C₆₁×#334	C₄	1	1	yes	characteristic, complemented
20	C₂×D₆₁⋊C₈	—	2	1	no	—
21	C₂×D₆₁⋊C₈	D₄	1	2	yes	—
22	C₂×C₄⋊#6923	C₂×C₄	1	4	yes	—
23	C₂×C₄×#6923	—	2	2	no	—
24	#14519846	—	2	4	no	—
25	#14519831	—	4	4	no	—
26	C₄×#92420	—	2	1	no	—
27	C₄×#92420	C₂×C₄	1	2	yes	—
28	C₄×#92420	C₂×C₄	1	2	yes	—
29	C₄×#92420	Q₈	1	2	yes	—
30	(C₂×#6923)⋊C₄	—	2	2	no	—
31	C₂×C₈×D₆₁	—	2	2	no	—
32	C₂×C₈×D₆₁	C₂×C₄	1	1	yes	characteristic
33	C₂×C₈×D₆₁	C₂×C₄	1	1	yes	characteristic
34	C₄²×D₆₁	C₂³	1	1	yes	characteristic
35	C₄⋊#92418	C₂×C₄	1	2	yes	—
36	#14519706	—	2	2	no	—
37	C₄×#92418	C₂×C₄	1	1	yes	characteristic
38	C₈×Dic₆₁	C₂×C₄	1	1	yes	characteristic
39	C₈×Dic₆₁	C₂×C₄	1	1	yes	characteristic
40	C₄⋊C₈×C₆₁	C₂×C₄	1	2	yes	—
41	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
42	C₂²×#6923	—	2	4	no	—
43	C₂×C₄×D₆₁	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₄×D₆₁	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×D₆₁	#41	1	1	yes	characteristic
46	C₂×#92420	—	2	2	no	—
47	C₂×#92420	C₄⋊C₄	1	4	yes	—
48	C₄⋊#6923	—	2	4	no	—
49	C₄⋊#6923	—	2	4	no	—
50	C₄×#6923	—	4	2	no	—
51	C₄×#6923	—	4	2	no	—
52	D₆₁⋊C₈	—	2	2	no	—
53	D₆₁⋊C₈	—	2	2	no	—
54	D₆₁⋊C₈	—	4	2	no	—
55	C₂×C₈×C₆₁	—	2	2	no	—
56	C₂×C₈×C₆₁	C₄²	1	1	yes	characteristic
57	C₂×C₈×C₆₁	C₄²	1	1	yes	characteristic
58	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
59	C₄×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
60	C₄×Dic₆₁	C₂×Q₈	1	1	yes	characteristic
61	C₄×Dic₆₁	#41	1	1	yes	characteristic
62	C₂×#92418	—	2	2	no	—
63	C₂×#92418	C₄²	1	1	yes	characteristic
64	C₂×#92418	C₄²	1	1	yes	characteristic
65	C₈×D₆₁	—	2	2	no	—
66	C₈×D₆₁	—	4	2	no	—
67	C₈×D₆₁	—	4	2	no	—
68	C₈×D₆₁	M₁₆	1	4	yes	—
69	C₂²×D₆₁	C₄×D₄	1	1	yes	characteristic
70	C₂×#6923	—	4	4	no	—
71	C₂×#6923	—	4	4	no	—
72	C₂×#6923	—	4	4	no	—
73	C₂×C₄×C₆₁	C₂×C₄²	1	1	yes	characteristic
74	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
75	C₂×C₄×C₆₁	C₄×Q₈	1	1	yes	characteristic
76	C₂×Dic₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
77	C₂×Dic₆₁	#117	1	1	yes	characteristic
78	C₂×Dic₆₁	C₄×Q₈	1	1	yes	characteristic
79	C₄×D₆₁	—	2	2	no	—
80	C₄×D₆₁	C₂×M₁₆	1	2	yes	—
81	C₄×D₆₁	#131	1	2	yes	—
82	C₄×D₆₁	AGL(1,ℤ/8)	1	2	yes	—
83	C₄×D₆₁	#137	1	2	yes	—
84	#92420	—	2	4	no	—
85	#92420	—	4	2	no	—
86	C₈×C₆₁	—	2	1	no	—
87	C₈×C₆₁	—	4	2	no	—
88	C₈×C₆₁	#97	1	2	yes	complemented
89	#92418	—	2	1	no	—
90	#92418	—	4	2	no	—
91	#92418	#97	1	2	yes	complemented
92	#8287	—	61	1	no	maximal, Sylow
93	C₂²×C₆₁	C₄×C₄⋊C₄	1	1	yes	characteristic
94	C₂×D₆₁	#436	1	2	yes	—
95	C₂×D₆₁	#441	1	1	yes	characteristic
96	C₂×D₆₁	#442	1	1	yes	characteristic
97	C₂×D₆₁	#444	1	2	yes	—
98	#6923	—	8	4	no	—
99	C₄×C₆₁	—	2	1	no	—
100	C₄×C₆₁	C₂×#97	1	1	yes	characteristic
101	C₄×C₆₁	#405	1	1	yes	characteristic
102	C₄×C₆₁	#440	1	2	yes	—
103	Dic₆₁	—	2	1	no	—
104	Dic₆₁	#428	1	2	yes	—
105	Dic₆₁	#446	1	2	yes	—
106	C₄×#107	—	61	1	no	—
107	#1770	—	61	2	no	—
108	C₂×#334	—	61	1	no	—
109	C₂×#322	—	61	1	no	—
110	#1279	—	61	2	no	—
111	C₂×C₆₁	#1770	1	1	yes	characteristic
112	C₂×C₆₁	#1773	1	1	yes	characteristic
113	C₂×C₆₁	#1776	1	1	yes	characteristic
114	D₆₁	#1593	1	2	yes	—
115	D₆₁	#1594	1	2	yes	—
116	C₂×#107	—	61	2	no	—
117	C₂×C₄⋊C₈	—	61	2	no	—
118	C₂×#97	—	122	2	no	—
119	C₂×C₄×C₈	—	61	1	no	—
120	C₂×C₄×C₈	—	61	1	no	—
121	C₂×C₄×C₈	—	61	1	no	—
122	C₄×C₄⋊C₄	—	61	2	no	—
123	#340	—	61	2	no	—
124	#336	—	122	2	no	—
125	#334	—	61	2	no	—
126	#334	—	61	1	no	—
127	#334	#6923	1	1	yes	characteristic, complemented
128	#322	—	61	2	no	—
129	#322	—	61	2	no	—
130	C₆₁	#8287	1	1	yes	characteristic, complemented, Sylow
131	C₂²×C₈	—	61	1	no	—
132	C₂²×C₈	—	61	2	no	—
133	C₂²×C₈	—	61	1	no	—
134	C₂²×C₈	—	122	2	no	—
135	C₂²×C₈	—	122	1	no	—
136	C₂×C₄⋊C₄	—	61	4	no	—
137	C₂×C₄²	—	61	1	no	—
138	C₂×C₄²	—	122	2	no	—
139	#107	—	122	4	no	—
140	C₄⋊C₈	—	61	2	no	—
141	C₄⋊C₈	—	122	2	no	—
142	C₄⋊C₈	C₂×#6923	1	2	yes	—
143	#97	—	244	4	no	—
144	C₄×C₈	—	61	2	no	—
145	C₄×C₈	—	61	2	no	—
146	C₄×C₈	—	61	1	no	—
147	C₄×C₈	—	61	1	no	—
148	C₄×C₈	—	61	2	no	—
149	C₄×C₈	—	61	1	no	—
150	C₄×C₈	—	122	1	no	—
151	C₄×C₈	C₂×#6923	1	1	yes	characteristic
152	#95	—	122	2	no	—
153	C₂²×C₄	—	61	1	no	—
154	C₂²×C₄	—	61	1	no	—
155	C₂²×C₄	—	61	1	no	—
156	C₂²×C₄	—	122	4	no	—
157	C₂×C₈	—	2	2	no	—
158	C₂×C₈	—	61	1	no	—
159	C₂×C₈	—	61	4	no	—
160	C₂×C₈	—	61	4	no	—
161	C₂×C₈	—	61	1	no	—
162	C₂×C₈	—	122	2	no	—
163	C₂×C₈	—	122	2	no	—
164	C₂×C₈	—	122	2	no	—
165	C₂×C₈	—	122	2	no	—
166	C₂×C₈	—	122	2	no	—
167	C₂×C₈	—	244	2	no	—
168	C₂×C₈	—	244	2	no	—
169	C₂×C₈	—	244	2	no	—
170	C₂×C₈	C₄×#6923	1	1	yes	characteristic
171	C₂×C₈	C₄×#6923	1	1	yes	characteristic
172	C₄⋊C₄	—	122	4	no	—
173	C₄⋊C₄	—	122	4	no	—
174	C₄²	—	61	1	no	—
175	C₄²	—	61	1	no	—
176	C₄²	—	61	1	no	—
177	C₄²	—	244	2	no	—
178	C₄²	—	244	2	no	—
179	C₄²	C₂²×#6923	1	1	yes	characteristic
180	C₂³	—	61	1	no	—
181	C₂×C₄	—	61	1	no	—
182	C₂×C₄	—	61	1	no	—
183	C₂×C₄	—	61	2	no	—
184	C₂×C₄	—	61	2	no	—
185	C₂×C₄	—	61	2	no	—
186	C₂×C₄	—	61	2	no	—
187	C₂×C₄	—	61	1	no	—
188	C₂×C₄	—	122	2	no	—
189	C₂×C₄	—	244	4	no	—
190	C₂×C₄	—	244	4	no	—
191	C₂×C₄	—	244	4	no	—
192	C₂×C₄	C₂×C₄×#6923	1	1	yes	characteristic
193	C₂×C₄	D₄×#6923	1	1	yes	characteristic
194	C₂×C₄	Q₈×#6923	1	1	yes	characteristic
195	C₈	—	2	1	no	—
196	C₈	—	4	2	no	—
197	C₈	—	61	2	no	—
198	C₈	—	122	4	no	—
199	C₈	—	122	1	no	—
200	C₈	—	244	2	no	—
201	C₈	—	244	2	no	—
202	C₈	#14519831	1	2	yes	complemented
203	C₂²	—	61	2	no	—
204	C₂²	—	61	2	no	—
205	C₂²	—	61	1	no	—
206	C₂²	—	61	1	no	—
207	C₂²	C₄⋊C₄×#6923	1	1	yes	characteristic
208	C₄	—	2	1	no	—
209	C₄	—	61	2	no	—
210	C₄	—	61	2	no	—
211	C₄	—	122	1	no	—
212	C₄	—	488	4	no	—
213	C₄	C₂×#14519831	1	1	yes	characteristic
214	C₄	#16840728	1	1	yes	characteristic
215	C₄	#16840801	1	2	yes	—
216	C₂	—	61	2	no	—
217	C₂	—	61	2	no	—
218	C₂	#21461616	1	1	yes	characteristic
219	C₂	#21461628	1	1	yes	characteristic
220	C₂	#21461635	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#14519846

C₂

yes

maximal, characteristic

#21461616

C₂

yes

maximal

#21461013

C₂

yes

maximal, characteristic

#21460653

C₂

yes

maximal

#334×D₆₁

C₂

yes

maximal, characteristic

C₂×#14519846

C₄

yes

—

C₂×#14519831

—

C₄×D₆₁⋊C₈

C₂²

yes

characteristic

C₄×D₆₁⋊C₈

C₂²

yes

characteristic

C₄×C₄⋊#6923

C₂²

yes

—

#16840636

C₄

yes

—

(C₂×#6923)⋊C₈

—

#16840603

C₄

yes

complemented

#16840603

C₄

yes

complemented

C₄⋊C₈×D₆₁

C₂²

yes

—

C₄×C₈×D₆₁

C₂²

yes

characteristic

#16839513

C₄

yes

characteristic, complemented

#16839512

C₄

yes

complemented

C₆₁×#334

C₄

yes

characteristic, complemented

C₂×D₆₁⋊C₈

—

C₂×D₆₁⋊C₈

D₄

yes

—

C₂×C₄⋊#6923

C₂×C₄

yes

—

C₂×C₄×#6923

—

—

—

—

C₂×C₄

yes

—

C₄×#92420

C₂×C₄

yes

—

C₄×#92420

Q₈

yes

—

(C₂×#6923)⋊C₄

—

C₂×C₈×D₆₁

—

C₂×C₈×D₆₁

C₂×C₄

yes

characteristic

C₂×C₈×D₆₁

C₂×C₄

yes

characteristic

C₄²×D₆₁

C₂³

yes

characteristic

C₄⋊#92418

C₂×C₄

yes

—

#14519706

—

C₄×#92418

C₂×C₄

yes

characteristic

C₈×Dic₆₁

C₂×C₄

yes

characteristic

C₈×Dic₆₁

C₂×C₄

yes

characteristic

C₄⋊C₈×C₆₁

C₂×C₄

yes

—

C₄×C₈×C₆₁

C₂×C₄

yes

characteristic

C₂²×#6923

—

C₂×C₄×D₆₁

C₂²×C₄

yes

characteristic

C₂×C₄×D₆₁

C₂×D₄

yes

C₂×C₄×D₆₁

yes

—

C₄⋊C₄

yes

—

—

—

—

—

D₆₁⋊C₈

—

D₆₁⋊C₈

—

D₆₁⋊C₈

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

C₄²

yes

characteristic

C₂×C₈×C₆₁

C₄²

yes

characteristic

C₄²×C₆₁

C₂²×C₄

yes

characteristic

C₄×Dic₆₁

C₂²×C₄

yes

characteristic

C₄×Dic₆₁

C₂×Q₈

yes

C₄×Dic₆₁

yes

—

C₄²

yes

characteristic

C₂×#92418

C₄²

yes

characteristic

C₈×D₆₁

—

C₈×D₆₁

—

C₈×D₆₁

—

C₈×D₆₁

M₁₆

yes

—

C₂²×D₆₁

C₄×D₄

yes

—

—

—

C₂×C₄×C₆₁

C₂×C₄²

yes

characteristic

C₂×C₄×C₆₁

C₄×D₄

yes

characteristic

C₂×C₄×C₆₁

C₄×Q₈

yes

characteristic

C₂×Dic₆₁

C₂×C₄⋊C₄

yes

characteristic

C₂×Dic₆₁

#117

yes

characteristic

C₂×Dic₆₁

C₄×Q₈

yes

characteristic

C₄×D₆₁

—

C₄×D₆₁

C₂×M₁₆

yes

—

C₄×D₆₁

#131

yes

—

C₄×D₆₁

AGL(1,ℤ/8)

yes

—

C₄×D₆₁

#137

yes

—

#92420

—

#92420

—

C₈×C₆₁

—

C₈×C₆₁

—

C₈×C₆₁

yes

—

—

yes

—

C₂²×C₆₁

C₄×C₄⋊C₄

yes

characteristic

C₂×D₆₁

#436

yes

—

C₂×D₆₁

yes

C₂×D₆₁

yes

C₂×D₆₁

yes

—

#6923

—

C₄×C₆₁

—

100

C₄×C₆₁

C₂×#97

yes

characteristic

101

C₄×C₆₁

#405

yes

characteristic

102

C₄×C₆₁

#440

yes

—

103

Dic₆₁

—

104

Dic₆₁

#428

yes

—

105

Dic₆₁

#446

yes

—

106

C₄×#107

—

107

#1770

—

108

C₂×#334

—

109

C₂×#322

—

110

#1279

—

111

C₂×C₆₁

#1770

yes

characteristic

112

C₂×C₆₁

#1773

yes

characteristic

113

C₂×C₆₁

#1776

yes

characteristic

114

D₆₁

#1593

yes

—

115

D₆₁

#1594

yes

—

116

C₂×#107

—

117

C₂×C₄⋊C₈

—

118

C₂×#97

—

122

—

119

C₂×C₄×C₈

—

120

C₂×C₄×C₈

—

121

C₂×C₄×C₈

—

122

C₄×C₄⋊C₄

—

123

#340

—

124

#336

—

122

—

125

#334

—

126

#334

—

127

#334

#6923

yes

characteristic, complemented

128

#322

—

129

#322

—

130

C₆₁

#8287

yes

characteristic, complemented, Sylow

131

C₂²×C₈

—

132

C₂²×C₈

—

133

C₂²×C₈

—

134

C₂²×C₈

—

122

—

135

C₂²×C₈

—

122

—

136

C₂×C₄⋊C₄

—

137

C₂×C₄²

—

138

C₂×C₄²

—

122

—

139

#107

—

122

—

140

C₄⋊C₈

—

141

C₄⋊C₈

—

122

—

142

C₄⋊C₈

C₂×#6923

yes

—

143

#97

—

244

—

144

C₄×C₈

—

145

C₄×C₈

—

146

C₄×C₈

—

147

C₄×C₈

—

148

C₄×C₈

—

149

C₄×C₈

—

150

C₄×C₈

—

122

—

151

C₄×C₈

C₂×#6923

yes

characteristic

152

#95

—

122

—

153

C₂²×C₄

—

154

C₂²×C₄

—

155

C₂²×C₄

—

156

C₂²×C₄

—

122

—

157

C₂×C₈

—

158

C₂×C₈

—

159

C₂×C₈

—

160

C₂×C₈

—

161

C₂×C₈

—

162

C₂×C₈

—

122

—

163

C₂×C₈

—

122

—

164

C₂×C₈

—

122

—

165

C₂×C₈

—

122

—

166

C₂×C₈

—

122

—

167

C₂×C₈

—

244

—

168

C₂×C₈

—

244

—

169

C₂×C₈

—

244

—

170

C₂×C₈

C₄×#6923

yes

characteristic

171

C₂×C₈

C₄×#6923

yes

characteristic

172

C₄⋊C₄

—

122

—

173

C₄⋊C₄

—

122

—

174

C₄²

—

175

C₄²

—

176

C₄²

—

177

C₄²

—

244

—

178

C₄²

—

244

—

179

C₄²

C₂²×#6923

yes

characteristic

180

C₂³

—

181

C₂×C₄

—

182

C₂×C₄

—

183

C₂×C₄

—

184

C₂×C₄

—

185

C₂×C₄

—

186

C₂×C₄

—

187

C₂×C₄

—

188

C₂×C₄

—

122

—

189

C₂×C₄

—

244

—

190

C₂×C₄

—

244

—

191

C₂×C₄

—

244

—

192

C₂×C₄

C₂×C₄×#6923

yes

characteristic

193

C₂×C₄

D₄×#6923

yes

characteristic

194

C₂×C₄

Q₈×#6923

yes

characteristic

195

C₈

—

196

C₈

—

197

C₈

—

198

C₈

—

122

—

199

C₈

—

122

—

200

C₈

—

244

—

201

C₈

—

244

—

202

C₈

#14519831

yes

complemented

203

C₂²

—

204

C₂²

—

205

C₂²

—

206

C₂²

—

207

C₂²

C₄⋊C₄×#6923

yes

characteristic

208

C₄

—

209

C₄

—

210

C₄

—

211

C₄

—

122

—

212

C₄

—

488

—

213

C₄

C₂×#14519831

yes

characteristic

214

C₄

#16840728

yes

characteristic

215

C₄

#16840801

yes

—

216

C₂

—

217

C₂

—

218

C₂

#21461616

yes

characteristic

219

C₂

#21461628

yes

characteristic

220

C₂

#21461635

yes

characteristic

	subgroup	G/H
center	C₂×C₄	D₄×#6923
commutator	C₂×C₄×C₆₁	C₂×C₄²
Frattini	C₄²	C₂²×#6923
Fitting	C₆₁×#334	C₄
socle	C₂²×C₆₁	C₄×C₄⋊C₄
prime radical	C₂²×D₆₁	C₄×D₄
cosocle	C₄²×D₆₁	C₂³
hypercenter	#334	#6923
hypocenter	C₆₁	#8287
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ #334

subgroup

G/H

center

C₂×C₄

D₄×#6923

commutator

C₂×C₄×C₆₁

C₂×C₄²

Frattini

C₄²

C₂²×#6923

Fitting

C₆₁×#334

C₄

socle

C₂²×C₆₁

C₄×C₄⋊C₄

prime radical

C₂²×D₆₁

C₄×D₄

cosocle

C₄²×D₆₁

C₂³

C₆₁

G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ #334

N	H ≅ G/N	type
13. #16840603	C₄	semidirect
14. #16840603	C₄	semidirect
17. #16839513	C₄	semidirect
18. #16839512	C₄	semidirect
19. C₆₁×#334	C₄	semidirect
88. C₈×C₆₁	#97	semidirect
91. #92418	#97	semidirect
127. #334	#6923	semidirect
130. C₆₁	#8287	semidirect
202. C₈	#14519831	semidirect

H ≅ G/N

type

13. #16840603

C₄

semidirect

14. #16840603

C₄

semidirect

17. #16839513

C₄

semidirect

18. #16839512

C₄

semidirect

19. C₆₁×#334

C₄

semidirect

88. C₈×C₆₁

#97

semidirect

91. #92418

#97

semidirect

127. #334

#6923

semidirect

130. C₆₁

#8287

semidirect

202. C₈

#14519831

semidirect

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

subgroup	index
C₄×#14519846	2
#21461616	2
#21461013	2
#21460653	2
#334×D₆₁	2
#8287	61

subgroup

index

#334×D₆₁

Subgroups — #30598062

Conjugacy classes of subgroups

Complements

Maximal subgroups