Group #30599514 — subgroups

Subgroups — #30599514

Order 15616.

382 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 140 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	#21461751	C₂	1	1	yes	maximal, characteristic
2	#21461736	C₂	1	1	yes	maximal, characteristic
3	C₂×#16840626	C₂	1	2	yes	maximal
4	C₂×#16840623	C₂	1	2	yes	maximal
5	#21445563	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#92420	—	2	2	no	—
7	C₂×C₄⋊#92420	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊#92420	C₂²	1	2	yes	—
9	C₂×C₄⋊#92420	C₂²	1	1	yes	characteristic
10	#16840628	—	2	2	no	—
11	#16840626	—	2	4	no	—
12	#16840623	—	2	4	no	—
13	C₂×C₈×Dic₆₁	C₂²	1	2	yes	—
14	C₄²×Dic₆₁	C₂²	1	1	yes	characteristic
15	#16839517	C₄	1	1	yes	characteristic
16	#16839516	C₄	1	2	yes	—
17	C₆₁×#336	C₄	1	1	yes	characteristic
18	C₂²×#92420	—	2	2	no	—
19	C₂²×#92420	—	2	1	no	—
20	C₂²×#92420	—	2	1	no	—
21	C₄⋊#92420	—	2	2	no	—
22	C₄⋊#92420	—	2	2	no	—
23	C₄⋊#92420	—	2	2	no	—
24	C₄⋊#92420	—	2	2	no	—
25	C₄⋊#92420	—	4	4	no	—
26	C₂²×#92418	C₂×C₄	1	2	yes	—
27	C₂×C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄×Dic₆₁	D₄	1	2	yes	—
29	C₂×C₄×Dic₆₁	Q₈	1	2	yes	—
30	C₂×C₄×Dic₆₁	C₂³	1	1	yes	characteristic
31	C₈×Dic₆₁	D₄	1	4	yes	—
32	C₈×Dic₆₁	Q₈	1	4	yes	—
33	C₂²×C₈×C₆₁	C₂×C₄	1	2	yes	—
34	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₄×C₆₁	C₄⋊C₄	1	2	yes	—
36	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
37	C₂²×Dic₆₁	C₄⋊C₄	1	2	yes	—
38	C₂²×Dic₆₁	C₂×D₄	1	1	yes	characteristic
39	C₂²×Dic₆₁	C₂×Q₈	1	1	yes	characteristic
40	C₂×#92420	—	4	2	no	—
41	C₂×#92420	—	4	1	no	—
42	C₂×#92420	—	4	1	no	—
43	C₂×#92420	—	4	4	no	—
44	C₂×#92420	—	4	2	no	—
45	C₂×#92420	—	4	2	no	—
46	C₂×C₈×C₆₁	—	2	2	no	—
47	C₂×C₈×C₆₁	—	2	2	no	—
48	C₂×C₈×C₆₁	C₄⋊C₄	1	4	yes	—
49	C₄²×C₆₁	—	2	2	no	—
50	C₄×Dic₆₁	—	2	4	no	—
51	C₄×Dic₆₁	—	2	1	no	—
52	C₄×Dic₆₁	—	2	1	no	—
53	C₄×Dic₆₁	—	2	2	no	—
54	C₄×Dic₆₁	—	2	2	no	—
55	C₄×Dic₆₁	C₂×D₄	1	2	yes	—
56	C₄×Dic₆₁	C₂×Q₈	1	2	yes	—
57	C₂×#92418	—	2	2	no	—
58	C₂×#92418	—	2	2	no	—
59	C₂×#92418	C₄⋊C₄	1	4	yes	—
60	C₂³×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
61	C₂×C₄×C₆₁	—	2	2	no	—
62	C₂×C₄×C₆₁	—	2	2	no	—
63	C₂×C₄×C₆₁	#108	1	4	yes	—
64	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	2	yes	—
65	C₂×C₄×C₆₁	#131	1	2	yes	—
66	C₂×C₄×C₆₁	#131	1	1	yes	characteristic
67	C₂×C₄×C₆₁	#131	1	1	yes	characteristic
68	C₂×Dic₆₁	—	2	2	no	—
69	C₂×Dic₆₁	—	2	2	no	—
70	C₂×Dic₆₁	—	2	2	no	—
71	C₂×Dic₆₁	—	2	2	no	—
72	C₂×Dic₆₁	#127	1	1	yes	characteristic
73	C₂×Dic₆₁	C₄⋊Q₈	1	2	yes	—
74	C₂×Dic₆₁	C₄⋊Q₈	1	1	yes	characteristic
75	C₂×Dic₆₁	AGL(1,ℤ/8)	1	2	yes	—
76	C₂×Dic₆₁	#137	1	2	yes	—
77	#92420	—	8	2	no	—
78	#92420	—	8	1	no	—
79	#92420	—	8	1	no	—
80	C₈×C₆₁	—	2	4	no	—
81	#92418	—	2	4	no	—
82	#8812	—	61	1	no	maximal, Sylow
83	C₂²×C₆₁	#384	1	1	yes	characteristic
84	C₂²×C₆₁	#424	1	1	yes	characteristic
85	C₂²×C₆₁	#424	1	1	yes	characteristic
86	C₂²×C₆₁	#428	1	2	yes	—
87	C₂²×C₆₁	C₂×#108	1	2	yes	—
88	C₄×C₆₁	—	2	4	no	—
89	C₄×C₆₁	#424	1	2	yes	—
90	C₄×C₆₁	#424	1	2	yes	—
91	Dic₆₁	—	2	2	no	—
92	Dic₆₁	—	4	2	no	—
93	Dic₆₁	#496	1	1	yes	characteristic
94	Dic₆₁	#497	1	1	yes	characteristic
95	Dic₆₁	#501	1	2	yes	—
96	#1842	—	61	1	no	—
97	#1831	—	61	1	no	—
98	#1744	—	61	1	no	—
99	C₂×#335	—	61	2	no	—
100	C₂×#334	—	61	2	no	—
101	C₂×C₆₁	#1573	1	2	yes	—
102	C₂×C₆₁	#1574	1	2	yes	—
103	C₂×C₆₁	#1843	1	1	yes	characteristic
104	C₂×C₆₁	#1847	1	1	yes	characteristic
105	C₂×C₆₁	#1848	1	1	yes	characteristic
106	C₂×C₄⋊C₈	—	61	1	no	—
107	C₂×C₄⋊C₈	—	61	2	no	—
108	C₂×C₄⋊C₈	—	61	1	no	—
109	C₂×C₄⋊C₈	—	122	2	no	—
110	C₂×C₄×C₈	—	61	2	no	—
111	C₄³	—	61	1	no	—
112	#336	—	61	2	no	—
113	#336	—	61	1	no	—
114	#336	—	122	2	no	—
115	#336	#6923	1	1	yes	characteristic
116	#335	—	122	4	no	—
117	#334	—	122	4	no	—
118	C₆₁	#8812	1	1	yes	characteristic, complemented, Sylow
119	C₂²×C₈	—	61	2	no	—
120	C₂²×C₈	—	122	1	no	—
121	C₂²×C₈	—	122	1	no	—
122	C₂²×C₈	—	122	2	no	—
123	C₂²×C₈	C₂×#6923	1	2	yes	—
124	C₂×C₄²	—	61	1	no	—
125	C₂×C₄²	—	61	2	no	—
126	C₂×C₄²	—	61	2	no	—
127	C₂×C₄²	—	61	1	no	—
128	C₂×C₄²	C₂×#6923	1	1	yes	characteristic
129	C₄⋊C₈	—	122	2	no	—
130	C₄⋊C₈	—	122	2	no	—
131	C₄⋊C₈	—	122	2	no	—
132	C₄⋊C₈	—	122	2	no	—
133	C₄⋊C₈	—	244	4	no	—
134	C₄×C₈	—	61	4	no	—
135	C₄×C₈	—	61	4	no	—
136	C₂²×C₄	—	61	1	no	—
137	C₂²×C₄	—	61	1	no	—
138	C₂²×C₄	—	61	2	no	—
139	C₂²×C₄	C₄⋊#6923	1	2	yes	—
140	C₂²×C₄	C₂²×#6923	1	1	yes	characteristic
141	C₂×C₈	—	2	2	no	—
142	C₂×C₈	—	2	2	no	—
143	C₂×C₈	—	61	4	no	—
144	C₂×C₈	—	122	2	no	—
145	C₂×C₈	—	122	2	no	—
146	C₂×C₈	—	244	1	no	—
147	C₂×C₈	—	244	2	no	—
148	C₂×C₈	—	244	1	no	—
149	C₂×C₈	—	244	4	no	—
150	C₂×C₈	—	244	2	no	—
151	C₂×C₈	—	244	2	no	—
152	C₂×C₈	C₄⋊#6923	1	4	yes	—
153	C₄²	—	2	2	no	—
154	C₄²	—	61	2	no	—
155	C₄²	—	61	2	no	—
156	C₄²	—	122	2	no	—
157	C₄²	—	122	2	no	—
158	C₄²	—	122	4	no	—
159	C₄²	—	122	1	no	—
160	C₄²	—	122	1	no	—
161	C₂³	C₂×C₄⋊#6923	1	1	yes	characteristic
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	61	1	no	—
165	C₂×C₄	—	61	1	no	—
166	C₂×C₄	—	61	2	no	—
167	C₂×C₄	—	61	2	no	—
168	C₂×C₄	—	61	2	no	—
169	C₂×C₄	—	122	2	no	—
170	C₂×C₄	—	122	2	no	—
171	C₂×C₄	—	122	2	no	—
172	C₂×C₄	—	122	2	no	—
173	C₂×C₄	#14519847	1	4	yes	—
174	C₂×C₄	C₂×C₄⋊#6923	1	2	yes	—
175	C₂×C₄	#14519864	1	1	yes	characteristic
176	C₂×C₄	#14519864	1	2	yes	—
177	C₂×C₄	#14519865	1	1	yes	characteristic
178	C₈	—	2	4	no	—
179	C₈	—	122	4	no	—
180	C₈	—	488	2	no	—
181	C₈	—	488	1	no	—
182	C₈	—	488	1	no	—
183	C₂²	#16840708	1	1	yes	characteristic
184	C₂²	#16840768	1	1	yes	characteristic
185	C₂²	#16840769	1	1	yes	characteristic
186	C₂²	#16840773	1	2	yes	—
187	C₂²	C₂×#14519847	1	2	yes	—
188	C₄	—	2	4	no	—
189	C₄	—	61	1	no	—
190	C₄	—	61	1	no	—
191	C₄	—	61	2	no	—
192	C₄	—	122	2	no	—
193	C₄	—	244	2	no	—
194	C₄	#16840768	1	2	yes	—
195	C₄	#16840769	1	2	yes	—
196	C₂	#21461392	1	2	yes	—
197	C₂	#21461396	1	2	yes	—
198	C₂	#21461753	1	1	yes	characteristic
199	C₂	#21461757	1	1	yes	characteristic
200	C₂	#21461758	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21461751

C₂

yes

maximal, characteristic

#21461736

C₂

yes

maximal, characteristic

C₂×#16840626

C₂

yes

maximal

C₂×#16840623

C₂

yes

maximal

#21445563

C₂

yes

maximal, characteristic

C₂×C₄⋊#92420

—

C₂×C₄⋊#92420

C₂²

yes

characteristic

C₂×C₄⋊#92420

C₂²

yes

—

C₂×C₄⋊#92420

C₂²

yes

—

—

—

C₂×C₈×Dic₆₁

C₂²

yes

—

C₄²×Dic₆₁

C₂²

yes

characteristic

#16839517

C₄

yes

characteristic

#16839516

C₄

yes

—

C₆₁×#336

C₄

yes

characteristic

C₂²×#92420

—

C₂²×#92420

—

C₂²×#92420

—

—

—

—

—

—

C₂²×#92418

C₂×C₄

yes

—

C₂×C₄×Dic₆₁

C₂×C₄

yes

characteristic

C₂×C₄×Dic₆₁

D₄

yes

—

C₂×C₄×Dic₆₁

Q₈

yes

—

C₂×C₄×Dic₆₁

C₂³

yes

characteristic

C₈×Dic₆₁

D₄

yes

—

C₈×Dic₆₁

Q₈

yes

—

C₂²×C₈×C₆₁

C₂×C₄

yes

—

C₂×C₄²×C₆₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₆₁

C₄⋊C₄

yes

—

C₂²×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₂²×Dic₆₁

C₄⋊C₄

yes

—

C₂²×Dic₆₁

C₂×D₄

yes

characteristic

C₂²×Dic₆₁

C₂×Q₈

yes

—

—

—

—

—

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

C₄⋊C₄

yes

—

C₄²×C₆₁

—

C₄×Dic₆₁

—

C₄×Dic₆₁

—

C₄×Dic₆₁

—

C₄×Dic₆₁

—

C₄×Dic₆₁

—

C₄×Dic₆₁

C₂×D₄

yes

—

C₄×Dic₆₁

C₂×Q₈

yes

—

C₂×#92418

—

C₂×#92418

—

C₂×#92418

C₄⋊C₄

yes

—

C₂³×C₆₁

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

#108

yes

—

C₂×C₄×C₆₁

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₆₁

#131

yes

—

C₂×C₄×C₆₁

yes

C₂×C₄×C₆₁

yes

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

#127

yes

characteristic

C₂×Dic₆₁

C₄⋊Q₈

yes

—

C₂×Dic₆₁

C₄⋊Q₈

yes

characteristic

C₂×Dic₆₁

AGL(1,ℤ/8)

yes

—

C₂×Dic₆₁

#137

yes

—

#92420

—

#92420

—

#92420

—

C₈×C₆₁

—

—

—

C₂²×C₆₁

yes

C₂²×C₆₁

yes

C₂²×C₆₁

yes

C₂²×C₆₁

yes

—

C₂²×C₆₁

C₂×#108

yes

—

C₄×C₆₁

—

C₄×C₆₁

#424

yes

—

C₄×C₆₁

#424

yes

—

Dic₆₁

—

Dic₆₁

—

Dic₆₁

yes

Dic₆₁

yes

Dic₆₁

yes

—

—

—

—

—

100

C₂×#334

—

101

C₂×C₆₁

#1573

yes

—

102

C₂×C₆₁

#1574

yes

—

103

C₂×C₆₁

#1843

yes

characteristic

104

C₂×C₆₁

#1847

yes

characteristic

105

C₂×C₆₁

#1848

yes

characteristic

106

C₂×C₄⋊C₈

—

107

C₂×C₄⋊C₈

—

108

C₂×C₄⋊C₈

—

109

C₂×C₄⋊C₈

—

122

—

110

C₂×C₄×C₈

—

111

C₄³

—

112

#336

—

113

#336

—

114

#336

—

122

—

115

yes

116

—

122

—

117

#334

—

122

—

118

C₆₁

#8812

yes

characteristic, complemented, Sylow

119

C₂²×C₈

—

120

C₂²×C₈

—

122

—

121

C₂²×C₈

—

122

—

122

C₂²×C₈

—

122

—

123

C₂²×C₈

C₂×#6923

yes

—

124

C₂×C₄²

—

125

C₂×C₄²

—

126

C₂×C₄²

—

127

C₂×C₄²

—

128

C₂×C₄²

C₂×#6923

yes

characteristic

129

C₄⋊C₈

—

122

—

130

C₄⋊C₈

—

122

—

131

C₄⋊C₈

—

122

—

132

C₄⋊C₈

—

122

—

133

C₄⋊C₈

—

244

—

134

C₄×C₈

—

135

C₄×C₈

—

136

C₂²×C₄

—

137

C₂²×C₄

—

138

C₂²×C₄

—

139

C₂²×C₄

C₄⋊#6923

yes

—

140

C₂²×C₄

C₂²×#6923

yes

characteristic

141

C₂×C₈

—

142

C₂×C₈

—

143

C₂×C₈

—

144

C₂×C₈

—

122

—

145

C₂×C₈

—

122

—

146

C₂×C₈

—

244

—

147

C₂×C₈

—

244

—

148

C₂×C₈

—

244

—

149

C₂×C₈

—

244

—

150

C₂×C₈

—

244

—

151

C₂×C₈

—

244

—

152

C₂×C₈

C₄⋊#6923

yes

—

153

C₄²

—

154

C₄²

—

155

C₄²

—

156

C₄²

—

122

—

157

C₄²

—

122

—

158

C₄²

—

122

—

159

C₄²

—

122

—

160

C₄²

—

122

—

161

C₂³

C₂×C₄⋊#6923

yes

characteristic

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

122

—

170

C₂×C₄

—

122

—

171

C₂×C₄

—

122

—

172

C₂×C₄

—

122

—

173

C₂×C₄

#14519847

yes

—

174

C₂×C₄

C₂×C₄⋊#6923

yes

—

175

C₂×C₄

#14519864

yes

characteristic

176

C₂×C₄

#14519864

yes

—

177

C₂×C₄

#14519865

yes

characteristic

178

C₈

—

179

C₈

—

122

—

180

C₈

—

488

—

181

C₈

—

488

—

182

C₈

—

488

—

183

C₂²

#16840708

yes

characteristic

184

C₂²

#16840768

yes

characteristic

185

C₂²

#16840769

yes

characteristic

186

C₂²

#16840773

yes

—

187

C₂²

C₂×#14519847

yes

—

188

C₄

—

189

C₄

—

190

C₄

—

191

C₄

—

192

C₄

—

122

—

193

C₄

—

244

—

194

C₄

#16840768

yes

—

195

C₄

#16840769

yes

—

196

C₂

#21461392

yes

—

197

C₂

#21461396

yes

—

198

C₂

#21461753

yes

characteristic

199

C₂

#21461757

yes

characteristic

200

C₂

#21461758

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₄⋊#6923
commutator	C₂²×C₄×C₆₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×#6923
Fitting	C₆₁×#336	C₄
socle, prime radical	C₂³×C₆₁	C₂×C₄⋊C₄
cosocle	C₂×C₄×Dic₆₁	C₂³
hypercenter	#336	#6923
hypocenter	C₆₁	#8812
LCS	G ▷ C₂²×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

subgroup

G/H

center

C₂³

C₂×C₄⋊#6923

commutator

C₂²×C₄×C₆₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×#6923

Fitting

C₆₁×#336

C₄

socle, prime radical

C₂³×C₆₁

C₂×C₄⋊C₄

cosocle

C₂×C₄×Dic₆₁

C₂³

C₆₁

G ▷ C₂²×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

N	H ≅ G/N	type
118. C₆₁	#8812	semidirect

H ≅ G/N

type

118. C₆₁

#8812

semidirect

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

subgroup	index
#21461751	2
#21461736	2
C₂×#16840626	2
C₂×#16840623	2
#21445563	2
#8812	61

subgroup

index

Subgroups — #30599514

Conjugacy classes of subgroups

Complements

Maximal subgroups