Group #29573685 — subgroups

Subgroups — #29573685

Order 15616.

322 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 131 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	#21445935	C₂	1	2	yes	maximal
2	#21445564	C₂	1	1	yes	maximal, characteristic
3	#21443654	C₂	1	1	yes	maximal, characteristic
4	#21443653	C₂	1	2	yes	maximal
5	C₆₁×#1277	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#92418	C₂²	1	2	yes	—
7	C₂×C₄×#92418	C₂²	1	1	yes	characteristic
8	C₄²×Dic₆₁	C₂²	1	1	yes	characteristic
9	#16839516	—	2	2	no	—
10	#16839516	—	2	2	no	—
11	#16839516	C₄	1	2	yes	—
12	#16839516	C₄	1	2	yes	—
13	C₂×C₄⋊C₈×C₆₁	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
15	C₂²×#92418	—	2	2	no	—
16	C₂²×#92418	C₂×C₄	1	1	yes	characteristic
17	C₂²×#92418	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×Dic₆₁	C₂×C₄	1	2	yes	—
19	C₂×C₄×Dic₆₁	D₄	1	1	yes	characteristic
20	C₂×C₄×Dic₆₁	D₄	1	1	yes	characteristic
21	C₂×C₄×Dic₆₁	Q₈	1	2	yes	—
22	C₄⋊#92418	—	2	2	no	—
23	C₄⋊#92418	—	2	2	no	—
24	C₄×#92418	—	2	2	no	—
25	C₂²×C₈×C₆₁	—	2	2	no	—
26	C₂²×C₈×C₆₁	C₂×C₄	1	2	yes	—
27	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
28	C₄⋊C₈×C₆₁	—	2	2	no	—
29	C₄⋊C₈×C₆₁	—	2	2	no	—
30	C₄×C₈×C₆₁	—	2	2	no	—
31	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
32	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
33	C₂²×C₄×C₆₁	C₂×Q₈	1	1	yes	characteristic
34	C₂²×Dic₆₁	C₄⋊C₄	1	2	yes	—
35	C₂²×Dic₆₁	C₄⋊C₄	1	2	yes	—
36	C₂×C₈×C₆₁	—	2	2	no	—
37	C₂×C₈×C₆₁	—	2	2	no	—
38	C₂×C₈×C₆₁	—	2	2	no	—
39	C₂×C₈×C₆₁	—	4	2	no	—
40	C₂×C₈×C₆₁	—	4	2	no	—
41	C₂×C₈×C₆₁	—	4	2	no	—
42	C₄²×C₆₁	#41	1	2	yes	—
43	C₄²×C₆₁	#41	1	2	yes	—
44	C₄×Dic₆₁	—	2	2	no	—
45	C₄×Dic₆₁	—	2	2	no	—
46	C₄×Dic₆₁	—	2	1	no	—
47	C₄×Dic₆₁	—	2	2	no	—
48	C₄×Dic₆₁	—	2	1	no	—
49	C₄×Dic₆₁	—	4	2	no	—
50	C₂×#92418	—	2	1	no	—
51	C₂×#92418	—	2	1	no	—
52	C₂×#92418	—	2	2	no	—
53	C₂×#92418	—	2	2	no	—
54	C₂×#92418	—	4	2	no	—
55	C₂×#92418	—	4	2	no	—
56	C₂×#92418	—	4	2	no	—
57	C₂³×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
58	C₂×C₄×C₆₁	—	2	2	no	—
59	C₂×C₄×C₆₁	#117	1	1	yes	characteristic
60	C₂×C₄×C₆₁	#117	1	1	yes	characteristic
61	C₂×C₄×C₆₁	#124	1	2	yes	—
62	C₂×C₄×C₆₁	#125	1	2	yes	—
63	C₂×C₄×C₆₁	#131	1	2	yes	—
64	C₂×C₄×C₆₁	#131	1	2	yes	—
65	C₂×C₄×C₆₁	#135	1	1	yes	characteristic
66	C₂×C₄×C₆₁	#135	1	2	yes	—
67	C₂×C₄×C₆₁	#135	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	—	2	2	no	—
73	C₂×Dic₆₁	—	2	2	no	—
74	C₈×C₆₁	—	4	2	no	—
75	C₈×C₆₁	—	8	2	no	—
76	#92418	—	4	1	no	—
77	#92418	—	4	1	no	—
78	#92418	—	8	2	no	—
79	#8788	—	61	1	no	maximal, Sylow
80	C₂²×C₆₁	#382	1	1	yes	characteristic
81	C₂²×C₆₁	#424	1	2	yes	—
82	C₂²×C₆₁	#427	1	1	yes	characteristic
83	C₂²×C₆₁	#427	1	1	yes	characteristic
84	C₂²×C₆₁	#430	1	2	yes	—
85	C₄×C₆₁	—	2	2	no	—
86	C₄×C₆₁	#433	1	2	yes	—
87	C₄×C₆₁	#433	1	1	yes	characteristic
88	C₄×C₆₁	#433	1	1	yes	characteristic
89	C₄×C₆₁	#488	1	2	yes	—
90	C₄×C₆₁	#488	1	2	yes	—
91	Dic₆₁	—	4	2	no	—
92	Dic₆₁	—	4	2	no	—
93	#1831	—	61	2	no	—
94	#1744	—	61	1	no	—
95	#1277	—	61	2	no	—
96	#1277	—	61	1	no	—
97	#1277	D₆₁	1	1	yes	characteristic
98	C₂×C₆₁	#1572	1	2	yes	—
99	C₂×C₆₁	#1572	1	2	yes	—
100	C₂×C₆₁	#1832	1	1	yes	characteristic
101	C₂×C₆₁	#1835	1	1	yes	characteristic
102	C₂×C₆₁	#1838	1	1	yes	characteristic
103	C₂×C₄⋊C₈	—	61	2	no	—
104	C₂×C₄⋊C₈	C₂×D₆₁	1	2	yes	—
105	C₂×C₄×C₈	—	61	1	no	—
106	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
107	C₄³	—	61	1	no	—
108	#336	—	61	2	no	—
109	#336	—	61	2	no	—
110	#336	—	122	2	no	—
111	#336	—	122	2	no	—
112	C₆₁	#8788	1	1	yes	characteristic, complemented, Sylow
113	C₂²×C₈	—	2	2	no	—
114	C₂²×C₈	—	61	1	no	—
115	C₂²×C₈	—	61	1	no	—
116	C₂²×C₈	—	122	2	no	—
117	C₂²×C₈	C₄×D₆₁	1	2	yes	—
118	C₂×C₄²	—	61	2	no	—
119	C₂×C₄²	—	61	1	no	—
120	C₂×C₄²	—	61	1	no	—
121	C₂×C₄²	—	61	2	no	—
122	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
123	C₄⋊C₈	—	2	2	no	—
124	C₄⋊C₈	—	2	2	no	—
125	C₄⋊C₈	—	122	2	no	—
126	C₄⋊C₈	—	122	2	no	—
127	C₄×C₈	—	2	2	no	—
128	C₄×C₈	—	122	2	no	—
129	C₂²×C₄	—	61	2	no	—
130	C₂²×C₄	—	61	2	no	—
131	C₂²×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
132	C₂²×C₄	D₄×D₆₁	1	1	yes	characteristic
133	C₂²×C₄	Q₈×D₆₁	1	1	yes	characteristic
134	C₂×C₈	—	2	2	no	—
135	C₂×C₈	—	2	2	no	—
136	C₂×C₈	—	2	2	no	—
137	C₂×C₈	—	4	2	no	—
138	C₂×C₈	—	4	2	no	—
139	C₂×C₈	—	4	2	no	—
140	C₂×C₈	—	122	1	no	—
141	C₂×C₈	—	122	1	no	—
142	C₂×C₈	—	122	2	no	—
143	C₂×C₈	—	122	2	no	—
144	C₂×C₈	—	244	2	no	—
145	C₂×C₈	—	244	2	no	—
146	C₂×C₈	—	244	2	no	—
147	C₄²	—	122	2	no	—
148	C₄²	—	122	2	no	—
149	C₄²	—	122	1	no	—
150	C₄²	—	122	2	no	—
151	C₄²	—	122	1	no	—
152	C₄²	—	244	2	no	—
153	C₄²	#11758222	1	2	yes	—
154	C₄²	#11758224	1	2	yes	—
155	C₂³	C₄⋊C₄×D₆₁	1	1	yes	characteristic
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	122	2	no	—
158	C₂×C₄	—	122	2	no	—
159	C₂×C₄	—	122	2	no	—
160	C₂×C₄	—	122	2	no	—
161	C₂×C₄	—	122	2	no	—
162	C₂×C₄	—	122	2	no	—
163	C₂×C₄	#14519733	1	1	yes	characteristic
164	C₂×C₄	#14519734	1	1	yes	characteristic
165	C₂×C₄	#14519763	1	2	yes	—
166	C₂×C₄	#14519767	1	2	yes	—
167	C₂×C₄	#14519783	1	2	yes	—
168	C₂×C₄	#14519785	1	2	yes	—
169	C₂×C₄	#14519796	1	1	yes	characteristic
170	C₂×C₄	#14519798	1	2	yes	—
171	C₂×C₄	#14519800	1	1	yes	characteristic
172	C₈	—	4	2	no	—
173	C₈	—	8	2	no	—
174	C₈	—	488	2	no	—
175	C₈	—	244	1	no	—
176	C₈	—	244	1	no	—
177	C₂²	#16839626	1	1	yes	characteristic
178	C₂²	#16839787	1	2	yes	—
179	C₂²	#16839797	1	1	yes	characteristic
180	C₂²	#16839799	1	1	yes	characteristic
181	C₂²	#16839809	1	2	yes	—
182	C₄	—	2	2	no	—
183	C₄	—	244	2	no	—
184	C₄	—	244	2	no	—
185	C₄	#16839820	1	2	yes	—
186	C₄	#16839821	1	1	yes	characteristic
187	C₄	#16839824	1	1	yes	characteristic
188	C₄	#16840115	1	2	yes	—
189	C₄	#16840118	1	2	yes	—
190	C₂	#21444596	1	2	yes	—
191	C₂	#21444598	1	2	yes	—
192	C₂	#21445937	1	1	yes	characteristic
193	C₂	#21445948	1	1	yes	characteristic
194	C₂	#21445958	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21445935

C₂

yes

maximal

#21445564

C₂

yes

maximal, characteristic

#21443654

C₂

yes

maximal, characteristic

#21443653

C₂

yes

maximal

C₆₁×#1277

C₂

yes

maximal, characteristic

C₂×C₄⋊#92418

C₂²

yes

—

C₂×C₄×#92418

C₂²

yes

characteristic

C₄²×Dic₆₁

C₂²

yes

—

—

C₄

yes

—

#16839516

C₄

yes

—

C₂×C₄⋊C₈×C₆₁

C₂²

yes

—

C₂×C₄×C₈×C₆₁

C₂²

yes

characteristic

C₂²×#92418

—

C₂²×#92418

C₂×C₄

yes

characteristic

C₂²×#92418

C₂×C₄

yes

characteristic

C₂×C₄×Dic₆₁

C₂×C₄

yes

—

C₂×C₄×Dic₆₁

D₄

yes

characteristic

C₂×C₄×Dic₆₁

D₄

yes

characteristic

C₂×C₄×Dic₆₁

Q₈

yes

—

C₄⋊#92418

—

C₄⋊#92418

—

C₄×#92418

—

C₂²×C₈×C₆₁

—

C₂²×C₈×C₆₁

C₂×C₄

yes

—

C₂×C₄²×C₆₁

C₂³

yes

characteristic

C₄⋊C₈×C₆₁

—

C₄⋊C₈×C₆₁

—

C₄×C₈×C₆₁

—

C₂²×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₂²×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂²×C₄×C₆₁

C₂×Q₈

yes

characteristic

C₂²×Dic₆₁

C₄⋊C₄

yes

—

C₂²×Dic₆₁

C₄⋊C₄

yes

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₄²×C₆₁

#41

yes

—

C₄²×C₆₁

#41

yes

—

C₄×Dic₆₁

—

C₄×Dic₆₁

—

C₄×Dic₆₁

—

C₄×Dic₆₁

—

C₄×Dic₆₁

—

C₄×Dic₆₁

—

—

—

—

—

—

—

—

C₂³×C₆₁

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

yes

C₂×C₄×C₆₁

yes

C₂×C₄×C₆₁

yes

—

C₂×C₄×C₆₁

#125

yes

—

C₂×C₄×C₆₁

#131

yes

—

C₂×C₄×C₆₁

#131

yes

—

C₂×C₄×C₆₁

#135

yes

characteristic

C₂×C₄×C₆₁

#135

yes

—

C₂×C₄×C₆₁

#135

yes

characteristic

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₈×C₆₁

—

C₈×C₆₁

—

—

—

—

—

C₂²×C₆₁

yes

C₂²×C₆₁

yes

—

C₂²×C₆₁

yes

C₂²×C₆₁

yes

C₂²×C₆₁

yes

—

C₄×C₆₁

—

C₄×C₆₁

#433

yes

—

C₄×C₆₁

yes

C₄×C₆₁

yes

C₄×C₆₁

yes

—

C₄×C₆₁

#488

yes

—

Dic₆₁

—

Dic₆₁

—

—

—

—

—

D₆₁

yes

characteristic

C₂×C₆₁

#1572

yes

—

C₂×C₆₁

#1572

yes

—

100

C₂×C₆₁

#1832

yes

characteristic

101

C₂×C₆₁

#1835

yes

characteristic

102

C₂×C₆₁

#1838

yes

characteristic

103

C₂×C₄⋊C₈

—

104

C₂×C₄⋊C₈

C₂×D₆₁

yes

—

105

C₂×C₄×C₈

—

106

C₂×C₄×C₈

C₂×D₆₁

yes

characteristic

107

C₄³

—

108

#336

—

109

#336

—

110

#336

—

122

—

111

#336

—

122

—

112

C₆₁

#8788

yes

characteristic, complemented, Sylow

113

C₂²×C₈

—

114

C₂²×C₈

—

115

C₂²×C₈

—

116

C₂²×C₈

—

122

—

117

C₂²×C₈

C₄×D₆₁

yes

—

118

C₂×C₄²

—

119

C₂×C₄²

—

120

C₂×C₄²

—

121

C₂×C₄²

—

122

C₂×C₄²

C₂²×D₆₁

yes

characteristic

123

C₄⋊C₈

—

124

C₄⋊C₈

—

125

C₄⋊C₈

—

122

—

126

C₄⋊C₈

—

122

—

127

C₄×C₈

—

128

C₄×C₈

—

122

—

129

C₂²×C₄

—

130

C₂²×C₄

—

131

C₂²×C₄

C₂×C₄×D₆₁

yes

characteristic

132

C₂²×C₄

D₄×D₆₁

yes

characteristic

133

C₂²×C₄

Q₈×D₆₁

yes

characteristic

134

C₂×C₈

—

135

C₂×C₈

—

136

C₂×C₈

—

137

C₂×C₈

—

138

C₂×C₈

—

139

C₂×C₈

—

140

C₂×C₈

—

122

—

141

C₂×C₈

—

122

—

142

C₂×C₈

—

122

—

143

C₂×C₈

—

122

—

144

C₂×C₈

—

244

—

145

C₂×C₈

—

244

—

146

C₂×C₈

—

244

—

147

C₄²

—

122

—

148

C₄²

—

122

—

149

C₄²

—

122

—

150

C₄²

—

122

—

151

C₄²

—

122

—

152

C₄²

—

244

—

153

C₄²

#11758222

yes

—

154

C₄²

#11758224

yes

—

155

C₂³

C₄⋊C₄×D₆₁

yes

characteristic

156

C₂×C₄

—

157

C₂×C₄

—

122

—

158

C₂×C₄

—

122

—

159

C₂×C₄

—

122

—

160

C₂×C₄

—

122

—

161

C₂×C₄

—

122

—

162

C₂×C₄

—

122

—

163

C₂×C₄

#14519733

yes

characteristic

164

C₂×C₄

#14519734

yes

characteristic

165

C₂×C₄

#14519763

yes

—

166

C₂×C₄

#14519767

yes

—

167

C₂×C₄

#14519783

yes

—

168

C₂×C₄

#14519785

yes

—

169

C₂×C₄

#14519796

yes

characteristic

170

C₂×C₄

#14519798

yes

—

171

C₂×C₄

#14519800

yes

characteristic

172

C₈

—

173

C₈

—

174

C₈

—

488

—

175

C₈

—

244

—

176

C₈

—

244

—

177

C₂²

#16839626

yes

characteristic

178

C₂²

#16839787

yes

—

179

C₂²

#16839797

yes

characteristic

180

C₂²

#16839799

yes

characteristic

181

C₂²

#16839809

yes

—

182

C₄

—

183

C₄

—

244

—

184

C₄

—

244

—

185

C₄

#16839820

yes

—

186

C₄

#16839821

yes

characteristic

187

C₄

#16839824

yes

characteristic

188

C₄

#16840115

yes

—

189

C₄

#16840118

yes

—

190

C₂

#21444596

yes

—

191

C₂

#21444598

yes

—

192

C₂

#21445937

yes

characteristic

193

C₂

#21445948

yes

characteristic

194

C₂

#21445958

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₆₁
commutator	C₂²×C₄×C₆₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₁
Fitting	C₆₁×#1277	C₂
socle, prime radical	C₂³×C₆₁	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₆₁	C₂³
hypercenter	#1277	D₆₁
hypocenter	C₆₁	#8788
LCS	G ▷ C₂²×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ #1277

subgroup

G/H

center

C₂²×C₄

D₄×D₆₁

commutator

C₂²×C₄×C₆₁

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₆₁

Fitting

C₆₁×#1277

C₂

socle, prime radical

C₂³×C₆₁

C₂×C₄⋊C₄

cosocle

C₂×C₄²×C₆₁

C₂³

D₆₁

C₆₁

G ▷ C₂²×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ #1277

N	H ≅ G/N	type
112. C₆₁	#8788	semidirect

H ≅ G/N

type

112. C₆₁

#8788

semidirect

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

subgroup	index
#21445935	2
#21445564	2
#21443654	2
#21443653	2
C₆₁×#1277	2
#8788	61

subgroup

index

C₆₁×#1277

Subgroups — #29573685

Conjugacy classes of subgroups

Complements

Maximal subgroups