Group #29573034 — subgroups

Subgroups — #29573034

Order 15616.

336 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 101 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	#21454749	C₂	1	1	yes	maximal, characteristic
2	#21444574	C₂	1	2	yes	maximal
3	#21444556	C₂	1	2	yes	maximal
4	#21443732	C₂	1	1	yes	maximal, characteristic
5	C₆₁×#1309	C₂	1	1	yes	maximal, characteristic
6	#16840319	C₂²	1	1	yes	characteristic
7	#16839889	—	2	2	no	—
8	#16839886	—	2	2	no	—
9	#16839790	C₄	1	2	yes	—
10	#16839786	C₂²	1	2	yes	—
11	#16839720	C₄	1	2	yes	—
12	C₂×C₄×#92418	C₂²	1	1	yes	characteristic
13	#16839512	—	2	2	no	—
14	#16839512	—	2	2	no	—
15	#16839509	—	2	2	no	—
16	#16839507	—	2	2	no	—
17	C₆₁×#423	C₂²	1	2	yes	—
18	C₆₁×#423	C₂²	1	1	yes	characteristic
19	C₂×#11758194	—	2	2	no	—
20	C₂²×#92418	C₂×C₄	1	1	yes	characteristic
21	C₂²×#92418	C₂×C₄	1	1	yes	characteristic
22	#14519777	D₄	1	2	yes	—
23	#14519769	Q₈	1	2	yes	—
24	#14519756	—	2	2	no	—
25	C₄×Dic₁₂₂	—	2	2	no	—
26	#14519736	C₂×C₄	1	2	yes	—
27	C₄⋊#92418	—	2	2	no	—
28	C₄⋊#92418	—	2	2	no	—
29	#14519706	—	2	4	no	—
30	#14519687	—	2	2	no	—
31	C₄×#92418	D₄	1	2	yes	—
32	C₄×#92418	D₄	1	2	yes	—
33	C₈×Dic₆₁	—	2	2	no	—
34	C₂×M₁₆×C₆₁	—	2	2	no	—
35	C₂×M₁₆×C₆₁	C₂×C₄	1	2	yes	—
36	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
37	C₄⋊C₈×C₆₁	—	2	2	no	—
38	C₄⋊C₈×C₆₁	—	2	2	no	—
39	C₄⋊C₈×C₆₁	—	2	2	no	—
40	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
41	C₂²×C₄×C₆₁	#41	1	2	yes	—
42	C₂×Dic₁₂₂	—	2	2	no	—
43	M₁₆×C₆₁	—	4	2	no	—
44	M₁₆×C₆₁	—	8	2	no	—
45	C₂×C₈×C₆₁	—	2	2	no	—
46	C₂×C₈×C₆₁	—	4	2	no	—
47	C₄²×C₆₁	C₂×D₄	1	2	yes	—
48	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
49	C₄²×C₆₁	C₂×Q₈	1	1	yes	characteristic
50	C₂²⋊Dic₆₁	—	2	2	no	—
51	C₄⋊Dic₆₁	—	2	2	no	—
52	C₄⋊Dic₆₁	C₄⋊C₄	1	4	yes	—
53	#11758196	—	2	4	no	—
54	C₄×Dic₆₁	—	2	2	no	—
55	#11758194	—	8	2	no	—
56	C₂×#92418	—	2	2	no	—
57	C₂×#92418	—	4	1	no	—
58	C₂×#92418	—	4	2	no	—
59	C₂×#92418	C₂²⋊C₄	1	1	yes	characteristic
60	C₂×#92418	C₂²⋊C₄	1	2	yes	—
61	C₂×#92418	C₂²⋊C₄	1	1	yes	characteristic
62	C₂³×C₆₁	#117	1	1	yes	characteristic
63	C₂×C₄×C₆₁	—	2	1	no	—
64	C₂×C₄×C₆₁	—	2	1	no	—
65	C₂×C₄×C₆₁	—	2	2	no	—
66	C₂×C₄×C₆₁	—	2	2	no	—
67	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
69	C₂×C₄×C₆₁	#121	1	2	yes	—
70	C₂×C₄×C₆₁	C₂²⋊Q₈	1	2	yes	—
71	C₂×Dic₆₁	—	2	4	no	—
72	Dic₁₂₂	—	4	2	no	—
73	Dic₁₂₂	—	4	2	no	—
74	C₈×C₆₁	—	4	2	no	—
75	C₈×C₆₁	—	8	2	no	—
76	#92418	—	2	1	no	—
77	#92418	—	2	2	no	—
78	#92418	—	8	2	no	—
79	#92418	#101	1	2	yes	—
80	#8637	—	61	1	no	maximal, Sylow
81	C₂²×C₆₁	—	2	1	no	—
82	C₂²×C₆₁	—	2	2	no	—
83	C₂²×C₆₁	#380	1	1	yes	characteristic
84	C₄×C₆₁	—	2	1	no	—
85	C₄×C₆₁	—	2	2	no	—
86	C₄×C₆₁	C₂×#101	1	1	yes	characteristic
87	C₄×C₆₁	#413	1	1	yes	characteristic
88	C₄×C₆₁	#471	1	2	yes	—
89	C₄×C₆₁	#471	1	2	yes	—
90	Dic₆₁	—	4	4	no	—
91	#2960	—	61	1	no	—
92	#1566	—	61	2	no	—
93	#1561	—	61	2	no	—
94	#1309	—	61	1	no	—
95	#1309	D₆₁	1	1	yes	characteristic
96	C₂×C₆₁	—	4	1	no	—
97	C₂×C₆₁	#1801	1	1	yes	characteristic
98	C₂×C₆₁	#1815	1	2	yes	—
99	#533	—	61	1	no	—
100	#446	—	122	2	no	—
101	C₈×Q₈	—	122	2	no	—
102	#424	—	61	2	no	—
103	#423	—	61	2	no	—
104	#423	C₂×D₆₁	1	2	yes	—
105	#423	C₂×D₆₁	1	1	yes	characteristic
106	#405	—	61	2	no	—
107	C₂×C₄×C₈	—	61	1	no	—
108	#334	—	122	2	no	—
109	#334	—	122	2	no	—
110	#332	—	122	2	no	—
111	#332	—	122	2	no	—
112	C₆₁	#8637	1	1	yes	characteristic, complemented, Sylow
113	C₂×M₁₆	—	2	2	no	—
114	C₂×M₁₆	—	122	2	no	—
115	C₂×M₁₆	C₄×D₆₁	1	2	yes	—
116	C₂²×C₈	—	61	1	no	—
117	C₂²×C₈	—	61	1	no	—
118	C₄⋊Q₈	—	61	2	no	—
119	#125	—	61	2	no	—
120	C₂²⋊Q₈	—	122	2	no	—
121	C₄×Q₈	—	122	2	no	—
122	#117	—	61	2	no	—
123	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
124	C₄⋊C₈	—	2	2	no	—
125	C₄⋊C₈	—	2	2	no	—
126	C₄⋊C₈	—	2	2	no	—
127	C₄⋊C₈	—	122	4	no	—
128	C₄⋊C₈	—	122	2	no	—
129	C₄⋊C₈	—	122	2	no	—
130	#97	—	122	2	no	—
131	C₄×C₈	—	61	2	no	—
132	C₄×C₈	—	61	2	no	—
133	C₄×C₈	—	122	2	no	—
134	C₂×Q₈	—	122	2	no	—
135	C₂²×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
136	C₂²×C₄	#11758222	1	2	yes	—
137	M₁₆	—	4	2	no	—
138	M₁₆	—	8	2	no	—
139	M₁₆	—	488	2	no	—
140	C₂×C₈	—	2	2	no	—
141	C₂×C₈	—	4	2	no	—
142	C₂×C₈	—	61	1	no	—
143	C₂×C₈	—	61	1	no	—
144	C₂×C₈	—	61	2	no	—
145	C₂×C₈	—	122	2	no	—
146	C₂×C₈	—	244	1	no	—
147	C₂×C₈	—	244	2	no	—
148	C₄⋊C₄	—	61	4	no	—
149	C₄⋊C₄	—	122	2	no	—
150	C₄⋊C₄	—	122	4	no	—
151	C₂²⋊C₄	—	122	2	no	—
152	C₄²	—	122	2	no	—
153	C₄²	D₄×D₆₁	1	2	yes	—
154	C₄²	D₄×D₆₁	1	1	yes	characteristic
155	C₄²	Q₈×D₆₁	1	1	yes	characteristic
156	C₂³	#14519734	1	1	yes	characteristic
157	Q₈	—	244	2	no	—
158	Q₈	—	244	2	no	—
159	C₂×C₄	—	2	1	no	—
160	C₂×C₄	—	2	1	no	—
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	122	4	no	—
164	C₂×C₄	C₂²⋊C₄×D₆₁	1	1	yes	characteristic
165	C₂×C₄	C₄⋊C₄×D₆₁	1	1	yes	characteristic
166	C₂×C₄	#14519750	1	2	yes	—
167	C₂×C₄	#14519755	1	2	yes	—
168	C₈	—	4	2	no	—
169	C₈	—	8	2	no	—
170	C₈	—	61	2	no	—
171	C₈	—	122	1	no	—
172	C₈	—	122	2	no	—
173	C₈	—	488	2	no	—
174	C₂²	—	2	1	no	—
175	C₂²	—	2	2	no	—
176	C₂²	#16839621	1	1	yes	characteristic
177	C₄	—	2	1	no	—
178	C₄	—	2	2	no	—
179	C₄	—	244	4	no	—
180	C₄	#101×D₆₁	1	1	yes	characteristic
181	C₄	#16839743	1	1	yes	characteristic
182	C₄	#16840024	1	2	yes	—
183	C₄	#16840028	1	2	yes	—
184	C₂	—	4	1	no	—
185	C₂	#21445814	1	1	yes	characteristic
186	C₂	#21445877	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21454749

C₂

yes

maximal, characteristic

#21444574

C₂

yes

maximal

#21444556

C₂

yes

maximal

#21443732

C₂

yes

maximal, characteristic

C₆₁×#1309

C₂

yes

maximal, characteristic

#16840319

C₂²

yes

—

—

C₄

yes

—

#16839786

C₂²

yes

—

#16839720

C₄

yes

—

C₂×C₄×#92418

C₂²

yes

—

—

—

—

C₆₁×#423

C₂²

yes

—

C₆₁×#423

C₂²

yes

characteristic

C₂×#11758194

—

C₂²×#92418

C₂×C₄

yes

characteristic

C₂²×#92418

C₂×C₄

yes

characteristic

#14519777

D₄

yes

—

#14519769

Q₈

yes

—

#14519756

—

C₄×Dic₁₂₂

—

#14519736

C₂×C₄

yes

—

—

—

—

—

D₄

yes

—

C₄×#92418

D₄

yes

—

C₈×Dic₆₁

—

C₂×M₁₆×C₆₁

—

C₂×M₁₆×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

#41

yes

—

C₂×Dic₁₂₂

—

M₁₆×C₆₁

—

M₁₆×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₄²×C₆₁

C₂×D₄

yes

—

C₄²×C₆₁

C₂×D₄

yes

characteristic

C₄²×C₆₁

C₂×Q₈

yes

characteristic

C₂²⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

C₄⋊C₄

yes

—

#11758196

—

C₄×Dic₆₁

—

—

—

—

—

C₂²⋊C₄

yes

characteristic

C₂×#92418

C₂²⋊C₄

yes

—

C₂×#92418

C₂²⋊C₄

yes

characteristic

C₂³×C₆₁

#117

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₁

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₁

#121

yes

—

C₂×C₄×C₆₁

C₂²⋊Q₈

yes

—

C₂×Dic₆₁

—

Dic₁₂₂

—

Dic₁₂₂

—

C₈×C₆₁

—

C₈×C₆₁

—

—

—

—

yes

—

#8637

—

maximal, Sylow

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

#380

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

yes

C₄×C₆₁

yes

C₄×C₆₁

yes

—

C₄×C₆₁

#471

yes

—

Dic₆₁

—

—

—

—

—

D₆₁

yes

characteristic

C₂×C₆₁

—

C₂×C₆₁

#1801

yes

characteristic

C₂×C₆₁

#1815

yes

—

#533

—

100

#446

—

122

—

101

C₈×Q₈

—

122

—

102

#424

—

103

#423

—

104

#423

C₂×D₆₁

yes

—

105

#423

C₂×D₆₁

yes

characteristic

106

#405

—

107

C₂×C₄×C₈

—

108

#334

—

122

—

109

#334

—

122

—

110

#332

—

122

—

111

#332

—

122

—

112

C₆₁

#8637

yes

characteristic, complemented, Sylow

113

C₂×M₁₆

—

114

C₂×M₁₆

—

122

—

115

C₂×M₁₆

C₄×D₆₁

yes

—

116

C₂²×C₈

—

117

C₂²×C₈

—

118

C₄⋊Q₈

—

119

#125

—

120

C₂²⋊Q₈

—

122

—

121

C₄×Q₈

—

122

—

122

#117

—

123

C₂×C₄²

C₂²×D₆₁

yes

characteristic

124

C₄⋊C₈

—

125

C₄⋊C₈

—

126

C₄⋊C₈

—

127

C₄⋊C₈

—

122

—

128

C₄⋊C₈

—

122

—

129

C₄⋊C₈

—

122

—

130

#97

—

122

—

131

C₄×C₈

—

132

C₄×C₈

—

133

C₄×C₈

—

122

—

134

C₂×Q₈

—

122

—

135

C₂²×C₄

C₂×C₄×D₆₁

yes

characteristic

136

C₂²×C₄

#11758222

yes

—

137

M₁₆

—

138

M₁₆

—

139

M₁₆

—

488

—

140

C₂×C₈

—

141

C₂×C₈

—

142

C₂×C₈

—

143

C₂×C₈

—

144

C₂×C₈

—

145

C₂×C₈

—

122

—

146

C₂×C₈

—

244

—

147

C₂×C₈

—

244

—

148

C₄⋊C₄

—

149

C₄⋊C₄

—

122

—

150

C₄⋊C₄

—

122

—

151

C₂²⋊C₄

—

122

—

152

C₄²

—

122

—

153

C₄²

D₄×D₆₁

yes

—

154

C₄²

D₄×D₆₁

yes

characteristic

155

C₄²

Q₈×D₆₁

yes

characteristic

156

C₂³

#14519734

yes

characteristic

157

Q₈

—

244

—

158

Q₈

—

244

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

122

—

164

C₂×C₄

C₂²⋊C₄×D₆₁

yes

characteristic

165

C₂×C₄

C₄⋊C₄×D₆₁

yes

characteristic

166

C₂×C₄

#14519750

yes

—

167

C₂×C₄

#14519755

yes

—

168

C₈

—

169

C₈

—

170

C₈

—

171

C₈

—

122

—

172

C₈

—

122

—

173

C₈

—

488

—

174

C₂²

—

175

C₂²

—

176

C₂²

#16839621

yes

characteristic

177

C₄

—

178

C₄

—

179

C₄

—

244

—

180

C₄

#101×D₆₁

yes

characteristic

181

C₄

#16839743

yes

characteristic

182

C₄

#16840024

yes

—

183

C₄

#16840028

yes

—

184

C₂

—

185

C₂

#21445814

yes

characteristic

186

C₂

#21445877

yes

—

	subgroup	G/H
center	C₂²	#16839621
commutator	C₂²×C₄×C₆₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₁
Fitting	C₆₁×#1309	C₂
socle	C₂²×C₆₁	#380
prime radical	C₂³×C₆₁	#117
cosocle	C₂×C₄²×C₆₁	C₂³
hypercenter	#1309	D₆₁
hypocenter	C₆₁	#8637
LCS	G ▷ C₂²×C₄×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1309

subgroup

G/H

center

C₂²

#16839621

commutator

C₂²×C₄×C₆₁

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₆₁

Fitting

C₆₁×#1309

C₂

C₂²×C₆₁

C₂³×C₆₁

C₂×C₄²×C₆₁

C₂³

D₆₁

C₆₁

G ▷ C₂²×C₄×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1309

N	H ≅ G/N	type
112. C₆₁	#8637	semidirect

H ≅ G/N

type

112. C₆₁

#8637

semidirect

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

subgroup	index
#21454749	2
#21444574	2
#21444556	2
#21443732	2
C₆₁×#1309	2
#8637	61

subgroup

index

C₆₁×#1309

Subgroups — #29573034

Conjugacy classes of subgroups

Complements

Maximal subgroups