Group #29573033 — subgroups

Subgroups — #29573033

Order 15616.

336 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 109 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	#21454754	C₂	1	1	yes	maximal, characteristic
2	#21444574	C₂	1	2	yes	maximal
3	#21444557	C₂	1	1	yes	maximal, characteristic
4	#21443732	C₂	1	2	yes	maximal
5	C₆₁×#1561	C₂	1	1	yes	maximal, characteristic
6	#16839893	—	2	2	no	—
7	Q₈×#92418	—	2	2	no	—
8	#16839789	C₄	1	2	yes	—
9	#16839786	C₂²	1	2	yes	—
10	#16839786	C₂²	1	1	yes	characteristic
11	#16839718	C₄	1	2	yes	—
12	C₂×C₄×#92418	C₂²	1	1	yes	characteristic
13	#16839512	—	2	2	no	—
14	#16839512	—	2	2	no	—
15	#16839508	—	2	2	no	—
16	C₆₁×#533	C₂²	1	1	yes	characteristic
17	C₆₁×#423	C₂²	1	2	yes	—
18	C₆₁×#332	—	2	2	no	—
19	C₂×#11758194	—	2	2	no	—
20	C₂×#11758194	C₂×C₄	1	2	yes	—
21	C₂²×#92418	C₂×C₄	1	1	yes	characteristic
22	C₂²×#92418	C₂×C₄	1	1	yes	characteristic
23	C₄⋊#92418	—	2	2	no	—
24	C₄⋊#92418	—	2	4	no	—
25	C₄⋊#92418	—	2	2	no	—
26	C₄⋊#92418	—	2	2	no	—
27	#14519688	—	2	2	no	—
28	C₄×#92418	—	2	2	no	—
29	C₄×#92418	D₄	1	2	yes	—
30	C₄×#92418	D₄	1	2	yes	—
31	C₂×M₁₆×C₆₁	—	2	2	no	—
32	C₄⋊Q₈×C₆₁	D₄	1	2	yes	—
33	#125×C₆₁	Q₈	1	2	yes	—
34	C₂²⋊Q₈×C₆₁	—	2	2	no	—
35	C₄×Q₈×C₆₁	—	2	2	no	—
36	#117×C₆₁	C₂×C₄	1	2	yes	—
37	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
38	C₄⋊C₈×C₆₁	—	2	2	no	—
39	C₄⋊C₈×C₆₁	—	2	2	no	—
40	C₂×Q₈×C₆₁	—	2	2	no	—
41	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₆₁	#41	1	2	yes	—
43	M₁₆×C₆₁	—	8	2	no	—
44	C₂×C₈×C₆₁	—	4	2	no	—
45	C₄⋊C₄×C₆₁	—	2	2	no	—
46	C₄⋊C₄×C₆₁	—	2	4	no	—
47	C₄⋊C₄×C₆₁	C₄⋊C₄	1	4	yes	—
48	C₂²⋊C₄×C₆₁	—	2	2	no	—
49	C₄²×C₆₁	—	2	2	no	—
50	C₄²×C₆₁	C₂×D₄	1	2	yes	—
51	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
52	C₄²×C₆₁	C₂×Q₈	1	1	yes	characteristic
53	#11758194	—	4	2	no	—
54	#11758194	—	8	2	no	—
55	C₂×#92418	—	2	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	Q₈×C₆₁	—	4	2	no	—
64	Q₈×C₆₁	—	4	2	no	—
65	C₂×C₄×C₆₁	—	2	1	no	—
66	C₂×C₄×C₆₁	—	2	1	no	—
67	C₂×C₄×C₆₁	—	2	2	no	—
68	C₂×C₄×C₆₁	—	2	2	no	—
69	C₂×C₄×C₆₁	—	2	4	no	—
70	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
72	C₂×C₄×C₆₁	#121	1	2	yes	—
73	C₂×C₄×C₆₁	C₂²⋊Q₈	1	2	yes	—
74	C₈×C₆₁	—	8	2	no	—
75	#92418	—	2	1	no	—
76	#92418	—	2	2	no	—
77	#92418	—	4	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	—	4	4	no	—
87	C₄×C₆₁	C₂×#101	1	1	yes	characteristic
88	C₄×C₆₁	#413	1	1	yes	characteristic
89	C₄×C₆₁	#471	1	2	yes	—
90	C₄×C₆₁	#471	1	2	yes	—
91	#2960	—	61	1	no	—
92	#1566	—	61	2	no	—
93	#1561	—	61	1	no	—
94	#1561	D₆₁	1	1	yes	characteristic
95	#1309	—	61	2	no	—
96	C₂×C₆₁	—	4	1	no	—
97	C₂×C₆₁	#1801	1	1	yes	characteristic
98	C₂×C₆₁	#1815	1	2	yes	—
99	#533	C₂×D₆₁	1	1	yes	characteristic
100	#446	—	122	2	no	—
101	C₈×Q₈	—	122	2	no	—
102	#424	—	61	2	no	—
103	#423	—	61	1	no	—
104	#423	—	61	2	no	—
105	#423	C₂×D₆₁	1	2	yes	—
106	#405	—	61	2	no	—
107	C₂×C₄×C₈	—	61	1	no	—
108	#334	—	122	2	no	—
109	#334	—	122	2	no	—
110	#332	—	2	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	—	61	2	no	—
115	C₂×M₁₆	—	122	2	no	—
116	C₂²×C₈	—	61	1	no	—
117	C₂²×C₈	—	61	1	no	—
118	C₄⋊Q₈	C₂²⋊D₆₁	1	2	yes	—
119	#125	Dic₁₂₂	1	2	yes	—
120	C₂²⋊Q₈	—	2	2	no	—
121	C₄×Q₈	—	2	2	no	—
122	#117	C₄×D₆₁	1	2	yes	—
123	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
124	C₄⋊C₈	—	2	2	no	—
125	C₄⋊C₈	—	2	2	no	—
126	C₄⋊C₈	—	122	4	no	—
127	C₄⋊C₈	—	122	2	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	—	2	2	no	—
135	C₂²×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
136	C₂²×C₄	#11758224	1	2	yes	—
137	M₁₆	—	8	2	no	—
138	M₁₆	—	488	2	no	—
139	M₁₆	—	244	2	no	—
140	C₂×C₈	—	4	2	no	—
141	C₂×C₈	—	61	1	no	—
142	C₂×C₈	—	61	1	no	—
143	C₂×C₈	—	61	2	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	C₄⋊C₄	—	2	2	no	—
149	C₄⋊C₄	—	2	4	no	—
150	C₄⋊C₄	#11758196	1	4	yes	—
151	C₂²⋊C₄	—	2	2	no	—
152	C₄²	—	2	2	no	—
153	C₄²	C₂×Dic₁₂₂	1	1	yes	characteristic
154	C₄²	D₄×D₆₁	1	2	yes	—
155	C₄²	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
156	C₂³	#14519733	1	1	yes	characteristic
157	Q₈	—	4	2	no	—
158	Q₈	—	4	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	—	2	4	no	—
164	C₂×C₄	C₂²⋊C₄×D₆₁	1	1	yes	characteristic
165	C₂×C₄	C₂×#11758196	1	1	yes	characteristic
166	C₂×C₄	#14519749	1	2	yes	—
167	C₂×C₄	#14519754	1	2	yes	—
168	C₈	—	8	2	no	—
169	C₈	—	61	2	no	—
170	C₈	—	122	1	no	—
171	C₈	—	122	2	no	—
172	C₈	—	488	2	no	—
173	C₈	—	244	2	no	—
174	C₂²	—	2	1	no	—
175	C₂²	—	2	2	no	—
176	C₂²	#16839620	1	1	yes	characteristic
177	C₄	—	2	1	no	—
178	C₄	—	2	2	no	—
179	C₄	—	4	4	no	—
180	C₄	#101×D₆₁	1	1	yes	characteristic
181	C₄	#16839745	1	1	yes	characteristic
182	C₄	#16840023	1	2	yes	—
183	C₄	#16840027	1	2	yes	—
184	C₂	—	4	1	no	—
185	C₂	#21445813	1	1	yes	characteristic
186	C₂	#21445876	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21454754

C₂

yes

maximal, characteristic

#21444574

C₂

yes

maximal

#21444557

C₂

yes

maximal, characteristic

#21443732

C₂

yes

maximal

C₆₁×#1561

C₂

yes

maximal, characteristic

#16839893

—

Q₈×#92418

—

#16839789

C₄

yes

—

#16839786

C₂²

yes

—

#16839786

C₂²

yes

characteristic

#16839718

C₄

yes

—

C₂×C₄×#92418

C₂²

yes

—

—

—

C₆₁×#533

C₂²

yes

characteristic

C₆₁×#423

C₂²

yes

—

C₆₁×#332

—

C₂×#11758194

—

C₂×#11758194

C₂×C₄

yes

—

C₂²×#92418

C₂×C₄

yes

characteristic

C₂²×#92418

C₂×C₄

yes

—

—

—

—

—

—

D₄

yes

—

C₄×#92418

D₄

yes

—

C₂×M₁₆×C₆₁

—

C₄⋊Q₈×C₆₁

D₄

yes

—

#125×C₆₁

Q₈

yes

—

C₂²⋊Q₈×C₆₁

—

C₄×Q₈×C₆₁

—

#117×C₆₁

C₂×C₄

yes

—

C₂×C₄²×C₆₁

C₂³

yes

characteristic

C₄⋊C₈×C₆₁

—

C₄⋊C₈×C₆₁

—

C₂×Q₈×C₆₁

—

C₂²×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₂²×C₄×C₆₁

#41

yes

—

M₁₆×C₆₁

—

C₂×C₈×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

C₄⋊C₄

yes

—

C₂²⋊C₄×C₆₁

—

C₄²×C₆₁

—

C₄²×C₆₁

C₂×D₄

yes

—

C₄²×C₆₁

C₂×D₄

yes

characteristic

C₄²×C₆₁

C₂×Q₈

yes

—

—

—

—

—

—

C₂²⋊C₄

yes

characteristic

C₂×#92418

C₂²⋊C₄

yes

—

C₂×#92418

C₂²⋊C₄

yes

characteristic

C₂³×C₆₁

#117

yes

characteristic

Q₈×C₆₁

—

Q₈×C₆₁

—

C₂×C₄×C₆₁

—

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

—

C₈×C₆₁

—

—

—

—

—

yes

—

#8637

—

maximal, Sylow

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

#380

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

yes

C₄×C₆₁

yes

C₄×C₆₁

yes

—

C₄×C₆₁

#471

yes

—

—

—

—

D₆₁

yes

characteristic

#1309

—

C₂×C₆₁

—

C₂×C₆₁

#1801

yes

characteristic

C₂×C₆₁

#1815

yes

—

#533

C₂×D₆₁

yes

characteristic

100

#446

—

122

—

101

C₈×Q₈

—

122

—

102

#424

—

103

#423

—

104

#423

—

105

#423

C₂×D₆₁

yes

—

106

#405

—

107

C₂×C₄×C₈

—

108

#334

—

122

—

109

#334

—

122

—

110

#332

—

111

#332

—

122

—

112

C₆₁

#8637

yes

characteristic, complemented, Sylow

113

C₂×M₁₆

—

114

C₂×M₁₆

—

115

C₂×M₁₆

—

122

—

116

C₂²×C₈

—

117

C₂²×C₈

—

118

C₄⋊Q₈

C₂²⋊D₆₁

yes

—

119

#125

Dic₁₂₂

yes

—

120

C₂²⋊Q₈

—

121

C₄×Q₈

—

122

#117

C₄×D₆₁

yes

—

123

C₂×C₄²

C₂²×D₆₁

yes

characteristic

124

C₄⋊C₈

—

125

C₄⋊C₈

—

126

C₄⋊C₈

—

122

—

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

—

135

C₂²×C₄

C₂×C₄×D₆₁

yes

characteristic

136

C₂²×C₄

#11758224

yes

—

137

M₁₆

—

138

M₁₆

—

488

—

139

M₁₆

—

244

—

140

C₂×C₈

—

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

C₄⋊C₄

—

149

C₄⋊C₄

—

150

C₄⋊C₄

#11758196

yes

—

151

C₂²⋊C₄

—

152

C₄²

—

153

C₄²

C₂×Dic₁₂₂

yes

characteristic

154

C₄²

D₄×D₆₁

yes

—

155

C₄²

C₂×C₂²⋊D₆₁

yes

characteristic

156

C₂³

#14519733

yes

characteristic

157

Q₈

—

158

Q₈

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

C₂²⋊C₄×D₆₁

yes

characteristic

165

C₂×C₄

C₂×#11758196

yes

characteristic

166

C₂×C₄

#14519749

yes

—

167

C₂×C₄

#14519754

yes

—

168

C₈

—

169

C₈

—

170

C₈

—

122

—

171

C₈

—

122

—

172

C₈

—

488

—

173

C₈

—

244

—

174

C₂²

—

175

C₂²

—

176

C₂²

#16839620

yes

characteristic

177

C₄

—

178

C₄

—

179

C₄

—

180

C₄

#101×D₆₁

yes

characteristic

181

C₄

#16839745

yes

characteristic

182

C₄

#16840023

yes

—

183

C₄

#16840027

yes

—

184

C₂

—

185

C₂

#21445813

yes

characteristic

186

C₂

#21445876

yes

—

	subgroup	G/H
center	C₂²	#16839620
commutator	C₂²×C₄×C₆₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₁
Fitting	C₆₁×#1561	C₂
socle	C₂²×C₆₁	#380
prime radical	C₂³×C₆₁	#117
cosocle	C₂×C₄²×C₆₁	C₂³
hypercenter	#1561	D₆₁
hypocenter	C₆₁	#8637
LCS	G ▷ C₂²×C₄×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1561

subgroup

G/H

center

C₂²

#16839620

commutator

C₂²×C₄×C₆₁

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₆₁

Fitting

C₆₁×#1561

C₂

C₂²×C₆₁

C₂³×C₆₁

C₂×C₄²×C₆₁

C₂³

D₆₁

C₆₁

G ▷ C₂²×C₄×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1561

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
#21454754	2
#21444574	2
#21444557	2
#21443732	2
C₆₁×#1561	2
#8637	61

subgroup

index

C₆₁×#1561

Subgroups — #29573033

Conjugacy classes of subgroups

Complements

Maximal subgroups