Group #29599562 — subgroups

Subgroups — #29599562

Order 15616.

378 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 95 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	#21447577	C₂	1	1	yes	maximal, characteristic
2	C₂×#16839578	C₂	1	2	yes	maximal
3	#21446735	C₂	1	2	yes	maximal
4	#21443913	C₂	1	1	yes	maximal, characteristic
5	C₂²×Q₃₂×C₆₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#14519717	C₂²	1	1	yes	characteristic
7	C₂×#14519713	C₂²	1	2	yes	—
8	#16839771	—	2	2	no	—
9	C₂×#14519696	—	2	2	no	—
10	#16839717	C₂²	1	1	yes	characteristic
11	#16839580	—	2	2	no	—
12	#16839578	—	2	4	no	—
13	C₂²×Q₁₆×C₆₁	C₂²	1	2	yes	—
14	C₂×Q₃₂×C₆₁	—	2	4	no	—
15	C₂×Q₃₂×C₆₁	—	2	2	no	—
16	C₂²×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
17	C₂×#11758194	—	2	2	no	—
18	C₂×#11758194	D₄	1	1	yes	characteristic
19	#14519736	D₄	1	1	yes	characteristic
20	#14519717	—	2	2	no	—
21	C₂×#11758185	—	2	1	no	—
22	#14519713	—	2	2	no	—
23	#14519713	—	2	2	no	—
24	#14519701	—	4	4	no	—
25	#14519696	—	4	4	no	—
26	#14519687	—	2	1	no	—
27	C₈×Dic₆₁	—	2	1	no	—
28	C₂²×Q₈×C₆₁	—	2	2	no	—
29	C₂×Q₁₆×C₆₁	—	2	2	no	—
30	C₂×Q₁₆×C₆₁	—	2	2	no	—
31	C₂×Q₁₆×C₆₁	—	2	2	no	—
32	C₂×Q₁₆×C₆₁	—	2	2	no	—
33	C₂×Q₁₆×C₆₁	—	2	2	no	—
34	C₂×Q₁₆×C₆₁	D₄	1	4	yes	—
35	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
36	Q₃₂×C₆₁	—	2	8	no	—
37	C₂×C₁₆×C₆₁	—	2	2	no	—
38	C₂×C₁₆×C₆₁	D₄	1	2	yes	—
39	C₂×Q₈×C₆₁	—	2	4	no	—
40	C₂×Q₈×C₆₁	—	4	2	no	—
41	C₂×Q₈×C₆₁	—	4	2	no	—
42	C₂×Q₈×C₆₁	—	4	2	no	—
43	C₂×Q₈×C₆₁	—	4	2	no	—
44	C₂×Q₈×C₆₁	—	4	2	no	—
45	C₂²×C₄×C₆₁	—	4	2	no	—
46	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
47	Q₁₆×C₆₁	—	2	4	no	—
48	Q₁₆×C₆₁	—	2	4	no	—
49	Q₁₆×C₆₁	—	4	2	no	—
50	Q₁₆×C₆₁	—	4	2	no	—
51	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
52	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
53	C₂×C₈×C₆₁	#41	1	1	yes	characteristic
54	C₂×C₈×C₆₁	#41	1	2	yes	—
55	C₂²⋊Dic₆₁	—	2	1	no	—
56	C₄⋊Dic₆₁	D₈	1	2	yes	—
57	C₄×Dic₆₁	—	2	1	no	—
58	#11758194	—	2	2	no	—
59	#11758194	—	4	2	no	—
60	#11758194	—	4	2	no	—
61	C₂×#92418	—	2	1	no	—
62	C₂×#92418	—	4	2	no	—
63	C₁₆×C₆₁	—	2	2	no	—
64	#11758185	—	4	1	no	—
65	C₂³×C₆₁	#135	1	1	yes	characteristic
66	Q₈×C₆₁	—	4	4	no	—
67	Q₈×C₆₁	—	4	4	no	—
68	Q₈×C₆₁	—	8	2	no	—
69	Q₈×C₆₁	—	8	2	no	—
70	C₂×C₄×C₆₁	—	4	4	no	—
71	C₂×C₄×C₆₁	—	8	2	no	—
72	C₂×C₄×C₆₁	—	8	2	no	—
73	C₂×C₄×C₆₁	#121	1	2	yes	—
74	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
75	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
76	C₂×C₄×C₆₁	AGL(1,ℤ/8)	1	2	yes	—
77	C₂×Dic₆₁	—	2	2	no	—
78	C₈×C₆₁	#120	1	1	yes	characteristic
79	C₈×C₆₁	#123	1	2	yes	—
80	C₈×C₆₁	#124	1	1	yes	characteristic
81	#92418	—	4	1	no	—
82	#92418	—	8	2	no	—
83	#13441	—	61	1	no	maximal, Sylow
84	C₂²×C₆₁	—	2	2	no	—
85	C₂²×C₆₁	#463	1	2	yes	—
86	C₂²×C₆₁	#466	1	1	yes	characteristic
87	C₄×C₆₁	—	8	4	no	—
88	C₄×C₆₁	#394	1	1	yes	characteristic
89	C₄×C₆₁	#459	1	2	yes	—
90	C₄×C₆₁	#469	1	1	yes	characteristic
91	Dic₆₁	—	4	2	no	—
92	C₂²×Q₃₂	D₆₁	1	1	yes	characteristic
93	#2151	—	61	1	no	—
94	C₂×#362	—	61	2	no	—
95	#1979	—	61	2	no	—
96	#1381	—	61	1	no	—
97	C₂×C₆₁	—	2	2	no	—
98	C₂×C₆₁	#2054	1	1	yes	characteristic
99	C₂×C₆₁	#2222	1	2	yes	—
100	C₂²×Q₁₆	C₂×D₆₁	1	2	yes	—
101	C₂×Q₃₂	—	2	4	no	—
102	C₂×Q₃₂	—	2	2	no	—
103	C₂×M₃₂	—	61	1	no	—
104	C₂²×C₁₆	C₂×D₆₁	1	1	yes	characteristic
105	C₂×#108	—	61	2	no	—
106	#419	—	122	2	no	—
107	C₂×#101	—	122	2	no	—
108	#405	—	61	1	no	—
109	C₄⋊C₁₆	—	122	2	no	—
110	#362	—	122	4	no	—
111	C₆₁	#13441	1	1	yes	characteristic, complemented, Sylow
112	C₂²×Q₈	—	2	2	no	—
113	C₂×Q₁₆	—	2	2	no	—
114	C₂×Q₁₆	—	2	2	no	—
115	C₂×Q₁₆	—	2	2	no	—
116	C₂×Q₁₆	—	2	2	no	—
117	C₂×Q₁₆	—	2	2	no	—
118	C₂×Q₁₆	C₂²⋊D₆₁	1	4	yes	—
119	C₂×M₁₆	—	61	1	no	—
120	C₂×M₁₆	—	122	2	no	—
121	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
122	#117	—	61	1	no	—
123	Q₃₂	—	2	8	no	—
124	M₃₂	—	122	2	no	—
125	C₂×C₁₆	—	2	2	no	—
126	C₂×C₁₆	—	122	1	no	—
127	C₂×C₁₆	C₂²⋊D₆₁	1	2	yes	—
128	#108	—	122	2	no	—
129	#108	—	122	2	no	—
130	#103	—	244	4	no	—
131	#101	—	244	4	no	—
132	#97	—	122	1	no	—
133	C₄×C₈	—	122	1	no	—
134	C₂×Q₈	—	2	4	no	—
135	C₂×Q₈	—	4	2	no	—
136	C₂×Q₈	—	4	2	no	—
137	C₂×Q₈	—	4	2	no	—
138	C₂×Q₈	—	4	2	no	—
139	C₂×Q₈	—	4	2	no	—
140	C₂²×C₄	—	4	2	no	—
141	C₂²×C₄	D₄×D₆₁	1	1	yes	characteristic
142	Q₁₆	—	2	4	no	—
143	Q₁₆	—	2	4	no	—
144	Q₁₆	—	4	2	no	—
145	Q₁₆	—	4	2	no	—
146	M₁₆	—	122	2	no	—
147	M₁₆	—	244	2	no	—
148	M₁₆	—	244	2	no	—
149	C₂×C₈	—	122	1	no	—
150	C₂×C₈	—	244	2	no	—
151	C₂×C₈	#11758224	1	1	yes	characteristic
152	C₂×C₈	#11758224	1	2	yes	—
153	C₂×C₈	C₂×C₂²⋊D₆₁	1	2	yes	—
154	C₂×C₈	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
155	C₄⋊C₄	—	61	2	no	—
156	C₂²⋊C₄	—	122	1	no	—
157	C₄²	—	122	1	no	—
158	C₁₆	—	2	2	no	—
159	C₁₆	—	244	1	no	—
160	C₂³	#14519796	1	1	yes	characteristic
161	Q₈	—	4	4	no	—
162	Q₈	—	4	4	no	—
163	Q₈	—	8	2	no	—
164	Q₈	—	8	2	no	—
165	C₂×C₄	—	4	4	no	—
166	C₂×C₄	—	8	2	no	—
167	C₂×C₄	—	8	2	no	—
168	C₂×C₄	—	122	2	no	—
169	C₂×C₄	#14519748	1	1	yes	characteristic
170	C₂×C₄	#14519749	1	2	yes	—
171	C₂×C₄	D₈×D₆₁	1	1	yes	characteristic
172	C₂×C₄	#14519801	1	2	yes	—
173	C₈	—	488	2	no	—
174	C₈	—	244	1	no	—
175	C₈	#14519746	1	1	yes	characteristic
176	C₈	#14519758	1	2	yes	—
177	C₈	#14519764	1	1	yes	characteristic
178	C₂²	—	2	2	no	—
179	C₂²	#16839975	1	2	yes	—
180	C₂²	#16839995	1	1	yes	characteristic
181	C₄	—	8	4	no	—
182	C₄	—	244	2	no	—
183	C₄	#16839676	1	1	yes	characteristic
184	C₄	#16839955	1	2	yes	—
185	C₄	#16840012	1	1	yes	characteristic
186	C₂	—	2	2	no	—
187	C₂	#21447127	1	1	yes	characteristic
188	C₂	#21447941	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21447577

C₂

yes

maximal, characteristic

C₂×#16839578

C₂

yes

maximal

#21446735

C₂

yes

maximal

#21443913

C₂

yes

maximal, characteristic

C₂²×Q₃₂×C₆₁

C₂

yes

maximal, characteristic

C₂×#14519717

C₂²

yes

characteristic

C₂×#14519713

C₂²

yes

—

#16839771

—

C₂×#14519696

—

#16839717

C₂²

yes

characteristic

#16839580

—

#16839578

—

C₂²×Q₁₆×C₆₁

C₂²

yes

—

C₂×Q₃₂×C₆₁

—

C₂×Q₃₂×C₆₁

—

C₂²×C₁₆×C₆₁

C₂²

yes

characteristic

C₂×#11758194

—

C₂×#11758194

D₄

yes

characteristic

#14519736

D₄

yes

—

—

—

—

—

—

—

C₈×Dic₆₁

—

C₂²×Q₈×C₆₁

—

C₂×Q₁₆×C₆₁

—

C₂×Q₁₆×C₆₁

—

C₂×Q₁₆×C₆₁

—

C₂×Q₁₆×C₆₁

—

C₂×Q₁₆×C₆₁

—

C₂×Q₁₆×C₆₁

D₄

yes

—

C₂²×C₈×C₆₁

C₂³

yes

characteristic

Q₃₂×C₆₁

—

C₂×C₁₆×C₆₁

—

C₂×C₁₆×C₆₁

D₄

yes

—

C₂×Q₈×C₆₁

—

C₂×Q₈×C₆₁

—

C₂×Q₈×C₆₁

—

C₂×Q₈×C₆₁

—

C₂×Q₈×C₆₁

—

C₂×Q₈×C₆₁

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

C₂×D₄

yes

characteristic

Q₁₆×C₆₁

—

Q₁₆×C₆₁

—

Q₁₆×C₆₁

—

Q₁₆×C₆₁

—

C₂×C₈×C₆₁

C₂×D₄

yes

—

C₂×C₈×C₆₁

C₂×D₄

yes

C₂×C₈×C₆₁

yes

C₂×C₈×C₆₁

yes

—

C₂²⋊Dic₆₁

—

C₄⋊Dic₆₁

D₈

yes

—

C₄×Dic₆₁

—

—

—

—

—

—

C₁₆×C₆₁

—

#11758185

—

C₂³×C₆₁

#135

yes

characteristic

Q₈×C₆₁

—

Q₈×C₆₁

—

Q₈×C₆₁

—

Q₈×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

#121

yes

—

C₂×C₄×C₆₁

#121

yes

characteristic

C₂×C₄×C₆₁

C₂×D₈

yes

characteristic

C₂×C₄×C₆₁

AGL(1,ℤ/8)

yes

—

C₂×Dic₆₁

—

C₈×C₆₁

#120

yes

characteristic

C₈×C₆₁

#123

yes

—

C₈×C₆₁

yes

—

—

—

C₂²×C₆₁

—

C₂²×C₆₁

#463

yes

—

C₂²×C₆₁

#466

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

#394

yes

characteristic

C₄×C₆₁

#459

yes

—

C₄×C₆₁

#469

yes

characteristic

Dic₆₁

—

C₂²×Q₃₂

D₆₁

yes

—

—

—

—

C₂×C₆₁

—

C₂×C₆₁

#2054

yes

characteristic

C₂×C₆₁

#2222

yes

—

100

C₂²×Q₁₆

C₂×D₆₁

yes

—

101

C₂×Q₃₂

—

102

C₂×Q₃₂

—

103

C₂×M₃₂

—

104

C₂²×C₁₆

C₂×D₆₁

yes

characteristic

105

C₂×#108

—

106

#419

—

122

—

107

C₂×#101

—

122

—

108

#405

—

109

C₄⋊C₁₆

—

122

—

110

#362

—

122

—

111

C₆₁

#13441

yes

characteristic, complemented, Sylow

112

C₂²×Q₈

—

113

C₂×Q₁₆

—

114

C₂×Q₁₆

—

115

C₂×Q₁₆

—

116

C₂×Q₁₆

—

117

C₂×Q₁₆

—

118

C₂×Q₁₆

C₂²⋊D₆₁

yes

—

119

C₂×M₁₆

—

120

C₂×M₁₆

—

122

—

121

C₂²×C₈

C₂²×D₆₁

yes

characteristic

122

#117

—

123

Q₃₂

—

124

M₃₂

—

122

—

125

C₂×C₁₆

—

126

C₂×C₁₆

—

122

—

127

C₂×C₁₆

C₂²⋊D₆₁

yes

—

128

#108

—

122

—

129

#108

—

122

—

130

#103

—

244

—

131

#101

—

244

—

132

#97

—

122

—

133

C₄×C₈

—

122

—

134

C₂×Q₈

—

135

C₂×Q₈

—

136

C₂×Q₈

—

137

C₂×Q₈

—

138

C₂×Q₈

—

139

C₂×Q₈

—

140

C₂²×C₄

—

141

C₂²×C₄

D₄×D₆₁

yes

characteristic

142

Q₁₆

—

143

Q₁₆

—

144

Q₁₆

—

145

Q₁₆

—

146

M₁₆

—

122

—

147

M₁₆

—

244

—

148

M₁₆

—

244

—

149

C₂×C₈

—

122

—

150

C₂×C₈

—

244

—

151

C₂×C₈

#11758224

yes

characteristic

152

C₂×C₈

#11758224

yes

—

153

C₂×C₈

C₂×C₂²⋊D₆₁

yes

—

154

C₂×C₈

C₂×C₂²⋊D₆₁

yes

characteristic

155

C₄⋊C₄

—

156

C₂²⋊C₄

—

122

—

157

C₄²

—

122

—

158

C₁₆

—

159

C₁₆

—

244

—

160

C₂³

#14519796

yes

characteristic

161

Q₈

—

162

Q₈

—

163

Q₈

—

164

Q₈

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

122

—

169

C₂×C₄

#14519748

yes

characteristic

170

C₂×C₄

#14519749

yes

—

171

C₂×C₄

D₈×D₆₁

yes

characteristic

172

C₂×C₄

#14519801

yes

—

173

C₈

—

488

—

174

C₈

—

244

—

175

C₈

#14519746

yes

characteristic

176

C₈

#14519758

yes

—

177

C₈

#14519764

yes

characteristic

178

C₂²

—

179

C₂²

#16839975

yes

—

180

C₂²

#16839995

yes

characteristic

181

C₄

—

182

C₄

—

244

—

183

C₄

#16839676

yes

characteristic

184

C₄

#16839955

yes

—

185

C₄

#16840012

yes

characteristic

186

C₂

—

187

C₂

#21447127

yes

characteristic

188

C₂

#21447941

yes

—

	subgroup	G/H
center	C₂²	#16839995
commutator, cosocle	C₂²×C₈×C₆₁	C₂³
Frattini	C₂²×C₈	C₂²×D₆₁
Fitting	C₂²×Q₃₂×C₆₁	C₂
socle	C₂²×C₆₁	#466
prime radical	C₂³×C₆₁	#135
hypercenter	C₂²×Q₃₂	D₆₁
hypocenter	C₆₁	#13441
LCS	G ▷ C₂²×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×Q₃₂

subgroup

G/H

center

C₂²

#16839995

commutator, cosocle

C₂²×C₈×C₆₁

C₂³

Frattini

C₂²×C₈

C₂²×D₆₁

Fitting

C₂²×Q₃₂×C₆₁

C₂

C₂²×C₆₁

C₂³×C₆₁

C₂²×Q₃₂

D₆₁

hypocenter

C₆₁

#13441

LCS

G ▷ C₂²×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×Q₃₂

N	H ≅ G/N	type
111. C₆₁	#13441	semidirect

H ≅ G/N

type

111. C₆₁

#13441

semidirect

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

subgroup	index
#21447577	2
C₂×#16839578	2
#21446735	2
#21443913	2
C₂²×Q₃₂×C₆₁	2
#13441	61

subgroup

index

C₂²×Q₃₂×C₆₁

Subgroups — #29599562

Conjugacy classes of subgroups

Complements

Maximal subgroups