Group #29599553 — subgroups

Subgroups — #29599553

Order 15616.

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

normal subgroups only

This group has 418 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	C₂²×D₉₇₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#16839576	C₂	1	2	yes	maximal, complemented
3	#21446729	C₂	1	2	yes	maximal, complemented
4	#21443913	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₁×#2151	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₄₈₈	C₂²	1	2	yes	—
7	C₂×D₉₇₆	—	2	4	no	—
8	C₂×D₉₇₆	—	2	2	no	—
9	C₂×#14519713	C₂²	1	2	yes	—
10	#16839766	—	2	2	no	—
11	C₂×#14519693	—	2	2	no	—
12	#16839576	—	2	4	no	—
13	C₂×M₃₂×C₆₁	C₂²	1	1	yes	characteristic
14	C₂²×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
15	C₆₁×#405	C₂²	1	1	yes	characteristic
16	C₆₁×C₄⋊C₁₆	—	2	2	no	—
17	C₂²×D₂₄₄	—	2	2	no	—
18	C₂×D₄₈₈	—	2	2	no	—
19	C₂×D₄₈₈	—	2	2	no	—
20	C₂×D₄₈₈	—	2	2	no	—
21	C₂×D₄₈₈	—	2	2	no	—
22	C₂×D₄₈₈	—	2	2	no	—
23	C₂×D₄₈₈	D₄	1	4	yes	—
24	C₂×#11758194	—	2	2	no	—
25	D₉₇₆	—	2	8	no	—
26	#14519713	—	2	2	no	—
27	#14519713	—	2	2	no	—
28	#14519697	—	4	4	no	—
29	#14519693	—	4	4	no	—
30	C₂×M₁₆×C₆₁	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
32	#117×C₆₁	D₄	1	1	yes	characteristic
33	M₃₂×C₆₁	—	2	2	no	—
34	C₂×C₁₆×C₆₁	—	2	2	no	—
35	C₂×C₁₆×C₆₁	—	2	1	no	—
36	C₂×C₁₆×C₆₁	D₄	1	2	yes	—
37	#97×C₆₁	—	2	1	no	—
38	C₄×C₈×C₆₁	—	2	1	no	—
39	C₂³×D₆₁	—	4	2	no	—
40	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
41	C₂×D₂₄₄	—	2	4	no	—
42	C₂×D₂₄₄	—	4	2	no	—
43	C₂×D₂₄₄	—	4	2	no	—
44	C₂×D₂₄₄	—	4	2	no	—
45	C₂×D₂₄₄	—	4	2	no	—
46	C₂×D₂₄₄	—	4	2	no	—
47	M₁₆×C₆₁	—	2	2	no	—
48	C₂×C₈×C₆₁	—	2	1	no	—
49	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
50	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
51	C₂×C₈×C₆₁	#41	1	1	yes	characteristic
52	C₂×C₈×C₆₁	#41	1	2	yes	—
53	C₄⋊C₄×C₆₁	D₈	1	2	yes	—
54	C₂²⋊C₄×C₆₁	—	2	1	no	—
55	C₄²×C₆₁	—	2	1	no	—
56	#11758194	—	4	2	no	—
57	#11758194	—	4	2	no	—
58	C₂×#92418	—	4	2	no	—
59	D₄₈₈	—	2	4	no	—
60	D₄₈₈	—	2	4	no	—
61	D₄₈₈	—	4	2	no	—
62	D₄₈₈	—	4	2	no	—
63	C₁₆×C₆₁	—	2	2	no	—
64	C₁₆×C₆₁	—	4	1	no	—
65	C₂³×C₆₁	#135	1	1	yes	characteristic
66	C₂²×D₆₁	—	4	4	no	—
67	C₂²×D₆₁	—	8	2	no	—
68	C₂²×D₆₁	—	8	2	no	—
69	C₂²×D₆₁	—	16	4	no	—
70	C₂×C₄×C₆₁	—	2	2	no	—
71	C₂×C₄×C₆₁	#121	1	2	yes	—
72	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
73	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
74	C₂×C₄×C₆₁	AGL(1,ℤ/8)	1	2	yes	—
75	D₂₄₄	—	4	4	no	—
76	D₂₄₄	—	4	4	no	—
77	D₂₄₄	—	8	2	no	—
78	D₂₄₄	—	8	2	no	—
79	C₈×C₆₁	—	4	1	no	—
80	C₈×C₆₁	#120	1	1	yes	characteristic
81	C₈×C₆₁	#123	1	2	yes	—
82	C₈×C₆₁	#124	1	1	yes	characteristic
83	#92418	—	8	2	no	—
84	#13439	—	61	1	no	maximal, Sylow
85	C₂²×C₆₁	—	2	2	no	—
86	C₂²×C₆₁	#463	1	2	yes	—
87	C₂²×C₆₁	#466	1	1	yes	characteristic
88	C₂×D₆₁	—	8	4	no	—
89	C₂×D₆₁	—	16	4	no	—
90	C₂×D₆₁	—	16	4	no	—
91	C₂×D₆₁	—	16	4	no	—
92	C₄×C₆₁	—	4	2	no	—
93	C₄×C₆₁	#394	1	1	yes	characteristic
94	C₄×C₆₁	#459	1	2	yes	—
95	C₄×C₆₁	#469	1	1	yes	characteristic
96	C₂²×D₁₆	—	61	1	no	—
97	#2151	D₆₁	1	1	yes	characteristic, complemented
98	C₂×#361	—	61	2	no	—
99	#1978	—	61	2	no	—
100	#1381	—	61	1	no	—
101	C₂×C₆₁	—	2	2	no	—
102	C₂×C₆₁	#2054	1	1	yes	characteristic
103	C₂×C₆₁	#2221	1	2	yes	—
104	D₆₁	—	16	4	no	—
105	C₂²×D₈	—	61	2	no	—
106	C₂×D₁₆	—	122	4	no	—
107	C₂×D₁₆	—	122	2	no	—
108	C₂×M₃₂	C₂×D₆₁	1	1	yes	characteristic
109	C₂²×C₁₆	C₂×D₆₁	1	1	yes	characteristic
110	C₂×#108	—	61	2	no	—
111	#418	—	122	2	no	—
112	C₂×#100	—	122	2	no	—
113	#405	C₂×D₆₁	1	1	yes	characteristic
114	C₄⋊C₁₆	—	2	2	no	—
115	#361	—	122	4	no	—
116	C₆₁	#13439	1	1	yes	characteristic, complemented, Sylow
117	C₂²×D₄	—	122	2	no	—
118	C₂×D₈	—	61	4	no	—
119	C₂×D₈	—	122	2	no	—
120	C₂×D₈	—	122	2	no	—
121	C₂×D₈	—	122	2	no	—
122	C₂×D₈	—	122	2	no	—
123	C₂×D₈	—	122	2	no	—
124	C₂×M₁₆	—	122	2	no	—
125	C₂×M₁₆	D₂₄₄	1	1	yes	characteristic
126	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
127	#117	D₂₄₄	1	1	yes	characteristic
128	D₁₆	—	122	8	no	—
129	M₃₂	—	2	2	no	—
130	C₂×C₁₆	—	2	2	no	—
131	C₂×C₁₆	—	2	1	no	—
132	C₂×C₁₆	C₂²⋊D₆₁	1	2	yes	—
133	#108	—	122	2	no	—
134	#108	—	122	2	no	—
135	#102	—	244	4	no	—
136	#100	—	244	4	no	—
137	#97	—	2	1	no	—
138	C₄×C₈	—	2	1	no	—
139	C₂⁴	—	244	2	no	—
140	C₂×D₄	—	122	4	no	—
141	C₂×D₄	—	244	2	no	—
142	C₂×D₄	—	244	2	no	—
143	C₂×D₄	—	244	2	no	—
144	C₂×D₄	—	244	2	no	—
145	C₂×D₄	—	244	2	no	—
146	C₂²×C₄	C₂×D₂₄₄	1	1	yes	characteristic
147	D₈	—	122	4	no	—
148	D₈	—	122	4	no	—
149	D₈	—	244	2	no	—
150	D₈	—	244	2	no	—
151	M₁₆	—	2	2	no	—
152	M₁₆	—	244	2	no	—
153	M₁₆	—	244	2	no	—
154	C₂×C₈	—	2	1	no	—
155	C₂×C₈	—	244	2	no	—
156	C₂×C₈	#11758222	1	1	yes	characteristic
157	C₂×C₈	D₄×D₆₁	1	2	yes	—
158	C₂×C₈	#11758226	1	2	yes	—
159	C₂×C₈	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
160	C₄⋊C₄	D₄₈₈	1	2	yes	—
161	C₂²⋊C₄	—	2	1	no	—
162	C₄²	—	2	1	no	—
163	C₁₆	—	2	2	no	—
164	C₁₆	—	4	1	no	—
165	C₂³	—	976	4	no	—
166	C₂³	—	488	2	no	—
167	C₂³	—	488	2	no	—
168	C₂³	—	244	4	no	—
169	C₂³	#14519799	1	1	yes	characteristic
170	D₄	—	488	2	no	—
171	D₄	—	488	2	no	—
172	D₄	—	244	4	no	—
173	D₄	—	244	4	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	#14519751	1	1	yes	characteristic
176	C₂×C₄	#14519752	1	2	yes	—
177	C₂×C₄	C₂×D₄₈₈	1	1	yes	characteristic
178	C₂×C₄	#14519805	1	2	yes	—
179	C₈	—	4	1	no	—
180	C₈	—	488	2	no	—
181	C₈	#14519745	1	1	yes	characteristic
182	C₈	#14519762	1	2	yes	—
183	C₈	#14519766	1	1	yes	characteristic
184	C₂²	—	2	2	no	—
185	C₂²	—	976	4	no	—
186	C₂²	—	976	4	no	—
187	C₂²	—	976	4	no	—
188	C₂²	—	488	4	no	—
189	C₂²	#16839978	1	2	yes	—
190	C₂²	#16839998	1	1	yes	characteristic
191	C₄	—	4	2	no	—
192	C₄	#16839679	1	1	yes	characteristic
193	C₄	#16839958	1	2	yes	—
194	C₄	#16840015	1	1	yes	characteristic
195	C₂	—	2	2	no	—
196	C₂	—	976	4	no	—
197	C₂	#21447130	1	1	yes	characteristic
198	C₂	#21447939	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×D₉₇₆

C₂

yes

maximal, characteristic

C₂×#16839576

C₂

yes

maximal, complemented

#21446729

C₂

yes

maximal, complemented

#21443913

C₂

yes

maximal, characteristic, complemented

C₆₁×#2151

C₂

yes

maximal, characteristic, complemented

C₂²×D₄₈₈

C₂²

yes

—

C₂×D₉₇₆

—

C₂×D₉₇₆

—

C₂×#14519713

C₂²

yes

—

#16839766

—

C₂×#14519693

—

#16839576

—

C₂×M₃₂×C₆₁

C₂²

yes

characteristic

C₂²×C₁₆×C₆₁

C₂²

yes

characteristic

C₆₁×#405

C₂²

yes

characteristic

C₆₁×C₄⋊C₁₆

—

C₂²×D₂₄₄

—

C₂×D₄₈₈

—

C₂×D₄₈₈

—

C₂×D₄₈₈

—

C₂×D₄₈₈

—

C₂×D₄₈₈

—

C₂×D₄₈₈

D₄

yes

—

C₂×#11758194

—

D₉₇₆

—

—

—

—

—

C₂×M₁₆×C₆₁

D₄

yes

characteristic

C₂²×C₈×C₆₁

C₂³

yes

characteristic

#117×C₆₁

D₄

yes

characteristic

M₃₂×C₆₁

—

C₂×C₁₆×C₆₁

—

C₂×C₁₆×C₆₁

—

C₂×C₁₆×C₆₁

D₄

yes

—

#97×C₆₁

—

C₄×C₈×C₆₁

—

C₂³×D₆₁

—

C₂²×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×D₂₄₄

—

C₂×D₂₄₄

—

C₂×D₂₄₄

—

C₂×D₂₄₄

—

C₂×D₂₄₄

—

C₂×D₂₄₄

—

M₁₆×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

C₂×D₄

yes

—

C₂×C₈×C₆₁

C₂×D₄

yes

C₂×C₈×C₆₁

yes

C₂×C₈×C₆₁

yes

—

C₄⋊C₄×C₆₁

D₈

yes

—

C₂²⋊C₄×C₆₁

—

C₄²×C₆₁

—

#11758194

—

#11758194

—

C₂×#92418

—

D₄₈₈

—

D₄₈₈

—

D₄₈₈

—

D₄₈₈

—

C₁₆×C₆₁

—

C₁₆×C₆₁

—

C₂³×C₆₁

#135

yes

characteristic

C₂²×D₆₁

—

C₂²×D₆₁

—

C₂²×D₆₁

—

C₂²×D₆₁

—

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

—

D₂₄₄

—

D₂₄₄

—

D₂₄₄

—

D₂₄₄

—

C₈×C₆₁

—

C₈×C₆₁

#120

yes

characteristic

C₈×C₆₁

#123

yes

—

C₈×C₆₁

yes

—

—

C₂²×C₆₁

—

C₂²×C₆₁

#463

yes

—

C₂²×C₆₁

#466

yes

characteristic

C₂×D₆₁

—

C₂×D₆₁

—

C₂×D₆₁

—

C₂×D₆₁

—

C₄×C₆₁

—

C₄×C₆₁

#394

yes

characteristic

C₄×C₆₁

#459

yes

—

C₄×C₆₁

#469

yes

characteristic

C₂²×D₁₆

—

#2151

D₆₁

yes

characteristic, complemented

C₂×#361

—

#1978

—

100

#1381

—

101

C₂×C₆₁

—

102

C₂×C₆₁

#2054

yes

characteristic

103

C₂×C₆₁

#2221

yes

—

104

D₆₁

—

105

C₂²×D₈

—

106

C₂×D₁₆

—

122

—

107

C₂×D₁₆

—

122

—

108

C₂×M₃₂

C₂×D₆₁

yes

characteristic

109

C₂²×C₁₆

C₂×D₆₁

yes

characteristic

110

C₂×#108

—

111

#418

—

122

—

112

C₂×#100

—

122

—

113

#405

C₂×D₆₁

yes

characteristic

114

C₄⋊C₁₆

—

115

#361

—

122

—

116

C₆₁

#13439

yes

characteristic, complemented, Sylow

117

C₂²×D₄

—

122

—

118

C₂×D₈

—

119

C₂×D₈

—

122

—

120

C₂×D₈

—

122

—

121

C₂×D₈

—

122

—

122

C₂×D₈

—

122

—

123

C₂×D₈

—

122

—

124

C₂×M₁₆

—

122

—

125

C₂×M₁₆

D₂₄₄

yes

characteristic

126

C₂²×C₈

C₂²×D₆₁

yes

characteristic

127

#117

D₂₄₄

yes

characteristic

128

D₁₆

—

122

—

129

M₃₂

—

130

C₂×C₁₆

—

131

C₂×C₁₆

—

132

C₂×C₁₆

C₂²⋊D₆₁

yes

—

133

#108

—

122

—

134

#108

—

122

—

135

#102

—

244

—

136

#100

—

244

—

137

#97

—

138

C₄×C₈

—

139

C₂⁴

—

244

—

140

C₂×D₄

—

122

—

141

C₂×D₄

—

244

—

142

C₂×D₄

—

244

—

143

C₂×D₄

—

244

—

144

C₂×D₄

—

244

—

145

C₂×D₄

—

244

—

146

C₂²×C₄

C₂×D₂₄₄

yes

characteristic

147

D₈

—

122

—

148

D₈

—

122

—

149

D₈

—

244

—

150

D₈

—

244

—

151

M₁₆

—

152

M₁₆

—

244

—

153

M₁₆

—

244

—

154

C₂×C₈

—

155

C₂×C₈

—

244

—

156

C₂×C₈

#11758222

yes

characteristic

157

C₂×C₈

D₄×D₆₁

yes

—

158

C₂×C₈

#11758226

yes

—

159

C₂×C₈

C₂×C₂²⋊D₆₁

yes

characteristic

160

C₄⋊C₄

D₄₈₈

yes

—

161

C₂²⋊C₄

—

162

C₄²

—

163

C₁₆

—

164

C₁₆

—

165

C₂³

—

976

—

166

C₂³

—

488

—

167

C₂³

—

488

—

168

C₂³

—

244

—

169

C₂³

#14519799

yes

characteristic

170

D₄

—

488

—

171

D₄

—

488

—

172

D₄

—

244

—

173

D₄

—

244

—

174

C₂×C₄

—

175

C₂×C₄

#14519751

yes

characteristic

176

C₂×C₄

#14519752

yes

—

177

C₂×C₄

C₂×D₄₈₈

yes

characteristic

178

C₂×C₄

#14519805

yes

—

179

C₈

—

180

C₈

—

488

—

181

C₈

#14519745

yes

characteristic

182

C₈

#14519762

yes

—

183

C₈

#14519766

yes

characteristic

184

C₂²

—

185

C₂²

—

976

—

186

C₂²

—

976

—

187

C₂²

—

976

—

188

C₂²

—

488

—

189

C₂²

#16839978

yes

—

190

C₂²

#16839998

yes

characteristic

191

C₄

—

192

C₄

#16839679

yes

characteristic

193

C₄

#16839958

yes

—

194

C₄

#16840015

yes

characteristic

195

C₂

—

196

C₂

—

976

—

197

C₂

#21447130

yes

characteristic

198

C₂

#21447939

yes

—

	subgroup	G/H
center	C₂²	#16839998
commutator, cosocle	C₂²×C₈×C₆₁	C₂³
Frattini	C₂²×C₈	C₂²×D₆₁
Fitting	C₆₁×#2151	C₂
socle	C₂²×C₆₁	#466
prime radical	C₂²×D₉₇₆	C₂
hypercenter	#2151	D₆₁
hypocenter	C₆₁	#13439
LCS	G ▷ C₂²×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #405 ◁ #2151

subgroup

G/H

center

C₂²

#16839998

commutator, cosocle

C₂²×C₈×C₆₁

C₂³

Frattini

C₂²×C₈

C₂²×D₆₁

Fitting

C₆₁×#2151

C₂

socle

C₂²×C₆₁

#466

prime radical

C₂²×D₉₇₆

C₂

D₆₁

C₆₁

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

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #405 ◁ #2151

N	H ≅ G/N	type
2. C₂×#16839576	C₂	semidirect
3. #21446729	C₂	semidirect
4. #21443913	C₂	semidirect
5. C₆₁×#2151	C₂	semidirect
97. #2151	D₆₁	semidirect
116. C₆₁	#13439	semidirect

H ≅ G/N

type

2. C₂×#16839576

C₂

semidirect

3. #21446729

C₂

semidirect

4. #21443913

C₂

semidirect

5. C₆₁×#2151

C₂

semidirect

97. #2151

D₆₁

semidirect

116. C₆₁

#13439

semidirect

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

subgroup	index
C₂²×D₉₇₆	2
C₂×#16839576	2
#21446729	2
#21443913	2
C₆₁×#2151	2
#13439	61

subgroup

index

C₂²×D₉₇₆

C₆₁×#2151

Subgroups — #29599553

Conjugacy classes of subgroups

Complements

Maximal subgroups