Group #29599844 — subgroups

Subgroups — #29599844

Order 15616.

302 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 302 conjugacy classes of subgroups — too many to draw. Only its 93 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	C₂×#16839577	C₂	1	2	yes	maximal
2	#21446690	C₂	1	1	yes	maximal, characteristic
3	#21443919	C₂	1	2	yes	maximal
4	#21443903	C₂	1	1	yes	maximal, characteristic
5	C₆₁×#1849	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₄₄	C₂²	1	1	yes	characteristic
7	C₂×#14519717	C₂²	1	2	yes	—
8	C₂×#14519709	C₂²	1	1	yes	characteristic
9	C₂×#14519702	—	2	1	no	—
10	#16839668	—	2	1	no	—
11	#16839577	—	2	4	no	—
12	C₂×#108×C₆₁	—	2	1	no	—
13	C₂×#108×C₆₁	C₂²	1	2	yes	—
14	C₆₁×#423	—	2	1	no	—
15	C₂×#97×C₆₁	C₂²	1	1	yes	characteristic
16	C₂²×Dic₁₂₂	—	2	1	no	—
17	C₂×Dic₂₄₄	—	2	2	no	—
18	C₂×Dic₂₄₄	—	2	2	no	—
19	C₂×Dic₂₄₄	—	2	1	no	—
20	C₂×Dic₂₄₄	D₄	1	2	yes	—
21	C₂×C₄⋊Dic₆₁	—	2	1	no	—
22	#14519717	—	2	2	no	—
23	#14519717	—	2	2	no	—
24	C₂×#11758185	—	2	2	no	—
25	#14519709	—	2	2	no	—
26	#14519702	—	4	2	no	—
27	#14519684	—	4	2	no	—
28	C₂×M₁₆×C₆₁	—	2	2	no	—
29	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
30	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
31	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
32	#108×C₆₁	—	2	2	no	—
33	#108×C₆₁	—	2	2	no	—
34	#108×C₆₁	—	4	2	no	—
35	C₄⋊C₈×C₆₁	—	4	2	no	—
36	#97×C₆₁	—	2	2	no	—
37	C₂²×C₄×C₆₁	SD₁₆	1	2	yes	—
38	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
39	C₂²×Dic₆₁	—	4	1	no	—
40	C₂²×Dic₆₁	—	4	1	no	—
41	C₂×Dic₁₂₂	—	2	2	no	—
42	C₂×Dic₁₂₂	—	4	1	no	—
43	C₂×Dic₁₂₂	—	4	1	no	—
44	C₂×Dic₁₂₂	—	4	1	no	—
45	C₂×Dic₁₂₂	—	4	1	no	—
46	C₂×Dic₁₂₂	—	4	1	no	—
47	M₁₆×C₆₁	—	4	2	no	—
48	M₁₆×C₆₁	—	4	2	no	—
49	C₂×C₈×C₆₁	—	2	1	no	—
50	C₂×C₈×C₆₁	—	2	1	no	—
51	C₂×C₈×C₆₁	—	4	2	no	—
52	C₂×C₈×C₆₁	D₈	1	2	yes	—
53	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
54	C₂×C₈×C₆₁	#41	1	2	yes	—
55	C₂×C₈×C₆₁	#41	1	2	yes	—
56	C₂×C₈×C₆₁	#41	1	1	yes	characteristic
57	C₄²×C₆₁	—	2	1	no	—
58	C₄²×C₆₁	—	2	1	no	—
59	C₄⋊Dic₆₁	—	4	1	no	—
60	C₄⋊Dic₆₁	—	4	1	no	—
61	Dic₂₄₄	—	2	4	no	—
62	Dic₂₄₄	—	4	2	no	—
63	#11758185	—	4	2	no	—
64	C₂³×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
65	C₂×C₄×C₆₁	—	2	2	no	—
66	C₂×C₄×C₆₁	—	4	2	no	—
67	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
68	C₂×C₄×C₆₁	#123	1	1	yes	characteristic
69	C₂×C₄×C₆₁	#124	1	1	yes	characteristic
70	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
71	C₂×C₄×C₆₁	AGL(1,ℤ/8)	1	1	yes	characteristic
72	C₂×C₄×C₆₁	#137	1	1	yes	characteristic
73	C₂×Dic₆₁	—	4	2	no	—
74	C₂×Dic₆₁	—	8	1	no	—
75	C₂×Dic₆₁	—	8	1	no	—
76	C₂×Dic₆₁	—	8	2	no	—
77	C₂×Dic₆₁	—	8	1	no	—
78	Dic₁₂₂	—	4	2	no	—
79	Dic₁₂₂	—	4	2	no	—
80	Dic₁₂₂	—	8	1	no	—
81	Dic₁₂₂	—	8	1	no	—
82	C₈×C₆₁	—	2	2	no	—
83	C₈×C₆₁	—	8	2	no	—
84	C₈×C₆₁	#123	1	2	yes	—
85	C₈×C₆₁	#126	1	2	yes	—
86	#13488	—	61	1	no	maximal, Sylow
87	C₂²×C₆₁	—	2	2	no	—
88	C₂²×C₆₁	#461	1	1	yes	characteristic
89	C₂²×C₆₁	#481	1	1	yes	characteristic
90	C₂²×C₆₁	#486	1	1	yes	characteristic
91	C₄×C₆₁	—	4	2	no	—
92	C₄×C₆₁	#397	1	1	yes	characteristic
93	C₄×C₆₁	#459	1	1	yes	characteristic
94	C₄×C₆₁	#480	1	1	yes	characteristic
95	C₄×C₆₁	#489	1	1	yes	characteristic
96	Dic₆₁	—	8	2	no	—
97	Dic₆₁	—	16	1	no	—
98	C₂×#362	—	61	2	no	—
99	#1971	—	61	1	no	—
100	#1849	D₆₁	1	1	yes	characteristic
101	#1383	—	61	2	no	—
102	#1377	—	61	1	no	—
103	C₂×C₆₁	—	2	1	no	—
104	C₂×C₆₁	—	2	1	no	—
105	C₂×C₆₁	#2071	1	1	yes	characteristic
106	C₂×C₆₁	#2239	1	2	yes	—
107	C₂²×Q₁₆	—	61	1	no	—
108	C₂×M₃₂	—	61	2	no	—
109	C₂×#108	—	2	1	no	—
110	C₂×#108	C₂×D₆₁	1	2	yes	—
111	C₂×#106	—	61	1	no	—
112	#423	—	2	1	no	—
113	C₂×#103	—	122	1	no	—
114	C₂×#97	C₂×D₆₁	1	1	yes	characteristic
115	#391	—	122	1	no	—
116	#362	—	122	4	no	—
117	C₆₁	#13488	1	1	yes	characteristic, complemented, Sylow
118	C₂²×Q₈	—	122	1	no	—
119	C₂×Q₁₆	—	61	2	no	—
120	C₂×Q₁₆	—	122	2	no	—
121	C₂×Q₁₆	—	122	1	no	—
122	C₂×Q₁₆	—	122	2	no	—
123	C₂×M₁₆	—	2	2	no	—
124	C₂²×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
125	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
126	C₂×C₄⋊C₄	—	122	1	no	—
127	C₂×C₄²	C₂²⋊D₆₁	1	1	yes	characteristic
128	M₃₂	—	122	2	no	—
129	M₃₂	—	122	2	no	—
130	C₂×C₁₆	—	122	2	no	—
131	#108	—	2	2	no	—
132	#108	—	2	2	no	—
133	#108	—	4	2	no	—
134	#106	—	122	2	no	—
135	C₄⋊C₈	—	4	2	no	—
136	#103	—	244	2	no	—
137	#97	—	2	2	no	—
138	#95	—	244	2	no	—
139	C₂×Q₈	—	122	2	no	—
140	C₂×Q₈	—	244	1	no	—
141	C₂×Q₈	—	244	1	no	—
142	C₂×Q₈	—	244	1	no	—
143	C₂×Q₈	—	244	1	no	—
144	C₂×Q₈	—	244	1	no	—
145	C₂²×C₄	—	244	1	no	—
146	C₂²×C₄	—	244	1	no	—
147	C₂²×C₄	#11758201	1	2	yes	—
148	C₂²×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
149	Q₁₆	—	122	4	no	—
150	Q₁₆	—	244	2	no	—
151	M₁₆	—	4	2	no	—
152	M₁₆	—	4	2	no	—
153	C₂×C₈	—	2	1	no	—
154	C₂×C₈	—	2	1	no	—
155	C₂×C₈	—	4	2	no	—
156	C₂×C₈	#11758199	1	2	yes	—
157	C₂×C₈	#11758222	1	2	yes	—
158	C₂×C₈	D₄×D₆₁	1	1	yes	characteristic
159	C₂×C₈	#11758224	1	1	yes	characteristic
160	C₂×C₈	#11758226	1	2	yes	—
161	C₄⋊C₄	—	244	1	no	—
162	C₄⋊C₄	—	244	1	no	—
163	C₄²	—	2	1	no	—
164	C₄²	—	2	1	no	—
165	C₁₆	—	244	2	no	—
166	C₂³	C₂×#11758201	1	1	yes	characteristic
167	Q₈	—	488	1	no	—
168	Q₈	—	488	1	no	—
169	Q₈	—	244	2	no	—
170	Q₈	—	244	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	4	2	no	—
173	C₂×C₄	—	488	2	no	—
174	C₂×C₄	—	488	1	no	—
175	C₂×C₄	—	488	1	no	—
176	C₂×C₄	—	488	1	no	—
177	C₂×C₄	—	244	2	no	—
178	C₂×C₄	#14519748	1	1	yes	characteristic
179	C₂×C₄	#14519760	1	1	yes	characteristic
180	C₂×C₄	#14519766	1	1	yes	characteristic
181	C₂×C₄	C₂×#11758199	1	1	yes	characteristic
182	C₂×C₄	#14519803	1	1	yes	characteristic
183	C₂×C₄	#14519808	1	1	yes	characteristic
184	C₈	—	2	2	no	—
185	C₈	—	8	2	no	—
186	C₈	#14519762	1	2	yes	—
187	C₈	#14519772	1	2	yes	—
188	C₂²	—	2	2	no	—
189	C₂²	#16839964	1	1	yes	characteristic
190	C₂²	#16840080	1	1	yes	characteristic
191	C₂²	#16840110	1	1	yes	characteristic
192	C₄	—	4	2	no	—
193	C₄	—	976	1	no	—
194	C₄	—	488	2	no	—
195	C₄	#16839688	1	1	yes	characteristic
196	C₄	#16839954	1	1	yes	characteristic
197	C₄	#16840075	1	1	yes	characteristic
198	C₄	#16840124	1	1	yes	characteristic
199	C₂	—	2	1	no	—
200	C₂	—	2	1	no	—
201	C₂	#21447223	1	1	yes	characteristic
202	C₂	#21448038	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16839577

C₂

yes

maximal

#21446690

C₂

yes

maximal, characteristic

#21443919

C₂

yes

maximal

#21443903

C₂

yes

maximal, characteristic

C₆₁×#1849

C₂

yes

maximal, characteristic

C₂²×Dic₂₄₄

C₂²

yes

characteristic

C₂×#14519717

C₂²

yes

—

C₂×#14519709

C₂²

yes

—

—

—

C₂×#108×C₆₁

—

C₂×#108×C₆₁

C₂²

yes

—

C₆₁×#423

—

C₂×#97×C₆₁

C₂²

yes

characteristic

C₂²×Dic₁₂₂

—

C₂×Dic₂₄₄

—

C₂×Dic₂₄₄

—

C₂×Dic₂₄₄

—

C₂×Dic₂₄₄

D₄

yes

—

C₂×C₄⋊Dic₆₁

—

—

—

—

—

—

—

C₂×M₁₆×C₆₁

—

C₂²×C₈×C₆₁

D₄

yes

characteristic

C₂²×C₈×C₆₁

C₂³

yes

characteristic

C₂×C₄²×C₆₁

D₄

yes

characteristic

#108×C₆₁

—

#108×C₆₁

—

#108×C₆₁

—

C₄⋊C₈×C₆₁

—

#97×C₆₁

—

C₂²×C₄×C₆₁

SD₁₆

yes

—

C₂²×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂²×Dic₆₁

—

C₂²×Dic₆₁

—

C₂×Dic₁₂₂

—

C₂×Dic₁₂₂

—

C₂×Dic₁₂₂

—

C₂×Dic₁₂₂

—

C₂×Dic₁₂₂

—

C₂×Dic₁₂₂

—

M₁₆×C₆₁

—

M₁₆×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

D₈

yes

—

C₂×C₈×C₆₁

C₂×D₄

yes

characteristic

C₂×C₈×C₆₁

#41

yes

—

C₂×C₈×C₆₁

#41

yes

—

C₂×C₈×C₆₁

#41

yes

characteristic

C₄²×C₆₁

—

C₄²×C₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

Dic₂₄₄

—

Dic₂₄₄

—

#11758185

—

C₂³×C₆₁

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

yes

C₂×C₄×C₆₁

yes

C₂×C₄×C₆₁

yes

C₂×C₄×C₆₁

C₂×D₈

yes

C₂×C₄×C₆₁

yes

C₂×C₄×C₆₁

yes

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

Dic₁₂₂

—

Dic₁₂₂

—

Dic₁₂₂

—

Dic₁₂₂

—

C₈×C₆₁

—

C₈×C₆₁

—

C₈×C₆₁

#123

yes

—

C₈×C₆₁

#126

yes

—

#13488

—

maximal, Sylow

C₂²×C₆₁

—

C₂²×C₆₁

yes

C₂²×C₆₁

yes

C₂²×C₆₁

yes

C₄×C₆₁

—

C₄×C₆₁

yes

C₄×C₆₁

yes

C₄×C₆₁

yes

C₄×C₆₁

yes

Dic₆₁

—

Dic₆₁

—

C₂×#362

—

#1971

—

100

#1849

D₆₁

yes

characteristic

101

#1383

—

102

#1377

—

103

C₂×C₆₁

—

104

C₂×C₆₁

—

105

C₂×C₆₁

#2071

yes

characteristic

106

C₂×C₆₁

#2239

yes

—

107

C₂²×Q₁₆

—

108

C₂×M₃₂

—

109

C₂×#108

—

110

C₂×#108

C₂×D₆₁

yes

—

111

C₂×#106

—

112

#423

—

113

C₂×#103

—

122

—

114

C₂×#97

C₂×D₆₁

yes

characteristic

115

#391

—

122

—

116

#362

—

122

—

117

C₆₁

#13488

yes

characteristic, complemented, Sylow

118

C₂²×Q₈

—

122

—

119

C₂×Q₁₆

—

120

C₂×Q₁₆

—

122

—

121

C₂×Q₁₆

—

122

—

122

C₂×Q₁₆

—

122

—

123

C₂×M₁₆

—

124

C₂²×C₈

C₂²⋊D₆₁

yes

characteristic

125

C₂²×C₈

C₂²×D₆₁

yes

characteristic

126

C₂×C₄⋊C₄

—

122

—

127

C₂×C₄²

C₂²⋊D₆₁

yes

characteristic

128

M₃₂

—

122

—

129

M₃₂

—

122

—

130

C₂×C₁₆

—

122

—

131

#108

—

132

#108

—

133

#108

—

134

#106

—

122

—

135

C₄⋊C₈

—

136

#103

—

244

—

137

#97

—

138

#95

—

244

—

139

C₂×Q₈

—

122

—

140

C₂×Q₈

—

244

—

141

C₂×Q₈

—

244

—

142

C₂×Q₈

—

244

—

143

C₂×Q₈

—

244

—

144

C₂×Q₈

—

244

—

145

C₂²×C₄

—

244

—

146

C₂²×C₄

—

244

—

147

C₂²×C₄

#11758201

yes

—

148

C₂²×C₄

C₂×C₂²⋊D₆₁

yes

characteristic

149

Q₁₆

—

122

—

150

Q₁₆

—

244

—

151

M₁₆

—

152

M₁₆

—

153

C₂×C₈

—

154

C₂×C₈

—

155

C₂×C₈

—

156

C₂×C₈

#11758199

yes

—

157

C₂×C₈

#11758222

yes

—

158

C₂×C₈

D₄×D₆₁

yes

characteristic

159

C₂×C₈

#11758224

yes

characteristic

160

C₂×C₈

#11758226

yes

—

161

C₄⋊C₄

—

244

—

162

C₄⋊C₄

—

244

—

163

C₄²

—

164

C₄²

—

165

C₁₆

—

244

—

166

C₂³

C₂×#11758201

yes

characteristic

167

Q₈

—

488

—

168

Q₈

—

488

—

169

Q₈

—

244

—

170

Q₈

—

244

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

488

—

174

C₂×C₄

—

488

—

175

C₂×C₄

—

488

—

176

C₂×C₄

—

488

—

177

C₂×C₄

—

244

—

178

C₂×C₄

#14519748

yes

characteristic

179

C₂×C₄

#14519760

yes

characteristic

180

C₂×C₄

#14519766

yes

characteristic

181

C₂×C₄

C₂×#11758199

yes

characteristic

182

C₂×C₄

#14519803

yes

characteristic

183

C₂×C₄

#14519808

yes

characteristic

184

C₈

—

185

C₈

—

186

C₈

#14519762

yes

—

187

C₈

#14519772

yes

—

188

C₂²

—

189

C₂²

#16839964

yes

characteristic

190

C₂²

#16840080

yes

characteristic

191

C₂²

#16840110

yes

characteristic

192

C₄

—

193

C₄

—

976

—

194

C₄

—

488

—

195

C₄

#16839688

yes

characteristic

196

C₄

#16839954

yes

characteristic

197

C₄

#16840075

yes

characteristic

198

C₄

#16840124

yes

characteristic

199

C₂

—

200

C₂

—

201

C₂

#21447223

yes

characteristic

202

C₂

#21448038

yes

—

	subgroup	G/H
center	C₂²	#16840080
commutator, cosocle	C₂²×C₈×C₆₁	C₂³
Frattini	C₂²×C₈	C₂²×D₆₁
Fitting	C₆₁×#1849	C₂
socle	C₂²×C₆₁	#481
prime radical	C₂³×C₆₁	C₂×SD₁₆
hypercenter	#1849	D₆₁
hypocenter	C₆₁	#13488
LCS	G ▷ C₂²×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#97 ◁ #1849

subgroup

G/H

center

C₂²

#16840080

commutator, cosocle

C₂²×C₈×C₆₁

C₂³

Frattini

C₂²×C₈

C₂²×D₆₁

Fitting

C₆₁×#1849

C₂

socle

C₂²×C₆₁

#481

prime radical

C₂³×C₆₁

C₂×SD₁₆

D₆₁

C₆₁

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

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#97 ◁ #1849

N	H ≅ G/N	type
117. C₆₁	#13488	semidirect

H ≅ G/N

type

117. C₆₁

#13488

semidirect

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

subgroup	index
C₂×#16839577	2
#21446690	2
#21443919	2
#21443903	2
C₆₁×#1849	2
#13488	61

subgroup

index

C₆₁×#1849

Subgroups — #29599844

Conjugacy classes of subgroups

Complements

Maximal subgroups