Group #29584787 — subgroups

Subgroups — #29584787

Order 15616.

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

normal subgroups only

This group has 292 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	#21446800	C₂	1	1	yes	maximal, characteristic
2	#21444477	C₂	1	2	yes	maximal
3	#21443716	C₂	1	2	yes	maximal
4	#21443648	C₂	1	1	yes	maximal, characteristic
5	C₆₁×#1843	C₂	1	1	yes	maximal, characteristic
6	#16840319	C₂²	1	1	yes	characteristic
7	#16839815	C₂²	1	2	yes	—
8	#16839786	C₂²	1	1	yes	characteristic
9	#16839522	—	2	1	no	—
10	#16839522	—	2	1	no	—
11	#16839510	—	2	2	no	—
12	#16839507	—	2	2	no	—
13	C₂×C₄⋊C₈×C₆₁	—	2	1	no	—
14	C₂×C₄⋊C₈×C₆₁	—	2	1	no	—
15	C₂×C₄⋊C₈×C₆₁	C₂²	1	2	yes	—
16	C₆₁×#376	C₂²	1	1	yes	characteristic
17	C₂×#11758194	—	2	1	no	—
18	#14519777	D₄	1	2	yes	—
19	#14519769	—	2	1	no	—
20	#14519756	—	2	2	no	—
21	C₄×Dic₁₂₂	—	2	2	no	—
22	#14519736	—	2	1	no	—
23	C₄⋊#92418	—	2	2	no	—
24	C₄⋊#92418	—	2	2	no	—
25	C₂²⋊#92418	—	2	2	no	—
26	C₄×#92418	—	2	2	no	—
27	C₂²×C₈×C₆₁	—	2	2	no	—
28	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
29	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
30	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
31	C₄⋊C₈×C₆₁	—	2	2	no	—
32	C₄⋊C₈×C₆₁	—	2	2	no	—
33	C₄⋊C₈×C₆₁	—	4	2	no	—
34	C₄⋊C₈×C₆₁	—	4	2	no	—
35	#95×C₆₁	—	2	2	no	—
36	C₂²×C₄×C₆₁	—	2	1	no	—
37	C₂²×C₄×C₆₁	—	2	1	no	—
38	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
39	C₂²×C₄×C₆₁	#41	1	2	yes	—
40	C₂×Dic₁₂₂	—	2	2	no	—
41	C₂×C₈×C₆₁	—	4	2	no	—
42	C₂×C₈×C₆₁	—	4	2	no	—
43	C₂×C₈×C₆₁	—	4	2	no	—
44	C₄²×C₆₁	—	2	1	no	—
45	C₄²×C₆₁	—	2	1	no	—
46	C₄²×C₆₁	SD₁₆	1	2	yes	—
47	C₄²×C₆₁	Q₁₆	1	2	yes	—
48	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
49	C₄²×C₆₁	#41	1	1	yes	characteristic
50	C₄²×C₆₁	#41	1	2	yes	—
51	C₂²⋊Dic₆₁	—	4	1	no	—
52	C₄⋊Dic₆₁	—	2	2	no	—
53	C₄⋊Dic₆₁	—	2	2	no	—
54	#11758196	—	4	2	no	—
55	C₄×Dic₆₁	—	4	1	no	—
56	#11758194	—	4	1	no	—
57	#11758194	—	4	1	no	—
58	C₂×#92418	—	4	2	no	—
59	C₂×#92418	—	4	1	no	—
60	C₂³×C₆₁	#123	1	1	yes	characteristic
61	C₂×C₄×C₆₁	—	2	2	no	—
62	C₂×C₄×C₆₁	—	2	2	no	—
63	C₂×C₄×C₆₁	—	4	2	no	—
64	C₂×C₄×C₆₁	—	4	2	no	—
65	C₂×C₄×C₆₁	—	4	2	no	—
66	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
67	C₂×C₄×C₆₁	#123	1	2	yes	—
68	C₂×C₄×C₆₁	#124	1	1	yes	characteristic
69	C₂×C₄×C₆₁	#126	1	2	yes	—
70	C₂×C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
71	C₂×C₄×C₆₁	C₂×Q₁₆	1	1	yes	characteristic
72	C₂×C₄×C₆₁	AGL(1,ℤ/8)	1	1	yes	characteristic
73	C₂×C₄×C₆₁	#137	1	1	yes	characteristic
74	C₂×Dic₆₁	—	4	2	no	—
75	Dic₁₂₂	—	8	2	no	—
76	C₈×C₆₁	—	8	2	no	—
77	#92418	—	8	2	no	—
78	#92418	—	8	1	no	—
79	#10628	—	61	1	no	maximal, Sylow
80	C₂²×C₆₁	—	2	2	no	—
81	C₂²×C₆₁	#397	1	1	yes	characteristic
82	C₂²×C₆₁	#483	1	1	yes	characteristic
83	C₂²×C₆₁	#484	1	1	yes	characteristic
84	C₄×C₆₁	—	2	2	no	—
85	C₄×C₆₁	—	2	2	no	—
86	C₄×C₆₁	—	4	2	no	—
87	C₄×C₆₁	—	4	2	no	—
88	C₄×C₆₁	#460	1	1	yes	characteristic
89	C₄×C₆₁	#462	1	1	yes	characteristic
90	C₄×C₆₁	#487	1	1	yes	characteristic
91	C₄×C₆₁	#489	1	1	yes	characteristic
92	Dic₆₁	—	8	2	no	—
93	#1995	—	61	1	no	—
94	#1843	D₆₁	1	1	yes	characteristic
95	#1542	—	61	2	no	—
96	#1301	—	61	2	no	—
97	#1274	—	61	1	no	—
98	C₂×C₆₁	—	2	1	no	—
99	C₂×C₆₁	—	2	1	no	—
100	C₂×C₆₁	#2072	1	1	yes	characteristic
101	C₂×C₆₁	#2080	1	2	yes	—
102	#533	—	61	1	no	—
103	#431	—	61	2	no	—
104	#423	—	61	1	no	—
105	C₂×C₄⋊C₈	—	2	1	no	—
106	C₂×C₄⋊C₈	—	2	1	no	—
107	C₂×C₄⋊C₈	C₂×D₆₁	1	2	yes	—
108	#376	C₂×D₆₁	1	1	yes	characteristic
109	#339	—	122	1	no	—
110	#339	—	122	1	no	—
111	#333	—	122	2	no	—
112	#332	—	122	2	no	—
113	C₆₁	#10628	1	1	yes	characteristic, complemented, Sylow
114	C₂×M₁₆	—	122	1	no	—
115	C₂²×C₈	—	2	2	no	—
116	C₄⋊Q₈	—	61	2	no	—
117	#125	—	122	1	no	—
118	C₂²⋊Q₈	—	122	2	no	—
119	C₄×Q₈	—	122	2	no	—
120	#117	—	122	1	no	—
121	C₂×C₄²	C₂²⋊D₆₁	1	1	yes	characteristic
122	C₂×C₄²	C₂²⋊D₆₁	1	1	yes	characteristic
123	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
124	C₄⋊C₈	—	2	2	no	—
125	C₄⋊C₈	—	2	2	no	—
126	C₄⋊C₈	—	4	2	no	—
127	C₄⋊C₈	—	4	2	no	—
128	C₄⋊C₈	—	122	2	no	—
129	C₄⋊C₈	—	122	2	no	—
130	C₂²⋊C₈	—	122	2	no	—
131	C₄×C₈	—	122	2	no	—
132	#95	—	2	2	no	—
133	C₂×Q₈	—	122	2	no	—
134	C₂²×C₄	—	2	1	no	—
135	C₂²×C₄	—	2	1	no	—
136	C₂²×C₄	#11758222	1	2	yes	—
137	C₂²×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
138	M₁₆	—	244	1	no	—
139	M₁₆	—	244	1	no	—
140	C₂×C₈	—	4	2	no	—
141	C₂×C₈	—	4	2	no	—
142	C₂×C₈	—	4	2	no	—
143	C₂×C₈	—	244	2	no	—
144	C₂×C₈	—	244	1	no	—
145	C₄⋊C₄	—	122	2	no	—
146	C₄⋊C₄	—	122	2	no	—
147	C₄⋊C₄	—	244	2	no	—
148	C₂²⋊C₄	—	244	1	no	—
149	C₄²	—	2	1	no	—
150	C₄²	—	2	1	no	—
151	C₄²	—	244	1	no	—
152	C₄²	#11758200	1	2	yes	—
153	C₄²	#11758202	1	2	yes	—
154	C₄²	D₄×D₆₁	1	1	yes	characteristic
155	C₄²	#11758224	1	1	yes	characteristic
156	C₄²	#11758226	1	2	yes	—
157	C₂³	#14519760	1	1	yes	characteristic
158	Q₈	—	488	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	4	2	no	—
162	C₂×C₄	—	4	2	no	—
163	C₂×C₄	—	4	2	no	—
164	C₂×C₄	—	244	2	no	—
165	C₂×C₄	#14519748	1	1	yes	characteristic
166	C₂×C₄	#14519762	1	2	yes	—
167	C₂×C₄	#14519766	1	1	yes	characteristic
168	C₂×C₄	#14519772	1	2	yes	—
169	C₂×C₄	C₂×#11758200	1	1	yes	characteristic
170	C₂×C₄	C₂×#11758202	1	1	yes	characteristic
171	C₂×C₄	#14519803	1	1	yes	characteristic
172	C₂×C₄	#14519808	1	1	yes	characteristic
173	C₈	—	8	2	no	—
174	C₈	—	488	2	no	—
175	C₈	—	488	1	no	—
176	C₂²	—	2	2	no	—
177	C₂²	#16839688	1	1	yes	characteristic
178	C₂²	#16840092	1	1	yes	characteristic
179	C₂²	#16840097	1	1	yes	characteristic
180	C₄	—	2	2	no	—
181	C₄	—	2	2	no	—
182	C₄	—	4	2	no	—
183	C₄	—	4	2	no	—
184	C₄	—	488	2	no	—
185	C₄	#16839959	1	1	yes	characteristic
186	C₄	#16839969	1	1	yes	characteristic
187	C₄	#16840114	1	1	yes	characteristic
188	C₄	#16840124	1	1	yes	characteristic
189	C₂	—	2	1	no	—
190	C₂	—	2	1	no	—
191	C₂	#21447228	1	1	yes	characteristic
192	C₂	#21447270	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21446800

C₂

yes

maximal, characteristic

#21444477

C₂

yes

maximal

#21443716

C₂

yes

maximal

#21443648

C₂

yes

maximal, characteristic

C₆₁×#1843

C₂

yes

maximal, characteristic

#16840319

C₂²

yes

characteristic

#16839815

C₂²

yes

—

#16839786

C₂²

yes

—

—

—

—

C₂×C₄⋊C₈×C₆₁

—

C₂×C₄⋊C₈×C₆₁

—

C₂×C₄⋊C₈×C₆₁

C₂²

yes

—

C₆₁×#376

C₂²

yes

characteristic

C₂×#11758194

—

#14519777

D₄

yes

—

#14519769

—

#14519756

—

C₄×Dic₁₂₂

—

—

—

—

—

—

C₂²×C₈×C₆₁

—

C₂×C₄²×C₆₁

D₄

yes

characteristic

C₂×C₄²×C₆₁

D₄

yes

characteristic

C₂×C₄²×C₆₁

C₂³

yes

characteristic

C₄⋊C₈×C₆₁

—

C₄⋊C₈×C₆₁

—

C₄⋊C₈×C₆₁

—

C₄⋊C₈×C₆₁

—

#95×C₆₁

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂²×C₄×C₆₁

#41

yes

—

C₂×Dic₁₂₂

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₄²×C₆₁

—

C₄²×C₆₁

—

C₄²×C₆₁

SD₁₆

yes

—

C₄²×C₆₁

Q₁₆

yes

—

C₄²×C₆₁

C₂×D₄

yes

C₄²×C₆₁

yes

C₄²×C₆₁

yes

—

C₂²⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

#11758196

—

C₄×Dic₆₁

—

—

—

—

—

C₂³×C₆₁

#123

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

#121

yes

characteristic

C₂×C₄×C₆₁

#123

yes

—

C₂×C₄×C₆₁

#124

yes

characteristic

C₂×C₄×C₆₁

#126

yes

—

C₂×C₄×C₆₁

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₆₁

C₂×Q₁₆

yes

C₂×C₄×C₆₁

yes

C₂×C₄×C₆₁

yes

C₂×Dic₆₁

—

Dic₁₂₂

—

C₈×C₆₁

—

—

—

—

C₂²×C₆₁

—

C₂²×C₆₁

yes

C₂²×C₆₁

yes

C₂²×C₆₁

yes

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

yes

C₄×C₆₁

yes

C₄×C₆₁

yes

C₄×C₆₁

yes

Dic₆₁

—

#1995

—

#1843

D₆₁

yes

—

—

—

C₂×C₆₁

—

C₂×C₆₁

—

100

C₂×C₆₁

#2072

yes

characteristic

101

C₂×C₆₁

#2080

yes

—

102

#533

—

103

#431

—

104

#423

—

105

C₂×C₄⋊C₈

—

106

C₂×C₄⋊C₈

—

107

C₂×C₄⋊C₈

C₂×D₆₁

yes

—

108

#376

C₂×D₆₁

yes

characteristic

109

#339

—

122

—

110

#339

—

122

—

111

#333

—

122

—

112

#332

—

122

—

113

C₆₁

#10628

yes

characteristic, complemented, Sylow

114

C₂×M₁₆

—

122

—

115

C₂²×C₈

—

116

C₄⋊Q₈

—

117

#125

—

122

—

118

C₂²⋊Q₈

—

122

—

119

C₄×Q₈

—

122

—

120

#117

—

122

—

121

C₂×C₄²

C₂²⋊D₆₁

yes

characteristic

122

C₂×C₄²

C₂²⋊D₆₁

yes

characteristic

123

C₂×C₄²

C₂²×D₆₁

yes

characteristic

124

C₄⋊C₈

—

125

C₄⋊C₈

—

126

C₄⋊C₈

—

127

C₄⋊C₈

—

128

C₄⋊C₈

—

122

—

129

C₄⋊C₈

—

122

—

130

C₂²⋊C₈

—

122

—

131

C₄×C₈

—

122

—

132

#95

—

133

C₂×Q₈

—

122

—

134

C₂²×C₄

—

135

C₂²×C₄

—

136

C₂²×C₄

#11758222

yes

—

137

C₂²×C₄

C₂×C₂²⋊D₆₁

yes

characteristic

138

M₁₆

—

244

—

139

M₁₆

—

244

—

140

C₂×C₈

—

141

C₂×C₈

—

142

C₂×C₈

—

143

C₂×C₈

—

244

—

144

C₂×C₈

—

244

—

145

C₄⋊C₄

—

122

—

146

C₄⋊C₄

—

122

—

147

C₄⋊C₄

—

244

—

148

C₂²⋊C₄

—

244

—

149

C₄²

—

150

C₄²

—

151

C₄²

—

244

—

152

C₄²

#11758200

yes

—

153

C₄²

#11758202

yes

—

154

C₄²

D₄×D₆₁

yes

characteristic

155

C₄²

#11758224

yes

characteristic

156

C₄²

#11758226

yes

—

157

C₂³

#14519760

yes

characteristic

158

Q₈

—

488

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

244

—

165

C₂×C₄

#14519748

yes

characteristic

166

C₂×C₄

#14519762

yes

—

167

C₂×C₄

#14519766

yes

characteristic

168

C₂×C₄

#14519772

yes

—

169

C₂×C₄

C₂×#11758200

yes

characteristic

170

C₂×C₄

C₂×#11758202

yes

characteristic

171

C₂×C₄

#14519803

yes

characteristic

172

C₂×C₄

#14519808

yes

characteristic

173

C₈

—

174

C₈

—

488

—

175

C₈

—

488

—

176

C₂²

—

177

C₂²

#16839688

yes

characteristic

178

C₂²

#16840092

yes

characteristic

179

C₂²

#16840097

yes

characteristic

180

C₄

—

181

C₄

—

182

C₄

—

183

C₄

—

184

C₄

—

488

—

185

C₄

#16839959

yes

characteristic

186

C₄

#16839969

yes

characteristic

187

C₄

#16840114

yes

characteristic

188

C₄

#16840124

yes

characteristic

189

C₂

—

190

C₂

—

191

C₂

#21447228

yes

characteristic

192

C₂

#21447270

yes

—

	subgroup	G/H
center	C₂×C₄	#14519766
commutator, cosocle	C₂×C₄²×C₆₁	C₂³
Frattini	C₂×C₄²	C₂²×D₆₁
Fitting	C₆₁×#1843	C₂
socle	C₂²×C₆₁	#397
prime radical	C₂³×C₆₁	#123
hypercenter	#1843	D₆₁
hypocenter	C₆₁	#10628
LCS	G ▷ C₂×C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1843

subgroup

G/H

center

C₂×C₄

#14519766

commutator, cosocle

C₂×C₄²×C₆₁

C₂³

Frattini

C₂×C₄²

C₂²×D₆₁

Fitting

C₆₁×#1843

C₂

C₂²×C₆₁

C₂³×C₆₁

D₆₁

C₆₁

G ▷ C₂×C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1843

N	H ≅ G/N	type
113. C₆₁	#10628	semidirect

H ≅ G/N

type

113. C₆₁

#10628

semidirect

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

subgroup	index
#21446800	2
#21444477	2
#21443716	2
#21443648	2
C₆₁×#1843	2
#10628	61

subgroup

index

C₆₁×#1843

Subgroups — #29584787

Conjugacy classes of subgroups

Complements

Maximal subgroups