Group #29580134 — subgroups

Subgroups — #29580134

Order 15616.

430 conjugacy classes of subgroups; only the 163 normal subgroups are shown.

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 163 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	#21454182	C₂	1	1	yes	maximal, characteristic
2	#21445936	C₂	1	1	yes	maximal, characteristic
3	#21443703	C₂	1	4	yes	maximal
4	C₄×C₄⋊C₈×C₆₁	C₂	1	1	yes	maximal, characteristic
5	C₂×#14519769	—	2	1	no	—
6	C₂×C₄⋊#92418	C₂²	1	2	yes	—
7	#16839660	—	2	2	no	—
8	#16839660	C₄	1	4	yes	—
9	#16839632	C₂²	1	2	yes	—
10	#16839628	—	2	1	no	—
11	#16839517	—	2	2	no	—
12	C₂×C₄⋊C₈×C₆₁	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₆₁	—	2	1	no	—
14	C₄³×C₆₁	C₂²	1	1	yes	characteristic
15	C₆₁×#336	—	2	1	no	—
16	C₂²×#92418	—	2	2	no	—
17	#14519769	—	4	4	no	—
18	C₂×C₄⋊Dic₆₁	—	2	2	no	—
19	C₂×C₄⋊Dic₆₁	—	2	2	no	—
20	C₂×C₄⋊Dic₆₁	C₂×C₄	1	4	yes	—
21	C₂×#11758196	—	2	4	no	—
22	C₄⋊#92418	—	2	4	no	—
23	#14519684	—	2	4	no	—
24	C₂²×C₈×C₆₁	—	2	2	no	—
25	C₂×C₄²×C₆₁	C₂×C₄	1	2	yes	—
26	C₂×C₄²×C₆₁	D₄	1	2	yes	—
27	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
28	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
29	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
30	C₄⋊C₈×C₆₁	—	2	2	no	—
31	C₄⋊C₈×C₆₁	Q₈	1	4	yes	—
32	C₄×C₈×C₆₁	—	4	2	no	—
33	C₂²×C₄×C₆₁	C₂²⋊C₄	1	2	yes	—
34	C₂²×C₄×C₆₁	C₂²⋊C₄	1	2	yes	—
35	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
36	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
37	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
38	C₂²×Dic₆₁	—	2	4	no	—
39	C₂×C₈×C₆₁	—	2	4	no	—
40	C₂×C₈×C₆₁	—	4	2	no	—
41	C₂×C₈×C₆₁	—	4	2	no	—
42	C₄²×C₆₁	—	2	4	no	—
43	C₄²×C₆₁	—	2	2	no	—
44	C₄²×C₆₁	—	4	2	no	—
45	C₄²×C₆₁	SD₁₆	1	2	yes	—
46	C₄²×C₆₁	Q₁₆	1	2	yes	—
47	C₄²×C₆₁	C₂×Q₈	1	2	yes	—
48	C₄²×C₆₁	#41	1	2	yes	—
49	C₄⋊Dic₆₁	—	4	4	no	—
50	C₄⋊Dic₆₁	—	4	4	no	—
51	C₄⋊Dic₆₁	—	4	4	no	—
52	#11758196	—	4	8	no	—
53	C₂×#92418	—	4	2	no	—
54	C₂×#92418	—	4	4	no	—
55	C₂³×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
56	C₂×C₄×C₆₁	—	2	2	no	—
57	C₂×C₄×C₆₁	—	2	2	no	—
58	C₂×C₄×C₆₁	—	2	2	no	—
59	C₂×C₄×C₆₁	—	2	2	no	—
60	C₂×C₄×C₆₁	—	2	2	no	—
61	C₂×C₄×C₆₁	—	2	2	no	—
62	C₂×C₄×C₆₁	#103	1	4	yes	—
63	C₂×C₄×C₆₁	C₄×Q₈	1	2	yes	—
64	C₂×C₄×C₆₁	C₂²⋊Q₈	1	2	yes	—
65	C₂×C₄×C₆₁	#124	1	1	yes	characteristic
66	C₂×C₄×C₆₁	C₄⋊Q₈	1	1	yes	characteristic
67	C₂×C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
68	C₂×C₄×C₆₁	C₂×Q₁₆	1	1	yes	characteristic
69	C₂×C₄×C₆₁	#135	1	4	yes	—
70	C₂×C₄×C₆₁	AGL(1,ℤ/8)	1	2	yes	—
71	C₂×Dic₆₁	—	4	4	no	—
72	C₂×Dic₆₁	—	4	4	no	—
73	C₂×Dic₆₁	—	4	4	no	—
74	C₈×C₆₁	—	4	4	no	—
75	#92418	—	8	2	no	—
76	#9897	—	61	1	no	maximal, Sylow
77	C₂²×C₆₁	#391	1	1	yes	characteristic
78	C₂²×C₆₁	#413	1	2	yes	—
79	C₂²×C₆₁	C₂×#103	1	1	yes	characteristic
80	C₂²×C₆₁	#416	1	2	yes	—
81	C₂²×C₆₁	#418	1	1	yes	characteristic
82	C₄×C₆₁	—	2	4	no	—
83	C₄×C₆₁	—	2	2	no	—
84	C₄×C₆₁	—	4	2	no	—
85	C₄×C₆₁	#476	1	2	yes	—
86	C₄×C₆₁	#477	1	2	yes	—
87	C₄×C₆₁	#488	1	2	yes	—
88	C₄×C₆₁	#499	1	2	yes	—
89	Dic₆₁	—	8	4	no	—
90	#2859	—	61	1	no	—
91	#1831	—	61	1	no	—
92	C₄×C₄⋊C₈	D₆₁	1	1	yes	characteristic
93	#1296	—	61	4	no	—
94	C₂×C₆₁	#1542	1	2	yes	—
95	C₂×C₆₁	#1553	1	2	yes	—
96	C₂×C₆₁	#1987	1	1	yes	characteristic
97	C₂×C₆₁	#1991	1	1	yes	characteristic
98	C₂×C₆₁	#1998	1	1	yes	characteristic
99	C₂×#125	—	122	1	no	—
100	C₂×C₄⋊C₈	—	61	2	no	—
101	C₂×C₄⋊C₈	C₂×D₆₁	1	2	yes	—
102	C₂×C₄×C₈	—	2	1	no	—
103	#389	—	61	4	no	—
104	#389	—	122	2	no	—
105	#384	—	61	2	no	—
106	#382	—	122	1	no	—
107	C₄³	C₂×D₆₁	1	1	yes	characteristic
108	#336	—	2	1	no	—
109	#336	—	122	2	no	—
110	C₆₁	#9897	1	1	yes	characteristic, complemented, Sylow
111	C₂²×C₈	—	2	2	no	—
112	C₂²×C₈	—	122	2	no	—
113	#125	—	244	4	no	—
114	C₂×C₄⋊C₄	—	61	4	no	—
115	C₂×C₄⋊C₄	—	122	4	no	—
116	C₂×C₄⋊C₄	—	122	2	no	—
117	C₂×C₄⋊C₄	—	122	2	no	—
118	C₂×C₄²	C₄×D₆₁	1	2	yes	—
119	C₂×C₄²	D₂₄₄	1	2	yes	—
120	C₂×C₄²	C₂²⋊D₆₁	1	1	yes	characteristic
121	C₂×C₄²	C₂²⋊D₆₁	1	1	yes	characteristic
122	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
123	C₄⋊C₈	—	2	2	no	—
124	C₄⋊C₈	—	122	4	no	—
125	C₄⋊C₈	Dic₁₂₂	1	4	yes	—
126	C₄×C₈	—	4	2	no	—
127	#95	—	122	4	no	—
128	C₂²×C₄	—	122	4	no	—
129	C₂²×C₄	#11758198	1	2	yes	—
130	C₂²×C₄	#11758198	1	2	yes	—
131	C₂²×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
132	C₂²×C₄	C₂×D₂₄₄	1	1	yes	characteristic
133	C₂²×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
134	C₂×C₈	—	2	4	no	—
135	C₂×C₈	—	4	2	no	—
136	C₂×C₈	—	4	2	no	—
137	C₂×C₈	—	244	2	no	—
138	C₂×C₈	—	244	4	no	—
139	C₄⋊C₄	—	244	4	no	—
140	C₄⋊C₄	—	244	8	no	—
141	C₄⋊C₄	—	244	4	no	—
142	C₄⋊C₄	—	244	4	no	—
143	C₄²	—	2	4	no	—
144	C₄²	—	2	2	no	—
145	C₄²	—	4	2	no	—
146	C₄²	#11758200	1	2	yes	—
147	C₄²	#11758202	1	2	yes	—
148	C₄²	C₂×Dic₁₂₂	1	2	yes	—
149	C₄²	#11758222	1	2	yes	—
150	C₂³	C₂×#11758198	1	1	yes	characteristic
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	244	4	no	—
158	C₂×C₄	—	244	4	no	—
159	C₂×C₄	—	244	4	no	—
160	C₂×C₄	#14519700	1	4	yes	—
161	C₂×C₄	C₄×Dic₁₂₂	1	2	yes	—
162	C₂×C₄	#14519756	1	2	yes	—
163	C₂×C₄	#14519765	1	1	yes	characteristic
164	C₂×C₄	#14519777	1	1	yes	characteristic
165	C₂×C₄	C₂×#11758200	1	1	yes	characteristic
166	C₂×C₄	C₂×#11758202	1	1	yes	characteristic
167	C₂×C₄	#14519799	1	4	yes	—
168	C₂×C₄	#14519802	1	2	yes	—
169	C₈	—	4	4	no	—
170	C₈	—	488	2	no	—
171	C₂²	#16839668	1	1	yes	characteristic
172	C₂²	#16839744	1	2	yes	—
173	C₂²	C₂×#14519700	1	1	yes	characteristic
174	C₂²	#16839759	1	2	yes	—
175	C₂²	#16839766	1	1	yes	characteristic
176	C₄	—	2	4	no	—
177	C₄	—	2	2	no	—
178	C₄	—	4	2	no	—
179	C₄	—	488	4	no	—
180	C₄	#16840052	1	2	yes	—
181	C₄	#16840057	1	2	yes	—
182	C₄	#16840117	1	2	yes	—
183	C₄	#16840164	1	2	yes	—
184	C₂	#21444477	1	2	yes	—
185	C₂	#21444522	1	2	yes	—
186	C₂	#21446767	1	1	yes	characteristic
187	C₂	#21446784	1	1	yes	characteristic
188	C₂	#21446812	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21454182

C₂

yes

maximal, characteristic

#21445936

C₂

yes

maximal, characteristic

#21443703

C₂

yes

maximal

C₄×C₄⋊C₈×C₆₁

C₂

yes

maximal, characteristic

C₂×#14519769

—

C₂×C₄⋊#92418

C₂²

yes

—

#16839660

—

#16839660

C₄

yes

—

#16839632

C₂²

yes

—

#16839628

—

#16839517

—

C₂×C₄⋊C₈×C₆₁

C₂²

yes

—

C₂×C₄×C₈×C₆₁

—

C₄³×C₆₁

C₂²

yes

characteristic

C₆₁×#336

—

C₂²×#92418

—

#14519769

—

C₂×C₄⋊Dic₆₁

—

C₂×C₄⋊Dic₆₁

—

C₂×C₄⋊Dic₆₁

C₂×C₄

yes

—

C₂×#11758196

—

C₄⋊#92418

—

#14519684

—

C₂²×C₈×C₆₁

—

C₂×C₄²×C₆₁

C₂×C₄

yes

—

C₂×C₄²×C₆₁

D₄

yes

—

C₂×C₄²×C₆₁

D₄

yes

characteristic

C₂×C₄²×C₆₁

D₄

yes

characteristic

C₂×C₄²×C₆₁

C₂³

yes

characteristic

C₄⋊C₈×C₆₁

—

C₄⋊C₈×C₆₁

Q₈

yes

—

C₄×C₈×C₆₁

—

C₂²×C₄×C₆₁

C₂²⋊C₄

yes

—

C₂²×C₄×C₆₁

C₂²⋊C₄

yes

—

C₂²×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₂²×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂²×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂²×Dic₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₄²×C₆₁

—

C₄²×C₆₁

—

C₄²×C₆₁

—

C₄²×C₆₁

SD₁₆

yes

—

C₄²×C₆₁

Q₁₆

yes

—

C₄²×C₆₁

C₂×Q₈

yes

—

C₄²×C₆₁

#41

yes

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

#11758196

—

C₂×#92418

—

C₂×#92418

—

C₂³×C₆₁

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

#103

yes

—

C₂×C₄×C₆₁

C₄×Q₈

yes

—

C₂×C₄×C₆₁

C₂²⋊Q₈

yes

—

C₂×C₄×C₆₁

#124

yes

characteristic

C₂×C₄×C₆₁

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₆₁

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₆₁

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₆₁

#135

yes

—

C₂×C₄×C₆₁

AGL(1,ℤ/8)

yes

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₈×C₆₁

—

—

—

C₂²×C₆₁

yes

C₂²×C₆₁

yes

—

C₂²×C₆₁

C₂×#103

yes

characteristic

C₂²×C₆₁

#416

yes

—

C₂²×C₆₁

#418

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

#476

yes

—

C₄×C₆₁

#477

yes

—

C₄×C₆₁

#488

yes

—

C₄×C₆₁

#499

yes

—

Dic₆₁

—

#2859

—

#1831

—

C₄×C₄⋊C₈

D₆₁

yes

characteristic

#1296

—

C₂×C₆₁

#1542

yes

—

C₂×C₆₁

#1553

yes

—

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

—

122

—

100

C₂×C₄⋊C₈

—

101

C₂×C₄⋊C₈

C₂×D₆₁

yes

—

102

C₂×C₄×C₈

—

103

#389

—

104

#389

—

122

—

105

#384

—

106

#382

—

122

—

107

C₄³

C₂×D₆₁

yes

characteristic

108

#336

—

109

#336

—

122

—

110

C₆₁

#9897

yes

characteristic, complemented, Sylow

111

C₂²×C₈

—

112

C₂²×C₈

—

122

—

113

#125

—

244

—

114

C₂×C₄⋊C₄

—

115

C₂×C₄⋊C₄

—

122

—

116

C₂×C₄⋊C₄

—

122

—

117

C₂×C₄⋊C₄

—

122

—

118

C₂×C₄²

C₄×D₆₁

yes

—

119

C₂×C₄²

D₂₄₄

yes

—

120

C₂×C₄²

C₂²⋊D₆₁

yes

characteristic

121

C₂×C₄²

C₂²⋊D₆₁

yes

characteristic

122

C₂×C₄²

C₂²×D₆₁

yes

characteristic

123

C₄⋊C₈

—

124

C₄⋊C₈

—

122

—

125

C₄⋊C₈

Dic₁₂₂

yes

—

126

C₄×C₈

—

127

#95

—

122

—

128

C₂²×C₄

—

122

—

129

C₂²×C₄

#11758198

yes

—

130

C₂²×C₄

#11758198

yes

—

131

C₂²×C₄

C₂×C₄×D₆₁

yes

characteristic

132

C₂²×C₄

C₂×D₂₄₄

yes

characteristic

133

C₂²×C₄

C₂×C₂²⋊D₆₁

yes

characteristic

134

C₂×C₈

—

135

C₂×C₈

—

136

C₂×C₈

—

137

C₂×C₈

—

244

—

138

C₂×C₈

—

244

—

139

C₄⋊C₄

—

244

—

140

C₄⋊C₄

—

244

—

141

C₄⋊C₄

—

244

—

142

C₄⋊C₄

—

244

—

143

C₄²

—

144

C₄²

—

145

C₄²

—

146

C₄²

#11758200

yes

—

147

C₄²

#11758202

yes

—

148

C₄²

C₂×Dic₁₂₂

yes

—

149

C₄²

#11758222

yes

—

150

C₂³

C₂×#11758198

yes

characteristic

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

244

—

158

C₂×C₄

—

244

—

159

C₂×C₄

—

244

—

160

C₂×C₄

#14519700

yes

—

161

C₂×C₄

C₄×Dic₁₂₂

yes

—

162

C₂×C₄

#14519756

yes

—

163

C₂×C₄

#14519765

yes

characteristic

164

C₂×C₄

#14519777

yes

characteristic

165

C₂×C₄

C₂×#11758200

yes

characteristic

166

C₂×C₄

C₂×#11758202

yes

characteristic

167

C₂×C₄

#14519799

yes

—

168

C₂×C₄

#14519802

yes

—

169

C₈

—

170

C₈

—

488

—

171

C₂²

#16839668

yes

characteristic

172

C₂²

#16839744

yes

—

173

C₂²

C₂×#14519700

yes

characteristic

174

C₂²

#16839759

yes

—

175

C₂²

#16839766

yes

characteristic

176

C₄

—

177

C₄

—

178

C₄

—

179

C₄

—

488

—

180

C₄

#16840052

yes

—

181

C₄

#16840057

yes

—

182

C₄

#16840117

yes

—

183

C₄

#16840164

yes

—

184

C₂

#21444477

yes

—

185

C₂

#21444522

yes

—

186

C₂

#21446767

yes

characteristic

187

C₂

#21446784

yes

characteristic

188

C₂

#21446812

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×#11758198
commutator	C₂²×C₄×C₆₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₁
Fitting	C₄×C₄⋊C₈×C₆₁	C₂
socle, prime radical	C₂³×C₆₁	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₆₁	C₂³
hypercenter	C₄×C₄⋊C₈	D₆₁
hypocenter	C₆₁	#9897
LCS	G ▷ C₂²×C₄×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×C₄⋊C₈

subgroup

G/H

center

C₂³

C₂×#11758198

commutator

C₂²×C₄×C₆₁

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₆₁

Fitting

C₄×C₄⋊C₈×C₆₁

C₂

socle, prime radical

C₂³×C₆₁

C₂×C₂²⋊C₄

cosocle

C₂×C₄²×C₆₁

C₂³

hypercenter

C₄×C₄⋊C₈

D₆₁

hypocenter

C₆₁

#9897

LCS

G ▷ C₂²×C₄×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×C₄⋊C₈

N	H ≅ G/N	type
110. C₆₁	#9897	semidirect

H ≅ G/N

type

110. C₆₁

#9897

semidirect

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

subgroup	index
#21454182	2
#21445936	2
#21443703	2
C₄×C₄⋊C₈×C₆₁	2
#9897	61

subgroup

index

C₄×C₄⋊C₈×C₆₁

Subgroups — #29580134

Conjugacy classes of subgroups

Complements

Maximal subgroups