Group #29599661 — subgroups

Subgroups — #29599661

Order 15616.

356 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 103 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×#16840160	C₂	1	1	yes	maximal, characteristic
2	C₂×#16839586	C₂	1	2	yes	maximal
3	#21443924	C₂	1	2	yes	maximal
4	#21443901	C₂	1	1	yes	maximal, characteristic
5	C₆₁×#1945	C₂	1	1	yes	maximal, characteristic
6	C₂×#14519777	—	2	1	no	—
7	C₂×#14519717	C₂²	1	2	yes	—
8	#16840160	—	2	4	no	—
9	C₂×#14519709	—	2	1	no	—
10	C₂×#14519709	C₂²	1	2	yes	—
11	#16839586	—	2	4	no	—
12	C₂×#108×C₆₁	C₂²	1	1	yes	characteristic
13	C₆₁×#427	C₂²	1	1	yes	characteristic
14	C₆₁×#424	—	2	1	no	—
15	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
16	C₆₁×#339	—	2	1	no	—
17	C₂²×Dic₁₂₂	—	4	1	no	—
18	#14519777	—	2	4	no	—
19	#14519777	—	4	2	no	—
20	C₂×C₄⋊Dic₆₁	—	2	2	no	—
21	C₂×C₄⋊Dic₆₁	—	4	1	no	—
22	#14519717	—	2	2	no	—
23	#14519717	Q₈	1	4	yes	complemented
24	C₂×#11758185	—	2	2	no	—
25	#14519709	—	2	4	no	—
26	#14519709	—	2	2	no	—
27	#14519709	D₄	1	4	yes	—
28	C₂×M₁₆×C₆₁	—	2	1	no	—
29	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
30	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
31	#117×C₆₁	—	2	1	no	—
32	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
33	#108×C₆₁	—	2	2	no	—
34	#107×C₆₁	Q₈	1	2	yes	complemented
35	#106×C₆₁	—	2	1	no	—
36	C₄⋊C₈×C₆₁	—	4	2	no	—
37	C₄×C₈×C₆₁	—	2	2	no	—
38	C₂²×C₄×C₆₁	—	2	1	no	—
39	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
40	C₂²×Dic₆₁	—	4	2	no	—
41	C₂×Dic₁₂₂	—	4	4	no	—
42	C₂×Dic₁₂₂	—	8	4	no	—
43	M₁₆×C₆₁	—	4	1	no	—
44	M₁₆×C₆₁	—	4	1	no	—
45	C₂×C₈×C₆₁	—	2	1	no	—
46	C₂×C₈×C₆₁	—	2	1	no	—
47	C₂×C₈×C₆₁	—	4	1	no	—
48	C₂×C₈×C₆₁	SD₁₆	1	2	yes	—
49	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
50	C₂×C₈×C₆₁	C₂×Q₈	1	2	yes	—
51	C₂×C₈×C₆₁	C₂×Q₈	1	1	yes	characteristic
52	C₂×C₈×C₆₁	#41	1	1	yes	characteristic
53	C₄⋊C₄×C₆₁	—	2	2	no	—
54	C₂²⋊C₄×C₆₁	—	4	1	no	—
55	C₄²×C₆₁	—	2	1	no	—
56	C₄²×C₆₁	—	4	1	no	—
57	C₄²×C₆₁	SD₁₆	1	2	yes	—
58	C₄⋊Dic₆₁	—	2	4	no	—
59	C₄⋊Dic₆₁	—	4	4	no	—
60	C₄⋊Dic₆₁	—	4	2	no	—
61	#11758185	—	2	4	no	—
62	C₂³×C₆₁	AGL(1,ℤ/8)	1	1	yes	characteristic
63	C₂×C₄×C₆₁	—	2	4	no	—
64	C₂×C₄×C₆₁	—	2	2	no	—
65	C₂×C₄×C₆₁	—	4	2	no	—
66	C₂×C₄×C₆₁	#120	1	1	yes	characteristic
67	C₂×C₄×C₆₁	C₂²⋊Q₈	1	1	yes	characteristic
68	C₂×C₄×C₆₁	C₄⋊Q₈	1	1	yes	characteristic
69	C₂×C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
70	C₂×C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
71	C₂×C₄×C₆₁	#137	1	1	yes	characteristic
72	C₂×Dic₆₁	—	4	4	no	—
73	C₂×Dic₆₁	—	8	4	no	—
74	Dic₁₂₂	—	8	8	no	—
75	C₈×C₆₁	—	2	2	no	—
76	C₈×C₆₁	—	8	1	no	—
77	C₈×C₆₁	C₂²⋊Q₈	1	2	yes	—
78	C₈×C₆₁	C₄⋊Q₈	1	2	yes	—
79	#13457	—	61	1	no	maximal, Sylow
80	C₂²×C₆₁	—	2	2	no	—
81	C₂²×C₆₁	#450	1	1	yes	characteristic
82	C₂²×C₆₁	#476	1	1	yes	characteristic
83	C₂²×C₆₁	#501	1	1	yes	characteristic
84	C₄×C₆₁	—	2	4	no	—
85	C₄×C₆₁	—	8	2	no	—
86	C₄×C₆₁	#395	1	1	yes	characteristic
87	C₄×C₆₁	#448	1	1	yes	characteristic
88	C₄×C₆₁	#475	1	1	yes	characteristic
89	C₄×C₆₁	#498	1	1	yes	characteristic
90	Dic₆₁	—	8	4	no	—
91	C₂×#498	—	61	1	no	—
92	C₂×#365	—	61	2	no	—
93	#1945	D₆₁	1	1	yes	characteristic
94	#1386	—	61	2	no	—
95	#1376	—	61	1	no	—
96	C₂×C₆₁	—	2	1	no	—
97	C₂×C₆₁	—	2	1	no	—
98	C₂×C₆₁	#2058	1	1	yes	characteristic
99	C₂×C₆₁	#2230	1	2	yes	—
100	C₂×C₄⋊Q₈	—	122	1	no	—
101	C₂×M₃₂	—	61	2	no	—
102	#498	—	122	4	no	—
103	C₂×#108	C₂×D₆₁	1	1	yes	characteristic
104	#427	C₂×D₆₁	1	1	yes	characteristic
105	C₂×#106	—	61	2	no	—
106	C₂×#106	—	122	1	no	—
107	#424	—	2	1	no	—
108	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
109	#365	—	122	4	no	—
110	#339	—	2	1	no	—
111	C₆₁	#13457	1	1	yes	characteristic, complemented, Sylow
112	C₂²×Q₈	—	244	1	no	—
113	C₂×M₁₆	—	2	1	no	—
114	C₂²×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
115	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
116	C₄⋊Q₈	—	122	4	no	—
117	C₄⋊Q₈	—	244	2	no	—
118	#117	—	2	1	no	—
119	C₂×C₄⋊C₄	—	122	2	no	—
120	C₂×C₄⋊C₄	—	244	1	no	—
121	C₂×C₄²	C₂²⋊D₆₁	1	1	yes	characteristic
122	M₃₂	—	61	4	no	—
123	M₃₂	—	122	2	no	—
124	C₂×C₁₆	—	122	2	no	—
125	#108	—	2	2	no	—
126	#107	Dic₁₂₂	1	2	yes	complemented
127	#106	—	2	1	no	—
128	#106	—	61	4	no	—
129	#106	—	122	4	no	—
130	#106	—	122	2	no	—
131	C₄⋊C₈	—	4	2	no	—
132	C₄×C₈	—	2	2	no	—
133	C₂×Q₈	—	488	4	no	—
134	C₂×Q₈	—	244	4	no	—
135	C₂²×C₄	—	2	1	no	—
136	C₂²×C₄	—	244	2	no	—
137	C₂²×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
138	M₁₆	—	4	1	no	—
139	M₁₆	—	4	1	no	—
140	C₂×C₈	—	2	1	no	—
141	C₂×C₈	—	2	1	no	—
142	C₂×C₈	—	4	1	no	—
143	C₂×C₈	#11758200	1	2	yes	—
144	C₂×C₈	C₂×Dic₁₂₂	1	1	yes	characteristic
145	C₂×C₈	#11758222	1	1	yes	characteristic
146	C₂×C₈	D₄×D₆₁	1	2	yes	—
147	C₂×C₈	Q₈×D₆₁	1	2	yes	—
148	C₄⋊C₄	—	2	2	no	—
149	C₄⋊C₄	—	122	4	no	—
150	C₄⋊C₄	—	244	4	no	—
151	C₄⋊C₄	—	244	2	no	—
152	C₂²⋊C₄	—	4	1	no	—
153	C₄²	—	2	1	no	—
154	C₄²	—	4	1	no	—
155	C₄²	#11758201	1	2	yes	—
156	C₁₆	—	122	4	no	—
157	C₂³	#14519802	1	1	yes	characteristic
158	Q₈	—	488	8	no	—
159	C₂×C₄	—	2	4	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	4	2	no	—
162	C₂×C₄	—	488	4	no	—
163	C₂×C₄	—	244	4	no	—
164	C₂×C₄	#14519745	1	1	yes	characteristic
165	C₂×C₄	#14519756	1	1	yes	characteristic
166	C₂×C₄	#14519775	1	1	yes	characteristic
167	C₂×C₄	C₂×#11758201	1	1	yes	characteristic
168	C₂×C₄	C₂×#11758200	1	1	yes	characteristic
169	C₂×C₄	#14519807	1	1	yes	characteristic
170	C₈	—	2	2	no	—
171	C₈	—	8	1	no	—
172	C₈	#14519755	1	2	yes	—
173	C₈	#14519776	1	2	yes	—
174	C₂²	—	2	2	no	—
175	C₂²	#16839912	1	1	yes	characteristic
176	C₂²	#16840052	1	1	yes	characteristic
177	C₂²	#16840172	1	1	yes	characteristic
178	C₄	—	2	4	no	—
179	C₄	—	8	2	no	—
180	C₄	—	488	4	no	—
181	C₄	#16839682	1	1	yes	characteristic
182	C₄	#16839900	1	1	yes	characteristic
183	C₄	#16840047	1	1	yes	characteristic
184	C₄	#16840158	1	1	yes	characteristic
185	C₂	—	2	1	no	—
186	C₂	—	2	1	no	—
187	C₂	#21447152	1	1	yes	characteristic
188	C₂	#21447990	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16840160

C₂

yes

maximal, characteristic

C₂×#16839586

C₂

yes

maximal

#21443924

C₂

yes

maximal

#21443901

C₂

yes

maximal, characteristic

C₆₁×#1945

C₂

yes

maximal, characteristic

C₂×#14519777

—

C₂×#14519717

C₂²

yes

—

#16840160

—

C₂×#14519709

—

C₂×#14519709

C₂²

yes

—

#16839586

—

C₂×#108×C₆₁

C₂²

yes

characteristic

C₆₁×#427

C₂²

yes

characteristic

C₆₁×#424

—

C₂×C₄×C₈×C₆₁

C₂²

yes

characteristic

C₆₁×#339

—

C₂²×Dic₁₂₂

—

#14519777

—

#14519777

—

C₂×C₄⋊Dic₆₁

—

C₂×C₄⋊Dic₆₁

—

#14519717

—

#14519717

Q₈

yes

—

—

—

D₄

yes

—

C₂×M₁₆×C₆₁

—

C₂²×C₈×C₆₁

D₄

yes

characteristic

C₂²×C₈×C₆₁

C₂³

yes

characteristic

#117×C₆₁

—

C₂×C₄²×C₆₁

D₄

yes

characteristic

#108×C₆₁

—

#107×C₆₁

Q₈

yes

complemented

#106×C₆₁

—

C₄⋊C₈×C₆₁

—

C₄×C₈×C₆₁

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂²×Dic₆₁

—

C₂×Dic₁₂₂

—

C₂×Dic₁₂₂

—

M₁₆×C₆₁

—

M₁₆×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

SD₁₆

yes

—

C₂×C₈×C₆₁

C₂×D₄

yes

—

C₂×C₈×C₆₁

C₂×Q₈

yes

—

C₂×C₈×C₆₁

C₂×Q₈

yes

characteristic

C₂×C₈×C₆₁

#41

yes

characteristic

C₄⋊C₄×C₆₁

—

C₂²⋊C₄×C₆₁

—

C₄²×C₆₁

—

C₄²×C₆₁

—

C₄²×C₆₁

SD₁₆

yes

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

#11758185

—

C₂³×C₆₁

AGL(1,ℤ/8)

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

#120

yes

characteristic

C₂×C₄×C₆₁

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₆₁

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₆₁

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₆₁

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₆₁

#137

yes

characteristic

C₂×Dic₆₁

—

C₂×Dic₆₁

—

Dic₁₂₂

—

C₈×C₆₁

—

C₈×C₆₁

—

C₈×C₆₁

C₂²⋊Q₈

yes

—

C₈×C₆₁

C₄⋊Q₈

yes

—

#13457

—

maximal, Sylow

C₂²×C₆₁

—

C₂²×C₆₁

yes

C₂²×C₆₁

yes

C₂²×C₆₁

yes

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

yes

C₄×C₆₁

yes

C₄×C₆₁

yes

C₄×C₆₁

yes

Dic₆₁

—

C₂×#498

—

C₂×#365

—

#1945

D₆₁

yes

characteristic

#1386

—

#1376

—

C₂×C₆₁

—

C₂×C₆₁

—

C₂×C₆₁

#2058

yes

characteristic

C₂×C₆₁

#2230

yes

—

100

C₂×C₄⋊Q₈

—

122

—

101

C₂×M₃₂

—

102

#498

—

122

—

103

C₂×#108

C₂×D₆₁

yes

characteristic

104

#427

C₂×D₆₁

yes

characteristic

105

C₂×#106

—

106

C₂×#106

—

122

—

107

#424

—

108

C₂×C₄×C₈

C₂×D₆₁

yes

characteristic

109

#365

—

122

—

110

#339

—

111

C₆₁

#13457

yes

characteristic, complemented, Sylow

112

C₂²×Q₈

—

244

—

113

C₂×M₁₆

—

114

C₂²×C₈

C₂²⋊D₆₁

yes

characteristic

115

C₂²×C₈

C₂²×D₆₁

yes

characteristic

116

C₄⋊Q₈

—

122

—

117

C₄⋊Q₈

—

244

—

118

#117

—

119

C₂×C₄⋊C₄

—

122

—

120

C₂×C₄⋊C₄

—

244

—

121

C₂×C₄²

C₂²⋊D₆₁

yes

characteristic

122

M₃₂

—

123

M₃₂

—

122

—

124

C₂×C₁₆

—

122

—

125

#108

—

126

#107

Dic₁₂₂

yes

complemented

127

#106

—

128

#106

—

129

#106

—

122

—

130

#106

—

122

—

131

C₄⋊C₈

—

132

C₄×C₈

—

133

C₂×Q₈

—

488

—

134

C₂×Q₈

—

244

—

135

C₂²×C₄

—

136

C₂²×C₄

—

244

—

137

C₂²×C₄

C₂×C₂²⋊D₆₁

yes

characteristic

138

M₁₆

—

139

M₁₆

—

140

C₂×C₈

—

141

C₂×C₈

—

142

C₂×C₈

—

143

C₂×C₈

#11758200

yes

—

144

C₂×C₈

C₂×Dic₁₂₂

yes

characteristic

145

C₂×C₈

#11758222

yes

characteristic

146

C₂×C₈

D₄×D₆₁

yes

—

147

C₂×C₈

Q₈×D₆₁

yes

—

148

C₄⋊C₄

—

149

C₄⋊C₄

—

122

—

150

C₄⋊C₄

—

244

—

151

C₄⋊C₄

—

244

—

152

C₂²⋊C₄

—

153

C₄²

—

154

C₄²

—

155

C₄²

#11758201

yes

—

156

C₁₆

—

122

—

157

C₂³

#14519802

yes

characteristic

158

Q₈

—

488

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

488

—

163

C₂×C₄

—

244

—

164

C₂×C₄

#14519745

yes

characteristic

165

C₂×C₄

#14519756

yes

characteristic

166

C₂×C₄

#14519775

yes

characteristic

167

C₂×C₄

C₂×#11758201

yes

characteristic

168

C₂×C₄

C₂×#11758200

yes

characteristic

169

C₂×C₄

#14519807

yes

characteristic

170

C₈

—

171

C₈

—

172

C₈

#14519755

yes

—

173

C₈

#14519776

yes

—

174

C₂²

—

175

C₂²

#16839912

yes

characteristic

176

C₂²

#16840052

yes

characteristic

177

C₂²

#16840172

yes

characteristic

178

C₄

—

179

C₄

—

180

C₄

—

488

—

181

C₄

#16839682

yes

characteristic

182

C₄

#16839900

yes

characteristic

183

C₄

#16840047

yes

characteristic

184

C₄

#16840158

yes

characteristic

185

C₂

—

186

C₂

—

187

C₂

#21447152

yes

characteristic

188

C₂

#21447990

yes

—

	subgroup	G/H
center	C₂²	#16840052
commutator, cosocle	C₂²×C₈×C₆₁	C₂³
Frattini	C₂²×C₈	C₂²×D₆₁
Fitting	C₆₁×#1945	C₂
socle	C₂²×C₆₁	#476
prime radical	C₂³×C₆₁	AGL(1,ℤ/8)
hypercenter	#1945	D₆₁
hypocenter	C₆₁	#13457
LCS	G ▷ C₂²×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1945

subgroup

G/H

center

C₂²

#16840052

commutator, cosocle

C₂²×C₈×C₆₁

C₂³

Frattini

C₂²×C₈

C₂²×D₆₁

Fitting

C₆₁×#1945

C₂

C₂²×C₆₁

C₂³×C₆₁

D₆₁

C₆₁

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

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1945

N	H ≅ G/N	type
23. #14519717	Q₈	semidirect
34. #107×C₆₁	Q₈	semidirect
111. C₆₁	#13457	semidirect
126. #107	Dic₁₂₂	semidirect

H ≅ G/N

type

23. #14519717

Q₈

semidirect

34. #107×C₆₁

Q₈

semidirect

111. C₆₁

#13457

semidirect

126. #107

Dic₁₂₂

semidirect

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

subgroup	index
C₂×#16840160	2
C₂×#16839586	2
#21443924	2
#21443901	2
C₆₁×#1945	2
#13457	61

subgroup

index

C₆₁×#1945

Subgroups — #29599661

Conjugacy classes of subgroups

Complements

Maximal subgroups