Group #29599566 — subgroups

Subgroups — #29599566

Order 15616.

378 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 91 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×Dic₄₈₈	C₂	1	1	yes	maximal, characteristic
2	#21447577	C₂	1	1	yes	maximal, characteristic
3	C₂×#16839577	C₂	1	2	yes	maximal
4	#21446732	C₂	1	2	yes	maximal
5	C₆₁×#1381	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₄₄	C₂²	1	2	yes	—
7	C₂×Dic₄₈₈	—	2	4	no	—
8	C₂×Dic₄₈₈	—	2	2	no	—
9	C₂×#14519717	C₂²	1	1	yes	characteristic
10	#16839772	—	2	2	no	—
11	C₂×#14519695	—	2	2	no	—
12	#16839717	C₂²	1	1	yes	characteristic
13	#16839580	—	2	2	no	—
14	#16839577	—	2	4	no	—
15	C₂²×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
16	C₂×#108×C₆₁	C₂²	1	2	yes	—
17	C₂²×Dic₁₂₂	—	2	2	no	—
18	C₂×Dic₂₄₄	—	2	2	no	—
19	C₂×Dic₂₄₄	—	2	2	no	—
20	C₂×Dic₂₄₄	—	2	2	no	—
21	C₂×Dic₂₄₄	—	2	2	no	—
22	C₂×Dic₂₄₄	—	2	2	no	—
23	C₂×Dic₂₄₄	D₄	1	4	yes	—
24	C₂×#11758194	D₄	1	1	yes	characteristic
25	#14519736	D₄	1	1	yes	characteristic
26	Dic₄₈₈	—	2	8	no	—
27	#14519717	—	2	2	no	—
28	C₂×#11758185	—	2	1	no	—
29	#14519702	—	4	4	no	—
30	#14519695	—	4	4	no	—
31	#14519687	—	2	1	no	—
32	C₈×Dic₆₁	—	2	1	no	—
33	C₂×M₁₆×C₆₁	—	2	2	no	—
34	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
35	C₂×C₁₆×C₆₁	—	2	2	no	—
36	C₂×C₁₆×C₆₁	D₄	1	2	yes	—
37	#108×C₆₁	—	2	2	no	—
38	#108×C₆₁	—	2	2	no	—
39	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
40	C₂²×Dic₆₁	—	4	2	no	—
41	C₂×Dic₁₂₂	—	2	4	no	—
42	C₂×Dic₁₂₂	—	4	2	no	—
43	C₂×Dic₁₂₂	—	4	2	no	—
44	C₂×Dic₁₂₂	—	4	2	no	—
45	C₂×Dic₁₂₂	—	4	2	no	—
46	C₂×Dic₁₂₂	—	4	2	no	—
47	M₁₆×C₆₁	—	4	2	no	—
48	M₁₆×C₆₁	—	4	2	no	—
49	C₂×C₈×C₆₁	—	4	2	no	—
50	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
51	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
52	C₂×C₈×C₆₁	#41	1	1	yes	characteristic
53	C₂×C₈×C₆₁	#41	1	2	yes	—
54	C₂²⋊Dic₆₁	—	2	1	no	—
55	C₄⋊Dic₆₁	D₈	1	2	yes	—
56	C₄×Dic₆₁	—	2	1	no	—
57	#11758194	—	2	2	no	—
58	C₂×#92418	—	2	1	no	—
59	Dic₂₄₄	—	2	4	no	—
60	Dic₂₄₄	—	2	4	no	—
61	Dic₂₄₄	—	4	2	no	—
62	Dic₂₄₄	—	4	2	no	—
63	C₁₆×C₆₁	—	2	2	no	—
64	#11758185	—	4	1	no	—
65	C₂³×C₆₁	#135	1	1	yes	characteristic
66	C₂×C₄×C₆₁	#121	1	2	yes	—
67	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
68	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
69	C₂×C₄×C₆₁	AGL(1,ℤ/8)	1	2	yes	—
70	C₂×Dic₆₁	—	2	2	no	—
71	C₂×Dic₆₁	—	4	4	no	—
72	C₂×Dic₆₁	—	8	2	no	—
73	C₂×Dic₆₁	—	8	2	no	—
74	Dic₁₂₂	—	4	4	no	—
75	Dic₁₂₂	—	4	4	no	—
76	Dic₁₂₂	—	8	2	no	—
77	Dic₁₂₂	—	8	2	no	—
78	C₈×C₆₁	—	8	2	no	—
79	C₈×C₆₁	#120	1	1	yes	characteristic
80	C₈×C₆₁	#123	1	2	yes	—
81	C₈×C₆₁	#124	1	1	yes	characteristic
82	#92418	—	4	1	no	—
83	#13441	—	61	1	no	maximal, Sylow
84	C₂²×C₆₁	—	2	2	no	—
85	C₂²×C₆₁	#463	1	2	yes	—
86	C₂²×C₆₁	#466	1	1	yes	characteristic
87	C₄×C₆₁	#394	1	1	yes	characteristic
88	C₄×C₆₁	#459	1	2	yes	—
89	C₄×C₆₁	#469	1	1	yes	characteristic
90	Dic₆₁	—	4	2	no	—
91	Dic₆₁	—	8	4	no	—
92	C₂²×Q₃₂	—	61	1	no	—
93	#2151	—	61	1	no	—
94	C₂×#362	—	61	2	no	—
95	#1979	—	61	2	no	—
96	#1381	D₆₁	1	1	yes	characteristic
97	C₂×C₆₁	—	2	2	no	—
98	C₂×C₆₁	#2054	1	1	yes	characteristic
99	C₂×C₆₁	#2222	1	2	yes	—
100	C₂²×Q₁₆	—	61	2	no	—
101	C₂×Q₃₂	—	122	4	no	—
102	C₂×Q₃₂	—	122	2	no	—
103	C₂×M₃₂	—	61	1	no	—
104	C₂²×C₁₆	C₂×D₆₁	1	1	yes	characteristic
105	C₂×#108	C₂×D₆₁	1	2	yes	—
106	#419	—	122	2	no	—
107	C₂×#101	—	122	2	no	—
108	#405	—	61	1	no	—
109	C₄⋊C₁₆	—	122	2	no	—
110	#362	—	122	4	no	—
111	C₆₁	#13441	1	1	yes	characteristic, complemented, Sylow
112	C₂²×Q₈	—	122	2	no	—
113	C₂×Q₁₆	—	61	4	no	—
114	C₂×Q₁₆	—	122	2	no	—
115	C₂×Q₁₆	—	122	2	no	—
116	C₂×Q₁₆	—	122	2	no	—
117	C₂×Q₁₆	—	122	2	no	—
118	C₂×Q₁₆	—	122	2	no	—
119	C₂×M₁₆	—	2	2	no	—
120	C₂×M₁₆	—	61	1	no	—
121	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
122	#117	—	61	1	no	—
123	Q₃₂	—	122	8	no	—
124	M₃₂	—	122	2	no	—
125	C₂×C₁₆	—	2	2	no	—
126	C₂×C₁₆	—	122	1	no	—
127	C₂×C₁₆	D₂₄₄	1	2	yes	—
128	#108	—	2	2	no	—
129	#108	—	2	2	no	—
130	#103	—	244	4	no	—
131	#101	—	244	4	no	—
132	#97	—	122	1	no	—
133	C₄×C₈	—	122	1	no	—
134	C₂×Q₈	—	122	4	no	—
135	C₂×Q₈	—	244	2	no	—
136	C₂×Q₈	—	244	2	no	—
137	C₂×Q₈	—	244	2	no	—
138	C₂×Q₈	—	244	2	no	—
139	C₂×Q₈	—	244	2	no	—
140	C₂²×C₄	—	244	2	no	—
141	C₂²×C₄	D₄×D₆₁	1	1	yes	characteristic
142	Q₁₆	—	122	4	no	—
143	Q₁₆	—	122	4	no	—
144	Q₁₆	—	244	2	no	—
145	Q₁₆	—	244	2	no	—
146	M₁₆	—	4	2	no	—
147	M₁₆	—	4	2	no	—
148	M₁₆	—	122	2	no	—
149	C₂×C₈	—	4	2	no	—
150	C₂×C₈	—	122	1	no	—
151	C₂×C₈	C₂×D₂₄₄	1	1	yes	characteristic
152	C₂×C₈	#11758222	1	2	yes	—
153	C₂×C₈	D₄×D₆₁	1	2	yes	—
154	C₂×C₈	#11758226	1	1	yes	characteristic
155	C₄⋊C₄	—	61	2	no	—
156	C₂²⋊C₄	—	122	1	no	—
157	C₄²	—	122	1	no	—
158	C₁₆	—	2	2	no	—
159	C₁₆	—	244	1	no	—
160	C₂³	#14519800	1	1	yes	characteristic
161	Q₈	—	488	2	no	—
162	Q₈	—	488	2	no	—
163	Q₈	—	244	4	no	—
164	Q₈	—	244	4	no	—
165	C₂×C₄	—	122	2	no	—
166	C₂×C₄	—	488	2	no	—
167	C₂×C₄	—	488	2	no	—
168	C₂×C₄	—	244	4	no	—
169	C₂×C₄	#14519750	1	2	yes	—
170	C₂×C₄	#14519752	1	1	yes	characteristic
171	C₂×C₄	D₈×D₆₁	1	1	yes	characteristic
172	C₂×C₄	#14519804	1	2	yes	—
173	C₈	—	8	2	no	—
174	C₈	—	244	1	no	—
175	C₈	#14519747	1	1	yes	characteristic
176	C₈	#14519762	1	2	yes	—
177	C₈	#14519765	1	1	yes	characteristic
178	C₂²	—	2	2	no	—
179	C₂²	#16839980	1	2	yes	—
180	C₂²	#16839999	1	1	yes	characteristic
181	C₄	—	488	4	no	—
182	C₄	—	244	2	no	—
183	C₄	#16839680	1	1	yes	characteristic
184	C₄	#16839956	1	2	yes	—
185	C₄	#16840016	1	1	yes	characteristic
186	C₂	—	2	2	no	—
187	C₂	#21447131	1	1	yes	characteristic
188	C₂	#21447945	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₄₈₈

C₂

yes

maximal, characteristic

#21447577

C₂

yes

maximal, characteristic

C₂×#16839577

C₂

yes

maximal

#21446732

C₂

yes

maximal

C₆₁×#1381

C₂

yes

maximal, characteristic

C₂²×Dic₂₄₄

C₂²

yes

—

C₂×Dic₄₈₈

—

C₂×Dic₄₈₈

—

C₂×#14519717

C₂²

yes

—

—

C₂²

yes

characteristic

#16839580

—

#16839577

—

C₂²×C₁₆×C₆₁

C₂²

yes

characteristic

C₂×#108×C₆₁

C₂²

yes

—

C₂²×Dic₁₂₂

—

C₂×Dic₂₄₄

—

C₂×Dic₂₄₄

—

C₂×Dic₂₄₄

—

C₂×Dic₂₄₄

—

C₂×Dic₂₄₄

—

C₂×Dic₂₄₄

D₄

yes

—

C₂×#11758194

D₄

yes

characteristic

#14519736

D₄

yes

characteristic

Dic₄₈₈

—

—

—

—

—

—

C₈×Dic₆₁

—

C₂×M₁₆×C₆₁

—

C₂²×C₈×C₆₁

C₂³

yes

characteristic

C₂×C₁₆×C₆₁

—

C₂×C₁₆×C₆₁

D₄

yes

—

#108×C₆₁

—

#108×C₆₁

—

C₂²×C₄×C₆₁

C₂×D₄

yes

characteristic

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×D₄

yes

—

C₂×C₈×C₆₁

C₂×D₄

yes

C₂×C₈×C₆₁

yes

C₂×C₈×C₆₁

yes

—

C₂²⋊Dic₆₁

—

C₄⋊Dic₆₁

D₈

yes

—

C₄×Dic₆₁

—

#11758194

—

C₂×#92418

—

Dic₂₄₄

—

Dic₂₄₄

—

Dic₂₄₄

—

Dic₂₄₄

—

C₁₆×C₆₁

—

#11758185

—

C₂³×C₆₁

#135

yes

characteristic

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

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

Dic₁₂₂

—

Dic₁₂₂

—

Dic₁₂₂

—

Dic₁₂₂

—

C₈×C₆₁

—

C₈×C₆₁

#120

yes

characteristic

C₈×C₆₁

#123

yes

—

C₈×C₆₁

yes

—

—

C₂²×C₆₁

—

C₂²×C₆₁

#463

yes

—

C₂²×C₆₁

yes

C₄×C₆₁

yes

C₄×C₆₁

yes

—

C₄×C₆₁

#469

yes

characteristic

Dic₆₁

—

Dic₆₁

—

C₂²×Q₃₂

—

—

—

—

D₆₁

yes

characteristic

C₂×C₆₁

—

C₂×C₆₁

#2054

yes

characteristic

C₂×C₆₁

#2222

yes

—

100

C₂²×Q₁₆

—

101

C₂×Q₃₂

—

122

—

102

C₂×Q₃₂

—

122

—

103

C₂×M₃₂

—

104

C₂²×C₁₆

C₂×D₆₁

yes

characteristic

105

C₂×#108

C₂×D₆₁

yes

—

106

#419

—

122

—

107

C₂×#101

—

122

—

108

#405

—

109

C₄⋊C₁₆

—

122

—

110

#362

—

122

—

111

C₆₁

#13441

yes

characteristic, complemented, Sylow

112

C₂²×Q₈

—

122

—

113

C₂×Q₁₆

—

114

C₂×Q₁₆

—

122

—

115

C₂×Q₁₆

—

122

—

116

C₂×Q₁₆

—

122

—

117

C₂×Q₁₆

—

122

—

118

C₂×Q₁₆

—

122

—

119

C₂×M₁₆

—

120

C₂×M₁₆

—

121

C₂²×C₈

C₂²×D₆₁

yes

characteristic

122

#117

—

123

Q₃₂

—

122

—

124

M₃₂

—

122

—

125

C₂×C₁₆

—

126

C₂×C₁₆

—

122

—

127

C₂×C₁₆

D₂₄₄

yes

—

128

#108

—

129

#108

—

130

#103

—

244

—

131

#101

—

244

—

132

#97

—

122

—

133

C₄×C₈

—

122

—

134

C₂×Q₈

—

122

—

135

C₂×Q₈

—

244

—

136

C₂×Q₈

—

244

—

137

C₂×Q₈

—

244

—

138

C₂×Q₈

—

244

—

139

C₂×Q₈

—

244

—

140

C₂²×C₄

—

244

—

141

C₂²×C₄

D₄×D₆₁

yes

characteristic

142

Q₁₆

—

122

—

143

Q₁₆

—

122

—

144

Q₁₆

—

244

—

145

Q₁₆

—

244

—

146

M₁₆

—

147

M₁₆

—

148

M₁₆

—

122

—

149

C₂×C₈

—

150

C₂×C₈

—

122

—

151

C₂×C₈

C₂×D₂₄₄

yes

characteristic

152

C₂×C₈

#11758222

yes

—

153

C₂×C₈

D₄×D₆₁

yes

—

154

C₂×C₈

#11758226

yes

characteristic

155

C₄⋊C₄

—

156

C₂²⋊C₄

—

122

—

157

C₄²

—

122

—

158

C₁₆

—

159

C₁₆

—

244

—

160

C₂³

#14519800

yes

characteristic

161

Q₈

—

488

—

162

Q₈

—

488

—

163

Q₈

—

244

—

164

Q₈

—

244

—

165

C₂×C₄

—

122

—

166

C₂×C₄

—

488

—

167

C₂×C₄

—

488

—

168

C₂×C₄

—

244

—

169

C₂×C₄

#14519750

yes

—

170

C₂×C₄

#14519752

yes

characteristic

171

C₂×C₄

D₈×D₆₁

yes

characteristic

172

C₂×C₄

#14519804

yes

—

173

C₈

—

174

C₈

—

244

—

175

C₈

#14519747

yes

characteristic

176

C₈

#14519762

yes

—

177

C₈

#14519765

yes

characteristic

178

C₂²

—

179

C₂²

#16839980

yes

—

180

C₂²

#16839999

yes

characteristic

181

C₄

—

488

—

182

C₄

—

244

—

183

C₄

#16839680

yes

characteristic

184

C₄

#16839956

yes

—

185

C₄

#16840016

yes

characteristic

186

C₂

—

187

C₂

#21447131

yes

characteristic

188

C₂

#21447945

yes

—

	subgroup	G/H
center	C₂²	#16839999
commutator, cosocle	C₂²×C₈×C₆₁	C₂³
Frattini	C₂²×C₈	C₂²×D₆₁
Fitting	C₆₁×#1381	C₂
socle	C₂²×C₆₁	#466
prime radical	C₂³×C₆₁	#135
hypercenter	#1381	D₆₁
hypocenter	C₆₁	#13441
LCS	G ▷ C₂²×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1381

subgroup

G/H

center

C₂²

#16839999

commutator, cosocle

C₂²×C₈×C₆₁

C₂³

Frattini

C₂²×C₈

C₂²×D₆₁

Fitting

C₆₁×#1381

C₂

C₂²×C₆₁

C₂³×C₆₁

D₆₁

C₆₁

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

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1381

N	H ≅ G/N	type
111. C₆₁	#13441	semidirect

H ≅ G/N

type

111. C₆₁

#13441

semidirect

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

subgroup	index
C₂²×Dic₄₈₈	2
#21447577	2
C₂×#16839577	2
#21446732	2
C₆₁×#1381	2
#13441	61

subgroup

index

C₂²×Dic₄₈₈

C₆₁×#1381

Subgroups — #29599566

Conjugacy classes of subgroups

Complements

Maximal subgroups