Group #29599888 — subgroups

Subgroups — #29599888

Order 15616.

284 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 87 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×#16839588	C₂	1	1	yes	maximal, characteristic, complemented
2	#21447432	C₂	1	1	yes	maximal, characteristic
3	#21445880	C₂	1	2	yes	maximal
4	#21443929	C₂	1	2	yes	maximal, complemented
5	C₆₁×#1381	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#14519783	C₂²	1	1	yes	characteristic
7	C₂×#14519717	C₂²	1	1	yes	characteristic
8	C₂×#14519711	C₂²	1	2	yes	—
9	#16839764	—	2	2	no	—
10	#16839588	—	2	2	no	—
11	#16839541	—	2	2	no	—
12	#16839530	—	2	2	no	—
13	C₂²×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
14	C₂×#108×C₆₁	C₂²	1	2	yes	—
15	C₂×#11758222	D₄	1	1	yes	characteristic
16	#14519783	—	2	2	no	—
17	#14519783	—	2	2	no	—
18	C₂×#11758194	D₄	1	1	yes	characteristic
19	C₂×#11758189	—	2	1	no	—
20	C₂×C₈×D₆₁	—	2	1	no	—
21	C₂×C₄⋊Dic₆₁	—	2	2	no	—
22	#14519717	—	2	2	no	—
23	C₂×#11758185	—	2	1	no	—
24	#14519711	—	2	2	no	—
25	#14519711	—	2	2	no	—
26	#14519702	—	4	2	no	—
27	#14519699	—	4	2	no	—
28	C₂×M₁₆×C₆₁	—	2	2	no	—
29	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
30	C₂×C₁₆×C₆₁	—	2	2	no	—
31	C₂×C₁₆×C₆₁	D₄	1	2	yes	—
32	#108×C₆₁	—	2	2	no	—
33	#108×C₆₁	—	2	2	no	—
34	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
35	C₂×C₂²⋊D₆₁	—	2	1	no	—
36	C₂²×Dic₆₁	—	4	2	no	—
37	#11758222	—	2	2	no	—
38	#11758222	—	2	2	no	—
39	C₂×D₂₄₄	D₈	1	1	yes	characteristic
40	C₂×C₄×D₆₁	—	2	1	no	—
41	C₂×Dic₁₂₂	D₈	1	1	yes	characteristic
42	M₁₆×C₆₁	—	4	2	no	—
43	M₁₆×C₆₁	—	4	2	no	—
44	C₂×C₈×C₆₁	—	4	2	no	—
45	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
46	C₂×C₈×C₆₁	#41	1	2	yes	—
47	C₂×C₈×C₆₁	#41	1	2	yes	—
48	C₂×C₈×C₆₁	#41	1	1	yes	characteristic
49	C₄⋊Dic₆₁	—	4	2	no	—
50	C₄⋊Dic₆₁	—	4	2	no	—
51	#11758194	—	2	2	no	—
52	C₂×#92418	—	2	1	no	—
53	#11758189	—	4	2	no	—
54	C₈×D₆₁	—	4	1	no	—
55	C₈×D₆₁	—	4	1	no	—
56	C₁₆×C₆₁	—	2	2	no	—
57	#11758185	—	4	1	no	—
58	C₂³×C₆₁	#135	1	1	yes	characteristic
59	C₂²×D₆₁	—	2	1	no	—
60	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
61	C₂×C₄×C₆₁	#123	1	2	yes	—
62	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
63	C₂×C₄×C₆₁	#137	1	2	yes	—
64	C₂²⋊D₆₁	—	4	2	no	—
65	C₂×Dic₆₁	—	2	1	no	—
66	C₂×Dic₆₁	—	8	2	no	—
67	C₂×Dic₆₁	—	16	2	no	—
68	D₂₄₄	—	2	2	no	—
69	C₄×D₆₁	—	4	1	no	—
70	C₄×D₆₁	—	4	1	no	—
71	Dic₁₂₂	—	2	2	no	—
72	C₈×C₆₁	—	8	2	no	—
73	C₈×C₆₁	#123	1	1	yes	characteristic
74	C₈×C₆₁	#124	1	1	yes	characteristic
75	C₈×C₆₁	#126	1	2	yes	—
76	#92418	—	4	1	no	—
77	#13495	—	61	1	no	maximal, Sylow
78	C₂²×C₆₁	—	2	1	no	—
79	C₂²×C₆₁	—	2	1	no	—
80	C₂²×C₆₁	#466	1	1	yes	characteristic
81	C₂²×C₆₁	#485	1	2	yes	—
82	C₂×D₆₁	—	4	1	no	—
83	C₂×D₆₁	—	4	1	no	—
84	C₂×D₆₁	—	4	1	no	—
85	C₄×C₆₁	#397	1	1	yes	characteristic
86	C₄×C₆₁	#470	1	1	yes	characteristic
87	C₄×C₆₁	#480	1	2	yes	—
88	Dic₆₁	—	4	1	no	—
89	Dic₆₁	—	16	2	no	—
90	C₂×#366	—	61	1	no	—
91	#2113	—	61	1	no	—
92	#1816	—	61	2	no	—
93	#1387	—	61	2	no	—
94	#1381	D₆₁	1	1	yes	characteristic, complemented
95	C₂×C₆₁	—	2	2	no	—
96	C₂×C₆₁	#2075	1	1	yes	characteristic
97	C₂×C₆₁	#2221	1	1	yes	characteristic
98	C₂×C₆₁	#2222	1	1	yes	characteristic
99	D₆₁	—	8	1	no	—
100	C₂×#131	—	61	1	no	—
101	C₂×M₃₂	—	61	1	no	—
102	C₂²×C₁₆	C₂×D₆₁	1	1	yes	characteristic
103	C₂×#108	C₂×D₆₁	1	2	yes	—
104	C₂×#107	—	61	2	no	—
105	#417	—	122	2	no	—
106	#366	—	122	2	no	—
107	C₂²⋊C₁₆	—	122	2	no	—
108	#343	—	122	2	no	—
109	C₆₁	#13495	1	1	yes	characteristic, complemented, Sylow
110	C₂×#41	—	61	1	no	—
111	#131	—	122	2	no	—
112	#131	—	122	2	no	—
113	C₂×M₁₆	—	2	2	no	—
114	C₂×M₁₆	—	61	1	no	—
115	C₂×M₁₆	—	122	1	no	—
116	C₂²×C₈	—	122	1	no	—
117	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
118	C₂×C₄⋊C₄	—	122	2	no	—
119	M₃₂	—	122	2	no	—
120	C₂×C₁₆	—	2	2	no	—
121	C₂×C₁₆	—	122	1	no	—
122	C₂×C₁₆	D₂₄₄	1	2	yes	—
123	#108	—	2	2	no	—
124	#108	—	2	2	no	—
125	#107	—	122	2	no	—
126	#107	—	122	2	no	—
127	#103	—	244	2	no	—
128	#102	—	244	2	no	—
129	#41	—	122	2	no	—
130	#41	—	122	2	no	—
131	C₂×Q₈	—	61	1	no	—
132	C₂×D₄	—	61	1	no	—
133	C₂×D₄	—	122	1	no	—
134	C₂²×C₄	—	122	1	no	—
135	C₂²×C₄	—	244	2	no	—
136	C₂²×C₄	D₄×D₆₁	1	1	yes	characteristic
137	M₁₆	—	4	2	no	—
138	M₁₆	—	4	2	no	—
139	M₁₆	—	122	2	no	—
140	M₁₆	—	244	2	no	—
141	C₂×C₈	—	4	2	no	—
142	C₂×C₈	—	122	1	no	—
143	C₂×C₈	—	244	1	no	—
144	C₂×C₈	—	244	1	no	—
145	C₂×C₈	C₂×D₂₄₄	1	1	yes	characteristic
146	C₂×C₈	#11758222	1	2	yes	—
147	C₂×C₈	#11758224	1	2	yes	—
148	C₂×C₈	#11758226	1	1	yes	characteristic
149	C₄⋊C₄	—	244	2	no	—
150	C₄⋊C₄	—	244	2	no	—
151	C₁₆	—	2	2	no	—
152	C₁₆	—	244	1	no	—
153	C₂³	—	122	1	no	—
154	C₂³	#14519800	1	1	yes	characteristic
155	Q₈	—	122	2	no	—
156	D₄	—	122	2	no	—
157	D₄	—	244	2	no	—
158	C₂×C₄	—	122	1	no	—
159	C₂×C₄	—	976	2	no	—
160	C₂×C₄	—	488	2	no	—
161	C₂×C₄	—	244	1	no	—
162	C₂×C₄	—	244	1	no	—
163	C₂×C₄	#14519752	1	1	yes	characteristic
164	C₂×C₄	#14519759	1	2	yes	—
165	C₂×C₄	D₈×D₆₁	1	1	yes	characteristic
166	C₂×C₄	#14519806	1	2	yes	—
167	C₈	—	8	2	no	—
168	C₈	—	244	1	no	—
169	C₈	#14519761	1	1	yes	characteristic
170	C₈	#14519765	1	1	yes	characteristic
171	C₈	#14519772	1	2	yes	—
172	C₂²	—	2	1	no	—
173	C₂²	—	2	1	no	—
174	C₂²	—	244	1	no	—
175	C₂²	—	244	1	no	—
176	C₂²	—	244	1	no	—
177	C₂²	#16839999	1	1	yes	characteristic
178	C₂²	#16840102	1	2	yes	—
179	C₄	—	976	2	no	—
180	C₄	—	244	1	no	—
181	C₄	#16839689	1	1	yes	characteristic
182	C₄	#16840021	1	1	yes	characteristic
183	C₄	#16840074	1	2	yes	—
184	C₂	—	2	2	no	—
185	C₂	—	488	1	no	—
186	C₂	#21447244	1	1	yes	characteristic
187	C₂	#21447940	1	1	yes	characteristic
188	C₂	#21447945	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16839588

C₂

yes

maximal, characteristic, complemented

#21447432

C₂

yes

maximal, characteristic

#21445880

C₂

yes

maximal

#21443929

C₂

yes

maximal, complemented

C₆₁×#1381

C₂

yes

maximal, characteristic, complemented

C₂×#14519783

C₂²

yes

characteristic

C₂×#14519717

C₂²

yes

characteristic

C₂×#14519711

C₂²

yes

—

—

—

—

—

C₂²×C₁₆×C₆₁

C₂²

yes

characteristic

C₂×#108×C₆₁

C₂²

yes

—

C₂×#11758222

D₄

yes

—

—

D₄

yes

characteristic

C₂×#11758189

—

C₂×C₈×D₆₁

—

C₂×C₄⋊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

C₂×C₂²⋊D₆₁

—

C₂²×Dic₆₁

—

#11758222

—

#11758222

—

C₂×D₂₄₄

D₈

yes

characteristic

C₂×C₄×D₆₁

—

C₂×Dic₁₂₂

D₈

yes

characteristic

M₁₆×C₆₁

—

M₁₆×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

C₂×D₄

yes

characteristic

C₂×C₈×C₆₁

#41

yes

—

C₂×C₈×C₆₁

#41

yes

—

C₂×C₈×C₆₁

#41

yes

characteristic

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

#11758194

—

C₂×#92418

—

#11758189

—

C₈×D₆₁

—

C₈×D₆₁

—

C₁₆×C₆₁

—

#11758185

—

C₂³×C₆₁

#135

yes

characteristic

C₂²×D₆₁

—

C₂×C₄×C₆₁

#121

yes

characteristic

C₂×C₄×C₆₁

#123

yes

—

C₂×C₄×C₆₁

C₂×D₈

yes

characteristic

C₂×C₄×C₆₁

#137

yes

—

C₂²⋊D₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

D₂₄₄

—

C₄×D₆₁

—

C₄×D₆₁

—

Dic₁₂₂

—

C₈×C₆₁

—

C₈×C₆₁

yes

C₈×C₆₁

yes

C₈×C₆₁

yes

—

#92418

—

#13495

—

maximal, Sylow

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

#466

yes

characteristic

C₂²×C₆₁

#485

yes

—

C₂×D₆₁

—

C₂×D₆₁

—

C₂×D₆₁

—

C₄×C₆₁

yes

C₄×C₆₁

yes

C₄×C₆₁

yes

—

Dic₆₁

—

Dic₆₁

—

—

—

—

—

D₆₁

yes

characteristic, complemented

C₂×C₆₁

—

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

D₆₁

—

100

C₂×#131

—

101

C₂×M₃₂

—

102

C₂²×C₁₆

C₂×D₆₁

yes

characteristic

103

C₂×#108

C₂×D₆₁

yes

—

104

C₂×#107

—

105

#417

—

122

—

106

#366

—

122

—

107

C₂²⋊C₁₆

—

122

—

108

#343

—

122

—

109

C₆₁

#13495

yes

characteristic, complemented, Sylow

110

C₂×#41

—

111

#131

—

122

—

112

#131

—

122

—

113

C₂×M₁₆

—

114

C₂×M₁₆

—

115

C₂×M₁₆

—

122

—

116

C₂²×C₈

—

122

—

117

C₂²×C₈

C₂²×D₆₁

yes

characteristic

118

C₂×C₄⋊C₄

—

122

—

119

M₃₂

—

122

—

120

C₂×C₁₆

—

121

C₂×C₁₆

—

122

—

122

C₂×C₁₆

D₂₄₄

yes

—

123

#108

—

124

#108

—

125

#107

—

122

—

126

#107

—

122

—

127

#103

—

244

—

128

#102

—

244

—

129

#41

—

122

—

130

#41

—

122

—

131

C₂×Q₈

—

132

C₂×D₄

—

133

C₂×D₄

—

122

—

134

C₂²×C₄

—

122

—

135

C₂²×C₄

—

244

—

136

C₂²×C₄

D₄×D₆₁

yes

characteristic

137

M₁₆

—

138

M₁₆

—

139

M₁₆

—

122

—

140

M₁₆

—

244

—

141

C₂×C₈

—

142

C₂×C₈

—

122

—

143

C₂×C₈

—

244

—

144

C₂×C₈

—

244

—

145

C₂×C₈

C₂×D₂₄₄

yes

characteristic

146

C₂×C₈

#11758222

yes

—

147

C₂×C₈

#11758224

yes

—

148

C₂×C₈

#11758226

yes

characteristic

149

C₄⋊C₄

—

244

—

150

C₄⋊C₄

—

244

—

151

C₁₆

—

152

C₁₆

—

244

—

153

C₂³

—

122

—

154

C₂³

#14519800

yes

characteristic

155

Q₈

—

122

—

156

D₄

—

122

—

157

D₄

—

244

—

158

C₂×C₄

—

122

—

159

C₂×C₄

—

976

—

160

C₂×C₄

—

488

—

161

C₂×C₄

—

244

—

162

C₂×C₄

—

244

—

163

C₂×C₄

#14519752

yes

characteristic

164

C₂×C₄

#14519759

yes

—

165

C₂×C₄

D₈×D₆₁

yes

characteristic

166

C₂×C₄

#14519806

yes

—

167

C₈

—

168

C₈

—

244

—

169

C₈

#14519761

yes

characteristic

170

C₈

#14519765

yes

characteristic

171

C₈

#14519772

yes

—

172

C₂²

—

173

C₂²

—

174

C₂²

—

244

—

175

C₂²

—

244

—

176

C₂²

—

244

—

177

C₂²

#16839999

yes

characteristic

178

C₂²

#16840102

yes

—

179

C₄

—

976

—

180

C₄

—

244

—

181

C₄

#16839689

yes

characteristic

182

C₄

#16840021

yes

characteristic

183

C₄

#16840074

yes

—

184

C₂

—

185

C₂

—

488

—

186

C₂

#21447244

yes

characteristic

187

C₂

#21447940

yes

characteristic

188

C₂

#21447945

yes

characteristic

	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	C₂×#11758222	D₄
hypercenter	#1381	D₆₁
hypocenter	C₆₁	#13495
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

D₄

D₆₁

C₆₁

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

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1381

N	H ≅ G/N	type
1. C₂×#16839588	C₂	semidirect
4. #21443929	C₂	semidirect
5. C₆₁×#1381	C₂	semidirect
94. #1381	D₆₁	semidirect
109. C₆₁	#13495	semidirect

H ≅ G/N

type

1. C₂×#16839588

C₂

semidirect

4. #21443929

C₂

semidirect

5. C₆₁×#1381

C₂

semidirect

94. #1381

D₆₁

semidirect

109. C₆₁

#13495

semidirect

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

subgroup	index
C₂×#16839588	2
#21447432	2
#21445880	2
#21443929	2
C₆₁×#1381	2
#13495	61

subgroup

index

C₆₁×#1381

Subgroups — #29599888

Conjugacy classes of subgroups

Complements

Maximal subgroups