Group #29596205 — subgroups

Subgroups — #29596205

Order 15616.

354 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 111 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×#16840180	C₂	1	1	yes	maximal, characteristic
2	C₂×#16839591	C₂	1	2	yes	maximal, complemented
3	#21445866	C₂	1	2	yes	maximal
4	#21443923	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₁×#1385	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#14519783	C₂²	1	1	yes	characteristic
7	#16840180	—	2	4	no	—
8	C₂×#14519716	C₄	1	2	yes	—
9	C₂²×#11758185	C₂²	1	1	yes	characteristic
10	C₂×C₁₆×D₆₁	C₄	1	2	yes	—
11	C₂×#14519713	C₂²	1	2	yes	—
12	#16839744	—	2	2	no	—
13	#16839591	—	2	4	no	—
14	#16839520	—	2	2	no	—
15	C₂×M₃₂×C₆₁	C₂²	1	1	yes	characteristic
16	C₂×#108×C₆₁	C₂²	1	2	yes	—
17	C₂×#11758222	D₄	1	1	yes	characteristic
18	#14519783	—	2	2	no	—
19	#14519783	—	2	2	no	—
20	C₂×#11758194	—	2	2	no	—
21	C₂×#11758194	D₄	1	1	yes	characteristic
22	C₂×#11758189	Q₈	1	2	yes	—
23	C₂×C₈×D₆₁	C₂×C₄	1	2	yes	—
24	#14519716	—	2	4	no	—
25	C₂×#11758185	C₂×C₄	1	1	yes	characteristic
26	C₂×#11758185	C₂×C₄	1	1	yes	characteristic
27	C₂×#11758185	D₄	1	4	yes	—
28	C₁₆×D₆₁	—	2	4	no	—
29	#14519713	—	2	2	no	—
30	#14519713	—	2	2	no	—
31	#14519695	—	4	2	no	—
32	#14519693	—	4	2	no	—
33	C₂×M₁₆×C₆₁	—	2	2	no	—
34	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
35	M₃₂×C₆₁	—	2	2	no	—
36	C₂×C₁₆×C₆₁	C₂×C₄	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	C₂×C₂²⋊D₆₁	Q₁₆	1	2	yes	—
41	#11758222	—	2	2	no	—
42	#11758222	—	2	2	no	—
43	C₂×D₂₄₄	D₈	1	1	yes	characteristic
44	C₂×C₄×D₆₁	C₄⋊C₄	1	2	yes	—
45	C₂×Dic₁₂₂	D₈	1	1	yes	characteristic
46	M₁₆×C₆₁	—	4	2	no	—
47	M₁₆×C₆₁	—	4	2	no	—
48	C₂×C₈×C₆₁	—	4	2	no	—
49	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
51	C₂×C₈×C₆₁	C₂×Q₈	1	1	yes	characteristic
52	C₂×C₈×C₆₁	#41	1	2	yes	—
53	#11758194	—	2	2	no	—
54	#11758194	—	4	2	no	—
55	#11758194	—	4	2	no	—
56	C₂×#92418	—	4	2	no	—
57	C₂×#92418	C₄⋊C₄	1	2	yes	—
58	#11758189	—	2	4	no	—
59	C₈×D₆₁	—	2	2	no	—
60	C₈×D₆₁	—	2	2	no	—
61	C₁₆×C₆₁	—	2	2	no	—
62	#11758185	C₂²⋊C₄	1	2	yes	—
63	#11758185	C₂²⋊C₄	1	2	yes	—
64	C₂³×C₆₁	C₂×Q₁₆	1	1	yes	characteristic
65	C₂²×D₆₁	#107	1	2	yes	—
66	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
67	C₂×C₄×C₆₁	#121	1	2	yes	—
68	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
69	C₂×C₄×C₆₁	#135	1	2	yes	—
70	C₂²⋊D₆₁	—	2	4	no	—
71	C₂×Dic₆₁	#107	1	2	yes	—
72	D₂₄₄	—	2	2	no	—
73	C₄×D₆₁	—	2	2	no	—
74	C₄×D₆₁	—	2	2	no	—
75	Dic₁₂₂	—	2	2	no	—
76	C₈×C₆₁	—	8	2	no	—
77	C₈×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
78	C₈×C₆₁	#117	1	1	yes	characteristic
79	C₈×C₆₁	C₂²⋊Q₈	1	2	yes	—
80	#92418	—	2	2	no	—
81	#92418	—	8	2	no	—
82	#12840	—	61	1	no	maximal, Sylow
83	C₂²×C₆₁	—	4	1	no	—
84	C₂²×C₆₁	C₂×#107	1	1	yes	characteristic
85	C₂²×C₆₁	#467	1	2	yes	—
86	C₂×D₆₁	—	2	2	no	—
87	C₂×D₆₁	—	4	2	no	—
88	C₄×C₆₁	#380	1	1	yes	characteristic
89	C₄×C₆₁	#427	1	1	yes	characteristic
90	C₄×C₆₁	#466	1	2	yes	—
91	Dic₆₁	—	2	2	no	—
92	C₂×#504	—	61	1	no	—
93	C₂×#368	—	61	2	no	—
94	#1813	—	61	2	no	—
95	#1385	—	61	1	no	—
96	#1385	D₆₁	1	1	yes	characteristic, complemented
97	C₂×C₆₁	—	2	1	no	—
98	C₂×C₆₁	—	2	2	no	—
99	C₂×C₆₁	#1808	1	1	yes	characteristic
100	D₆₁	—	4	2	no	—
101	C₂×#131	—	61	1	no	—
102	#504	—	122	4	no	—
103	C₂×M₃₂	—	61	2	no	—
104	C₂×M₃₂	C₂×D₆₁	1	1	yes	characteristic
105	C₂²×C₁₆	—	61	1	no	—
106	C₂²×C₁₆	—	61	2	no	—
107	C₂×#108	—	61	2	no	—
108	C₂×#108	C₂×D₆₁	1	2	yes	—
109	#413	—	122	2	no	—
110	#368	—	122	4	no	—
111	#338	—	122	2	no	—
112	C₆₁	#12840	1	1	yes	characteristic, complemented, Sylow
113	C₂×#41	—	61	1	no	—
114	#131	—	122	2	no	—
115	#131	—	122	2	no	—
116	C₂×M₁₆	—	2	2	no	—
117	C₂×M₁₆	—	61	1	no	—
118	C₂×M₁₆	—	61	2	no	—
119	C₂×M₁₆	—	122	2	no	—
120	C₂²×C₈	—	61	2	no	—
121	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
122	M₃₂	—	2	2	no	—
123	M₃₂	—	122	4	no	—
124	C₂×C₁₆	—	61	1	no	—
125	C₂×C₁₆	—	61	1	no	—
126	C₂×C₁₆	—	61	4	no	—
127	C₂×C₁₆	—	122	4	no	—
128	C₂×C₁₆	C₄×D₆₁	1	2	yes	—
129	#108	—	2	2	no	—
130	#108	—	2	2	no	—
131	#108	—	122	2	no	—
132	#108	—	122	2	no	—
133	#101	—	244	2	no	—
134	#100	—	244	2	no	—
135	#41	—	122	2	no	—
136	#41	—	122	2	no	—
137	C₂×Q₈	—	61	1	no	—
138	C₂×D₄	—	61	2	no	—
139	C₂×D₄	—	61	1	no	—
140	C₂²×C₄	—	61	2	no	—
141	C₂²×C₄	D₄×D₆₁	1	1	yes	characteristic
142	M₁₆	—	4	2	no	—
143	M₁₆	—	4	2	no	—
144	M₁₆	—	122	2	no	—
145	M₁₆	—	122	4	no	—
146	M₁₆	—	244	2	no	—
147	M₁₆	—	244	2	no	—
148	C₂×C₈	—	4	2	no	—
149	C₂×C₈	—	61	2	no	—
150	C₂×C₈	—	122	2	no	—
151	C₂×C₈	—	122	2	no	—
152	C₂×C₈	—	244	2	no	—
153	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
154	C₂×C₈	#11758222	1	2	yes	—
155	C₂×C₈	D₄×D₆₁	1	2	yes	—
156	C₂×C₈	Q₈×D₆₁	1	1	yes	characteristic
157	C₁₆	—	2	2	no	—
158	C₁₆	—	61	2	no	—
159	C₁₆	—	61	2	no	—
160	C₂³	—	61	2	no	—
161	C₂³	Q₁₆×D₆₁	1	1	yes	characteristic
162	Q₈	—	122	2	no	—
163	D₄	—	122	4	no	—
164	D₄	—	122	2	no	—
165	C₂×C₄	—	61	2	no	—
166	C₂×C₄	—	122	2	no	—
167	C₂×C₄	—	122	2	no	—
168	C₂×C₄	C₄⋊C₄×D₆₁	1	1	yes	characteristic
169	C₂×C₄	#14519750	1	2	yes	—
170	C₂×C₄	D₈×D₆₁	1	1	yes	characteristic
171	C₂×C₄	#14519798	1	2	yes	—
172	C₈	—	8	2	no	—
173	C₈	—	122	2	no	—
174	C₈	—	488	2	no	—
175	C₈	C₂²⋊C₄×D₆₁	1	1	yes	characteristic
176	C₈	#14519734	1	1	yes	characteristic
177	C₈	#14519755	1	2	yes	—
178	C₂²	—	4	1	no	—
179	C₂²	—	122	2	no	—
180	C₂²	—	244	2	no	—
181	C₂²	#107×D₆₁	1	1	yes	characteristic
182	C₂²	#16840002	1	2	yes	—
183	C₄	—	122	2	no	—
184	C₄	#16839621	1	1	yes	characteristic
185	C₄	#16839797	1	1	yes	characteristic
186	C₄	#16839997	1	2	yes	—
187	C₂	—	2	1	no	—
188	C₂	—	2	2	no	—
189	C₂	—	244	2	no	—
190	C₂	#21445847	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16840180

C₂

yes

maximal, characteristic

C₂×#16839591

C₂

yes

maximal, complemented

#21445866

C₂

yes

maximal

#21443923

C₂

yes

maximal, characteristic, complemented

C₆₁×#1385

C₂

yes

maximal, characteristic, complemented

C₂×#14519783

C₂²

yes

characteristic

#16840180

—

C₂×#14519716

C₄

yes

—

C₂²×#11758185

C₂²

yes

characteristic

C₂×C₁₆×D₆₁

C₄

yes

—

C₂×#14519713

C₂²

yes

—

#16839744

—

#16839591

—

#16839520

—

C₂×M₃₂×C₆₁

C₂²

yes

characteristic

C₂×#108×C₆₁

C₂²

yes

—

C₂×#11758222

D₄

yes

—

—

—

D₄

yes

characteristic

C₂×#11758189

Q₈

yes

—

C₂×C₈×D₆₁

C₂×C₄

yes

—

#14519716

—

C₂×#11758185

C₂×C₄

yes

characteristic

C₂×#11758185

C₂×C₄

yes

characteristic

C₂×#11758185

D₄

yes

—

C₁₆×D₆₁

—

—

—

—

—

C₂×M₁₆×C₆₁

—

C₂²×C₈×C₆₁

C₂³

yes

characteristic

M₃₂×C₆₁

—

C₂×C₁₆×C₆₁

C₂×C₄

yes

—

#108×C₆₁

—

#108×C₆₁

—

C₂²×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₂²⋊D₆₁

Q₁₆

yes

—

#11758222

—

#11758222

—

C₂×D₂₄₄

D₈

yes

characteristic

C₂×C₄×D₆₁

C₄⋊C₄

yes

—

C₂×Dic₁₂₂

D₈

yes

characteristic

M₁₆×C₆₁

—

M₁₆×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

C₂²×C₄

yes

characteristic

C₂×C₈×C₆₁

C₂×D₄

yes

—

C₂×C₈×C₆₁

C₂×Q₈

yes

characteristic

C₂×C₈×C₆₁

#41

yes

—

—

—

—

—

C₄⋊C₄

yes

—

#11758189

—

C₈×D₆₁

—

C₈×D₆₁

—

C₁₆×C₆₁

—

#11758185

C₂²⋊C₄

yes

—

#11758185

C₂²⋊C₄

yes

—

C₂³×C₆₁

C₂×Q₁₆

yes

characteristic

C₂²×D₆₁

#107

yes

—

C₂×C₄×C₆₁

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₁

#121

yes

—

C₂×C₄×C₆₁

C₂×D₈

yes

characteristic

C₂×C₄×C₆₁

#135

yes

—

C₂²⋊D₆₁

—

C₂×Dic₆₁

#107

yes

—

D₂₄₄

—

C₄×D₆₁

—

C₄×D₆₁

—

Dic₁₂₂

—

C₈×C₆₁

—

C₈×C₆₁

C₂×C₂²⋊C₄

yes

characteristic

C₈×C₆₁

#117

yes

characteristic

C₈×C₆₁

C₂²⋊Q₈

yes

—

—

—

—

C₂²×C₆₁

—

C₂²×C₆₁

C₂×#107

yes

characteristic

C₂²×C₆₁

#467

yes

—

C₂×D₆₁

—

C₂×D₆₁

—

C₄×C₆₁

yes

C₄×C₆₁

yes

C₄×C₆₁

yes

—

Dic₆₁

—

—

—

—

—

D₆₁

yes

characteristic, complemented

C₂×C₆₁

—

C₂×C₆₁

—

C₂×C₆₁

#1808

yes

characteristic

100

D₆₁

—

101

C₂×#131

—

102

#504

—

122

—

103

C₂×M₃₂

—

104

C₂×M₃₂

C₂×D₆₁

yes

characteristic

105

C₂²×C₁₆

—

106

C₂²×C₁₆

—

107

C₂×#108

—

108

C₂×#108

C₂×D₆₁

yes

—

109

#413

—

122

—

110

#368

—

122

—

111

#338

—

122

—

112

C₆₁

#12840

yes

characteristic, complemented, Sylow

113

C₂×#41

—

114

#131

—

122

—

115

#131

—

122

—

116

C₂×M₁₆

—

117

C₂×M₁₆

—

118

C₂×M₁₆

—

119

C₂×M₁₆

—

122

—

120

C₂²×C₈

—

121

C₂²×C₈

C₂²×D₆₁

yes

characteristic

122

M₃₂

—

123

M₃₂

—

122

—

124

C₂×C₁₆

—

125

C₂×C₁₆

—

126

C₂×C₁₆

—

127

C₂×C₁₆

—

122

—

128

C₂×C₁₆

C₄×D₆₁

yes

—

129

#108

—

130

#108

—

131

#108

—

122

—

132

#108

—

122

—

133

#101

—

244

—

134

#100

—

244

—

135

#41

—

122

—

136

#41

—

122

—

137

C₂×Q₈

—

138

C₂×D₄

—

139

C₂×D₄

—

140

C₂²×C₄

—

141

C₂²×C₄

D₄×D₆₁

yes

characteristic

142

M₁₆

—

143

M₁₆

—

144

M₁₆

—

122

—

145

M₁₆

—

122

—

146

M₁₆

—

244

—

147

M₁₆

—

244

—

148

C₂×C₈

—

149

C₂×C₈

—

150

C₂×C₈

—

122

—

151

C₂×C₈

—

122

—

152

C₂×C₈

—

244

—

153

C₂×C₈

C₂×C₄×D₆₁

yes

characteristic

154

C₂×C₈

#11758222

yes

—

155

C₂×C₈

D₄×D₆₁

yes

—

156

C₂×C₈

Q₈×D₆₁

yes

characteristic

157

C₁₆

—

158

C₁₆

—

159

C₁₆

—

160

C₂³

—

161

C₂³

Q₁₆×D₆₁

yes

characteristic

162

Q₈

—

122

—

163

D₄

—

122

—

164

D₄

—

122

—

165

C₂×C₄

—

166

C₂×C₄

—

122

—

167

C₂×C₄

—

122

—

168

C₂×C₄

C₄⋊C₄×D₆₁

yes

characteristic

169

C₂×C₄

#14519750

yes

—

170

C₂×C₄

D₈×D₆₁

yes

characteristic

171

C₂×C₄

#14519798

yes

—

172

C₈

—

173

C₈

—

122

—

174

C₈

—

488

—

175

C₈

C₂²⋊C₄×D₆₁

yes

characteristic

176

C₈

#14519734

yes

characteristic

177

C₈

#14519755

yes

—

178

C₂²

—

179

C₂²

—

122

—

180

C₂²

—

244

—

181

C₂²

#107×D₆₁

yes

characteristic

182

C₂²

#16840002

yes

—

183

C₄

—

122

—

184

C₄

#16839621

yes

characteristic

185

C₄

#16839797

yes

characteristic

186

C₄

#16839997

yes

—

187

C₂

—

188

C₂

—

189

C₂

—

244

—

190

C₂

#21445847

yes

characteristic

	subgroup	G/H
center	C₄	#16839797
commutator	C₂×C₈×C₆₁	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₆₁
Fitting	C₆₁×#1385	C₂
socle	C₂×C₆₁	#1808
prime radical	C₂×#11758222	D₄
cosocle	C₂²×C₈×C₆₁	C₂³
hypercenter	#1385	D₆₁
hypocenter	C₆₁	#12840
LCS	G ▷ C₂×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1385

subgroup

G/H

center

C₄

#16839797

commutator

C₂×C₈×C₆₁

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₆₁

Fitting

C₆₁×#1385

C₂

C₂×C₆₁

D₄

C₂²×C₈×C₆₁

C₂³

D₆₁

C₆₁

G ▷ C₂×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1385

N	H ≅ G/N	type
2. C₂×#16839591	C₂	semidirect
4. #21443923	C₂	semidirect
5. C₆₁×#1385	C₂	semidirect
96. #1385	D₆₁	semidirect
112. C₆₁	#12840	semidirect

H ≅ G/N

type

2. C₂×#16839591

C₂

semidirect

4. #21443923

C₂

semidirect

5. C₆₁×#1385

C₂

semidirect

96. #1385

D₆₁

semidirect

112. C₆₁

#12840

semidirect

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

subgroup	index
C₂×#16840180	2
C₂×#16839591	2
#21445866	2
#21443923	2
C₆₁×#1385	2
#12840	61

subgroup

index

C₆₁×#1385

Subgroups — #29596205

Conjugacy classes of subgroups

Complements

Maximal subgroups