Group #35320561 — subgroups

Subgroups — #35320561

Order 21248.

358 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 358 conjugacy classes of subgroups — too many to draw. Only its 127 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×#19128118	C₂	1	1	yes	maximal, characteristic
2	#24909237	C₂	1	2	yes	maximal, complemented
3	#24907509	C₂	1	1	yes	maximal, characteristic
4	#24905560	C₂	1	2	yes	maximal, complemented
5	C₈₃×#1813	C₂	1	1	yes	maximal, characteristic
6	#19128118	—	2	4	no	—
7	C₂×#14581732	C₄	1	2	yes	complemented
8	C₂×#14581732	C₂²	1	1	yes	characteristic
9	C₂²×C₈₃⋊C₁₆	C₄	1	2	yes	complemented
10	C₂²×C₈₃⋊C₁₆	C₂²	1	1	yes	characteristic
11	#19127739	C₂²	1	2	yes	—
12	#19127675	—	2	1	no	—
13	#19127525	—	2	4	no	—
14	#19127458	—	2	1	no	—
15	C₂×#131×C₈₃	C₂²	1	1	yes	characteristic
16	C₂×#108×C₈₃	C₂²	1	2	yes	—
17	#413×C₈₃	—	2	1	no	—
18	#338×C₈₃	—	2	1	no	—
19	#14581751	—	2	2	no	—
20	#14581732	—	2	4	no	—
21	#14581732	D₄	1	4	yes	—
22	C₂×C₈₃⋊C₁₆	—	2	4	no	—
23	C₂×C₈₃⋊C₁₆	—	2	2	no	—
24	C₂×C₈₃⋊C₁₆	C₂×C₄	1	1	yes	characteristic
25	C₂×C₈₃⋊C₁₆	C₂×C₄	1	1	yes	characteristic
26	C₂×C₈₃⋊C₁₆	C₂×C₄	1	2	yes	—
27	#14581726	—	2	2	no	—
28	#14581724	—	2	2	no	—
29	#14581707	—	4	2	no	—
30	C₂×#41×C₈₃	D₄	1	1	yes	characteristic
31	#131×C₈₃	—	2	2	no	—
32	#131×C₈₃	—	2	2	no	—
33	C₂×M₁₆×C₈₃	—	2	2	no	—
34	C₂×M₁₆×C₈₃	D₄	1	1	yes	characteristic
35	C₂×M₁₆×C₈₃	Q₈	1	2	yes	—
36	C₂²×C₈×C₈₃	C₂×C₄	1	2	yes	—
37	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
38	#108×C₈₃	—	2	2	no	—
39	#108×C₈₃	—	2	2	no	—
40	#101×C₈₃	—	4	1	no	—
41	#100×C₈₃	—	4	1	no	—
42	C₄⋊Dic₈₃	—	4	2	no	—
43	C₂²⋊Dic₈₃	—	4	2	no	—
44	C₄×Dic₈₃	—	4	2	no	—
45	C₈₃⋊C₁₆	—	2	1	no	—
46	C₈₃⋊C₁₆	—	2	1	no	—
47	C₈₃⋊C₁₆	C₂²⋊C₄	1	4	yes	—
48	#41×C₈₃	—	2	2	no	—
49	#41×C₈₃	—	2	2	no	—
50	C₂×Q₈×C₈₃	SD₁₆	1	1	yes	characteristic
51	C₂×D₄×C₈₃	SD₁₆	1	2	yes	—
52	C₂×D₄×C₈₃	SD₁₆	1	1	yes	characteristic
53	C₂²×C₄×C₈₃	C₄⋊C₄	1	2	yes	—
54	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
55	M₁₆×C₈₃	—	2	2	no	—
56	M₁₆×C₈₃	—	2	4	no	—
57	M₁₆×C₈₃	—	4	2	no	—
58	M₁₆×C₈₃	—	4	2	no	—
59	C₂×C₈×C₈₃	—	2	2	no	—
60	C₂×C₈×C₈₃	—	2	2	no	—
61	C₂×C₈×C₈₃	—	4	2	no	—
62	C₂×C₈×C₈₃	C₄⋊C₄	1	2	yes	—
63	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₈×C₈₃	C₂×D₄	1	2	yes	—
65	C₂×C₈×C₈₃	C₂×Q₈	1	1	yes	characteristic
66	C₂×C₈×C₈₃	#41	1	2	yes	—
67	C₂³×C₈₃	#106	1	2	yes	—
68	C₂³×C₈₃	C₂×SD₁₆	1	1	yes	characteristic
69	Q₈×C₈₃	—	2	2	no	—
70	D₄×C₈₃	—	2	4	no	—
71	D₄×C₈₃	—	2	2	no	—
72	C₂×C₄×C₈₃	—	2	2	no	—
73	C₂×C₄×C₈₃	—	2	2	no	—
74	C₂×C₄×C₈₃	#106	1	2	yes	—
75	C₂×C₄×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₂×C₄×C₈₃	#121	1	2	yes	—
77	C₂×C₄×C₈₃	C₂×SD₁₆	1	1	yes	characteristic
78	C₂×C₄×C₈₃	#135	1	2	yes	—
79	C₂×Dic₈₃	—	8	2	no	—
80	C₈×C₈₃	—	2	2	no	—
81	C₈×C₈₃	—	8	2	no	—
82	C₈×C₈₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
83	C₈×C₈₃	#117	1	1	yes	characteristic
84	C₈×C₈₃	C₂²⋊Q₈	1	2	yes	—
85	C₂²×C₈₃	—	2	2	no	—
86	C₂²×C₈₃	—	4	1	no	—
87	C₂²×C₈₃	—	4	2	no	—
88	C₂²×C₈₃	C₂×#106	1	1	yes	characteristic
89	C₂²×C₈₃	#465	1	2	yes	—
90	C₄×C₈₃	—	2	2	no	—
91	C₄×C₈₃	#380	1	1	yes	characteristic
92	C₄×C₈₃	#427	1	1	yes	characteristic
93	C₄×C₈₃	#465	1	2	yes	—
94	Dic₈₃	—	16	2	no	—
95	#12832	—	83	1	no	maximal, Sylow
96	C₂×C₈₃	—	2	2	no	—
97	C₂×C₈₃	—	2	1	no	—
98	C₂×C₈₃	—	4	2	no	—
99	C₂×C₈₃	#1807	1	1	yes	characteristic
100	C₂×#504	—	83	1	no	—
101	#2155	—	83	2	no	—
102	#1813	D₈₃	1	1	yes	characteristic
103	#1812	—	83	1	no	—
104	#1382	—	83	2	no	—
105	C₈₃	#12832	1	1	yes	characteristic, complemented, Sylow
106	C₂×#131	C₂×D₈₃	1	1	yes	characteristic
107	#504	—	166	4	no	—
108	C₂×M₃₂	—	83	1	no	—
109	C₂×M₃₂	—	83	2	no	—
110	C₂²×C₁₆	—	83	2	no	—
111	C₂²×C₁₆	—	83	1	no	—
112	C₂×#108	C₂×D₈₃	1	2	yes	—
113	#427	—	83	2	no	—
114	#413	—	2	1	no	—
115	#410	—	166	1	no	—
116	#365	—	166	4	no	—
117	#338	—	2	1	no	—
118	H(3,ℤ/4)	—	166	1	no	—
119	C₂×#41	C₂²⋊D₈₃	1	1	yes	characteristic
120	#131	—	2	2	no	—
121	#131	—	2	2	no	—
122	C₂×M₁₆	—	2	2	no	—
123	C₂×M₁₆	Dic₁₆₆	1	2	yes	—
124	C₂×M₁₆	C₂²⋊D₈₃	1	1	yes	characteristic
125	C₂²×C₈	C₄×D₈₃	1	2	yes	—
126	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
127	#117	—	166	2	no	—
128	M₃₂	—	83	4	no	—
129	M₃₂	—	166	4	no	—
130	C₂×C₁₆	—	83	2	no	—
131	C₂×C₁₆	—	83	1	no	—
132	C₂×C₁₆	—	83	1	no	—
133	C₂×C₁₆	—	166	4	no	—
134	C₂×C₁₆	—	166	2	no	—
135	#108	—	2	2	no	—
136	#108	—	2	2	no	—
137	#107	—	166	2	no	—
138	#106	—	166	2	no	—
139	#101	—	4	1	no	—
140	#100	—	4	1	no	—
141	#99	—	332	2	no	—
142	#41	—	2	2	no	—
143	#41	—	2	2	no	—
144	C₂×Q₈	#13050578	1	1	yes	characteristic
145	C₂×D₄	#13050577	1	2	yes	—
146	C₂×D₄	#13050578	1	1	yes	characteristic
147	C₂²×C₄	Dic₈₃⋊C₄	1	2	yes	—
148	C₂²×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
149	M₁₆	—	2	2	no	—
150	M₁₆	—	2	4	no	—
151	M₁₆	—	4	2	no	—
152	M₁₆	—	4	2	no	—
153	C₂×C₈	—	2	2	no	—
154	C₂×C₈	—	2	2	no	—
155	C₂×C₈	—	4	2	no	—
156	C₂×C₈	Dic₈₃⋊C₄	1	2	yes	—
157	C₂×C₈	C₂×C₄×D₈₃	1	1	yes	characteristic
158	C₂×C₈	D₄×D₈₃	1	2	yes	—
159	C₂×C₈	C₂×Dic₁₆₆	1	1	yes	characteristic
160	C₂×C₈	#13050589	1	2	yes	—
161	C₄⋊C₄	—	332	2	no	—
162	C₂²⋊C₄	—	332	2	no	—
163	C₄²	—	332	2	no	—
164	C₁₆	—	83	4	no	—
165	C₁₆	—	166	1	no	—
166	C₁₆	—	166	1	no	—
167	C₂³	#14581723	1	2	yes	—
168	C₂³	C₂×#13050577	1	1	yes	characteristic
169	Q₈	—	2	2	no	—
170	D₄	—	2	4	no	—
171	D₄	—	2	2	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	664	2	no	—
175	C₂×C₄	#14581723	1	2	yes	—
176	C₂×C₄	C₂×Dic₈₃⋊C₄	1	1	yes	characteristic
177	C₂×C₄	#14581764	1	2	yes	—
178	C₂×C₄	C₂×#13050578	1	1	yes	characteristic
179	C₂×C₄	#14581812	1	2	yes	—
180	C₈	—	2	2	no	—
181	C₈	—	8	2	no	—
182	C₈	C₂²⋊C₄×D₈₃	1	1	yes	characteristic
183	C₈	#14581748	1	1	yes	characteristic
184	C₈	#14581769	1	2	yes	—
185	C₂²	—	2	2	no	—
186	C₂²	—	4	1	no	—
187	C₂²	—	4	2	no	—
188	C₂²	C₂×#14581723	1	1	yes	characteristic
189	C₂²	#19127932	1	2	yes	—
190	C₄	—	2	2	no	—
191	C₄	—	1328	2	no	—
192	C₄	#19127559	1	1	yes	characteristic
193	C₄	#19127737	1	1	yes	characteristic
194	C₄	#19127928	1	2	yes	—
195	C₂	—	2	2	no	—
196	C₂	—	2	1	no	—
197	C₂	—	4	2	no	—
198	C₂	#24907486	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19128118

C₂

yes

maximal, characteristic

#24909237

C₂

yes

maximal, complemented

#24907509

C₂

yes

maximal, characteristic

#24905560

C₂

yes

maximal, complemented

C₈₃×#1813

C₂

yes

maximal, characteristic

#19128118

—

C₂×#14581732

C₄

yes

complemented

C₂×#14581732

C₂²

yes

characteristic

C₂²×C₈₃⋊C₁₆

C₄

yes

complemented

C₂²×C₈₃⋊C₁₆

C₂²

yes

characteristic

#19127739

C₂²

yes

—

#19127675

—

#19127525

—

#19127458

—

C₂×#131×C₈₃

C₂²

yes

characteristic

C₂×#108×C₈₃

C₂²

yes

—

#413×C₈₃

—

#338×C₈₃

—

#14581751

—

#14581732

—

#14581732

D₄

yes

—

C₂×C₈₃⋊C₁₆

—

C₂×C₈₃⋊C₁₆

—

C₂×C₈₃⋊C₁₆

C₂×C₄

yes

characteristic

C₂×C₈₃⋊C₁₆

C₂×C₄

yes

characteristic

C₂×C₈₃⋊C₁₆

C₂×C₄

yes

—

#14581726

—

#14581724

—

#14581707

—

C₂×#41×C₈₃

D₄

yes

characteristic

#131×C₈₃

—

#131×C₈₃

—

C₂×M₁₆×C₈₃

—

C₂×M₁₆×C₈₃

D₄

yes

characteristic

C₂×M₁₆×C₈₃

Q₈

yes

—

C₂²×C₈×C₈₃

C₂×C₄

yes

—

C₂²×C₈×C₈₃

C₂³

yes

characteristic

#108×C₈₃

—

#108×C₈₃

—

#101×C₈₃

—

#100×C₈₃

—

C₄⋊Dic₈₃

—

C₂²⋊Dic₈₃

—

C₄×Dic₈₃

—

C₈₃⋊C₁₆

—

C₈₃⋊C₁₆

—

C₈₃⋊C₁₆

C₂²⋊C₄

yes

—

#41×C₈₃

—

#41×C₈₃

—

C₂×Q₈×C₈₃

SD₁₆

yes

characteristic

C₂×D₄×C₈₃

SD₁₆

yes

—

C₂×D₄×C₈₃

SD₁₆

yes

characteristic

C₂²×C₄×C₈₃

C₄⋊C₄

yes

—

C₂²×C₄×C₈₃

C₂×D₄

yes

characteristic

M₁₆×C₈₃

—

M₁₆×C₈₃

—

M₁₆×C₈₃

—

M₁₆×C₈₃

—

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

C₄⋊C₄

yes

—

C₂×C₈×C₈₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₈₃

C₂×D₄

yes

—

C₂×C₈×C₈₃

C₂×Q₈

yes

characteristic

C₂×C₈×C₈₃

#41

yes

—

C₂³×C₈₃

#106

yes

—

C₂³×C₈₃

C₂×SD₁₆

yes

characteristic

Q₈×C₈₃

—

D₄×C₈₃

—

D₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

#106

yes

—

C₂×C₄×C₈₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₈₃

#121

yes

—

C₂×C₄×C₈₃

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₈₃

#135

yes

—

C₂×Dic₈₃

—

C₈×C₈₃

—

C₈×C₈₃

—

C₈×C₈₃

C₂×C₂²⋊C₄

yes

characteristic

C₈×C₈₃

#117

yes

characteristic

C₈×C₈₃

C₂²⋊Q₈

yes

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

C₂×#106

yes

characteristic

C₂²×C₈₃

#465

yes

—

C₄×C₈₃

—

C₄×C₈₃

yes

C₄×C₈₃

yes

C₄×C₈₃

yes

—

Dic₈₃

—

#12832

—

maximal, Sylow

C₂×C₈₃

—

C₂×C₈₃

—

C₂×C₈₃

—

C₂×C₈₃

#1807

yes

characteristic

100

C₂×#504

—

101

#2155

—

102

#1813

D₈₃

yes

characteristic

103

#1812

—

104

#1382

—

105

C₈₃

#12832

yes

characteristic, complemented, Sylow

106

C₂×#131

C₂×D₈₃

yes

characteristic

107

#504

—

166

—

108

C₂×M₃₂

—

109

C₂×M₃₂

—

110

C₂²×C₁₆

—

111

C₂²×C₁₆

—

112

C₂×#108

C₂×D₈₃

yes

—

113

#427

—

114

#413

—

115

#410

—

166

—

116

#365

—

166

—

117

#338

—

118

H(3,ℤ/4)

—

166

—

119

C₂×#41

C₂²⋊D₈₃

yes

characteristic

120

#131

—

121

#131

—

122

C₂×M₁₆

—

123

C₂×M₁₆

Dic₁₆₆

yes

—

124

C₂×M₁₆

C₂²⋊D₈₃

yes

characteristic

125

C₂²×C₈

C₄×D₈₃

yes

—

126

C₂²×C₈

C₂²×D₈₃

yes

characteristic

127

#117

—

166

—

128

M₃₂

—

129

M₃₂

—

166

—

130

C₂×C₁₆

—

131

C₂×C₁₆

—

132

C₂×C₁₆

—

133

C₂×C₁₆

—

166

—

134

C₂×C₁₆

—

166

—

135

#108

—

136

#108

—

137

#107

—

166

—

138

#106

—

166

—

139

#101

—

140

#100

—

141

#99

—

332

—

142

#41

—

143

#41

—

144

C₂×Q₈

#13050578

yes

characteristic

145

C₂×D₄

#13050577

yes

—

146

C₂×D₄

#13050578

yes

characteristic

147

C₂²×C₄

Dic₈₃⋊C₄

yes

—

148

C₂²×C₄

C₂×C₂²⋊D₈₃

yes

characteristic

149

M₁₆

—

150

M₁₆

—

151

M₁₆

—

152

M₁₆

—

153

C₂×C₈

—

154

C₂×C₈

—

155

C₂×C₈

—

156

C₂×C₈

Dic₈₃⋊C₄

yes

—

157

C₂×C₈

C₂×C₄×D₈₃

yes

characteristic

158

C₂×C₈

D₄×D₈₃

yes

—

159

C₂×C₈

C₂×Dic₁₆₆

yes

characteristic

160

C₂×C₈

#13050589

yes

—

161

C₄⋊C₄

—

332

—

162

C₂²⋊C₄

—

332

—

163

C₄²

—

332

—

164

C₁₆

—

165

C₁₆

—

166

—

166

C₁₆

—

166

—

167

C₂³

#14581723

yes

—

168

C₂³

C₂×#13050577

yes

characteristic

169

Q₈

—

170

D₄

—

171

D₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

664

—

175

C₂×C₄

#14581723

yes

—

176

C₂×C₄

C₂×Dic₈₃⋊C₄

yes

characteristic

177

C₂×C₄

#14581764

yes

—

178

C₂×C₄

C₂×#13050578

yes

characteristic

179

C₂×C₄

#14581812

yes

—

180

C₈

—

181

C₈

—

182

C₈

C₂²⋊C₄×D₈₃

yes

characteristic

183

C₈

#14581748

yes

characteristic

184

C₈

#14581769

yes

—

185

C₂²

—

186

C₂²

—

187

C₂²

—

188

C₂²

C₂×#14581723

yes

characteristic

189

C₂²

#19127932

yes

—

190

C₄

—

191

C₄

—

1328

—

192

C₄

#19127559

yes

characteristic

193

C₄

#19127737

yes

characteristic

194

C₄

#19127928

yes

—

195

C₂

—

196

C₂

—

197

C₂

—

198

C₂

#24907486

yes

characteristic

	subgroup	G/H
center	C₄	#19127737
commutator	C₂×C₈×C₈₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₈₃
Fitting	C₈₃×#1813	C₂
socle	C₂×C₈₃	#1807
prime radical	C₂×#41×C₈₃	D₄
cosocle	C₂²×C₈×C₈₃	C₂³
hypercenter	#1813	D₈₃
hypocenter	C₈₃	#12832
LCS	G ▷ C₂×C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ C₂×#131 ◁ #1813

subgroup

G/H

center

C₄

#19127737

commutator

C₂×C₈×C₈₃

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₈₃

Fitting

C₈₃×#1813

C₂

socle

C₂×C₈₃

#1807

prime radical

C₂×#41×C₈₃

D₄

cosocle

C₂²×C₈×C₈₃

C₂³

D₈₃

C₈₃

G ▷ C₂×C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ C₂×#131 ◁ #1813

N	H ≅ G/N	type
2. #24909237	C₂	semidirect
4. #24905560	C₂	semidirect
7. C₂×#14581732	C₄	semidirect
9. C₂²×C₈₃⋊C₁₆	C₄	semidirect
105. C₈₃	#12832	semidirect

H ≅ G/N

type

2. #24909237

C₂

semidirect

4. #24905560

C₂

semidirect

7. C₂×#14581732

C₄

semidirect

9. C₂²×C₈₃⋊C₁₆

C₄

semidirect

105. C₈₃

#12832

semidirect

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

subgroup	index
C₂×#19128118	2
#24909237	2
#24907509	2
#24905560	2
C₈₃×#1813	2
#12832	83

subgroup

index

C₈₃×#1813

Subgroups — #35320561

Conjugacy classes of subgroups

Complements

Maximal subgroups