Group #35323956 — subgroups

Subgroups — #35323956

Order 21248.

378 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 95 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×Dic₆₆₄	C₂	1	1	yes	maximal, characteristic
2	C₂×#19127518	C₂	1	2	yes	maximal
3	#24908378	C₂	1	2	yes	maximal
4	#24905557	C₂	1	1	yes	maximal, characteristic
5	C₈₃×#2151	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₃₃₂	C₂²	1	2	yes	—
7	C₂×Dic₆₆₄	—	2	2	no	—
8	C₂×Dic₆₆₄	—	2	4	no	—
9	C₂×#14581728	C₂²	1	2	yes	—
10	#19127709	—	2	2	no	—
11	C₂×#14581710	—	2	2	no	—
12	#19127518	—	2	4	no	—
13	C₂×M₃₂×C₈₃	C₂²	1	1	yes	characteristic
14	C₂²×C₁₆×C₈₃	C₂²	1	1	yes	characteristic
15	#405×C₈₃	C₂²	1	1	yes	characteristic
16	C₄⋊C₁₆×C₈₃	—	2	2	no	—
17	C₂²×Dic₁₆₆	—	2	2	no	—
18	C₂×Dic₃₃₂	—	2	2	no	—
19	C₂×Dic₃₃₂	—	2	2	no	—
20	C₂×Dic₃₃₂	—	2	2	no	—
21	C₂×Dic₃₃₂	—	2	2	no	—
22	C₂×Dic₃₃₂	—	2	2	no	—
23	C₂×Dic₃₃₂	D₄	1	4	yes	—
24	C₂×#13050574	—	2	2	no	—
25	Dic₆₆₄	—	2	8	no	—
26	#14581728	—	2	2	no	—
27	#14581728	—	2	2	no	—
28	#14581715	—	4	4	no	—
29	#14581710	—	4	4	no	—
30	C₂×M₁₆×C₈₃	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
32	#117×C₈₃	D₄	1	1	yes	characteristic
33	M₃₂×C₈₃	—	2	2	no	—
34	C₂×C₁₆×C₈₃	—	2	2	no	—
35	C₂×C₁₆×C₈₃	—	2	1	no	—
36	C₂×C₁₆×C₈₃	D₄	1	2	yes	—
37	#97×C₈₃	—	2	1	no	—
38	C₄×C₈×C₈₃	—	2	1	no	—
39	C₂×Dic₁₆₆	—	2	4	no	—
40	C₂×Dic₁₆₆	—	4	2	no	—
41	C₂×Dic₁₆₆	—	4	2	no	—
42	C₂×Dic₁₆₆	—	4	2	no	—
43	C₂×Dic₁₆₆	—	4	2	no	—
44	C₂×Dic₁₆₆	—	4	2	no	—
45	C₂²×Dic₈₃	—	4	2	no	—
46	Dic₃₃₂	—	2	4	no	—
47	Dic₃₃₂	—	2	4	no	—
48	Dic₃₃₂	—	4	2	no	—
49	Dic₃₃₂	—	4	2	no	—
50	#13050574	—	4	2	no	—
51	#13050574	—	4	2	no	—
52	C₂×C₈₃⋊C₈	—	4	2	no	—
53	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
54	M₁₆×C₈₃	—	2	2	no	—
55	C₂×C₈×C₈₃	—	2	1	no	—
56	C₂×C₈×C₈₃	C₂×D₄	1	2	yes	—
57	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
58	C₂×C₈×C₈₃	#41	1	1	yes	characteristic
59	C₂×C₈×C₈₃	#41	1	2	yes	—
60	C₄⋊C₄×C₈₃	D₈	1	2	yes	—
61	C₂²⋊C₄×C₈₃	—	2	1	no	—
62	C₄²×C₈₃	—	2	1	no	—
63	C₁₆×C₈₃	—	2	2	no	—
64	C₁₆×C₈₃	—	4	1	no	—
65	C₂³×C₈₃	#135	1	1	yes	characteristic
66	C₂×C₄×C₈₃	—	2	2	no	—
67	C₂×C₄×C₈₃	#121	1	1	yes	characteristic
68	C₂×C₄×C₈₃	#121	1	2	yes	—
69	C₂×C₄×C₈₃	C₂×D₈	1	1	yes	characteristic
70	C₂×C₄×C₈₃	AGL(1,ℤ/8)	1	2	yes	—
71	C₂×Dic₈₃	—	4	4	no	—
72	C₂×Dic₈₃	—	8	2	no	—
73	C₂×Dic₈₃	—	8	2	no	—
74	Dic₁₆₆	—	4	4	no	—
75	Dic₁₆₆	—	4	4	no	—
76	Dic₁₆₆	—	8	2	no	—
77	Dic₁₆₆	—	8	2	no	—
78	C₈×C₈₃	—	4	1	no	—
79	C₈×C₈₃	#120	1	1	yes	characteristic
80	C₈×C₈₃	#123	1	2	yes	—
81	C₈×C₈₃	#124	1	1	yes	characteristic
82	C₈₃⋊C₈	—	8	2	no	—
83	C₂²×C₈₃	—	2	2	no	—
84	C₂²×C₈₃	#463	1	2	yes	—
85	C₂²×C₈₃	#466	1	1	yes	characteristic
86	C₄×C₈₃	—	4	2	no	—
87	C₄×C₈₃	#394	1	1	yes	characteristic
88	C₄×C₈₃	#459	1	2	yes	—
89	C₄×C₈₃	#469	1	1	yes	characteristic
90	Dic₈₃	—	8	4	no	—
91	#13441	—	83	1	no	maximal, Sylow
92	C₂×C₈₃	—	2	2	no	—
93	C₂×C₈₃	#2054	1	1	yes	characteristic
94	C₂×C₈₃	#2222	1	2	yes	—
95	C₂²×Q₃₂	—	83	1	no	—
96	#2151	D₈₃	1	1	yes	characteristic
97	C₂×#362	—	83	2	no	—
98	#1979	—	83	2	no	—
99	#1381	—	83	1	no	—
100	C₈₃	#13441	1	1	yes	characteristic, complemented, Sylow
101	C₂²×Q₁₆	—	83	2	no	—
102	C₂×Q₃₂	—	166	4	no	—
103	C₂×Q₃₂	—	166	2	no	—
104	C₂×M₃₂	C₂×D₈₃	1	1	yes	characteristic
105	C₂²×C₁₆	C₂×D₈₃	1	1	yes	characteristic
106	C₂×#108	—	83	2	no	—
107	#419	—	166	2	no	—
108	C₂×#101	—	166	2	no	—
109	#405	C₂×D₈₃	1	1	yes	characteristic
110	C₄⋊C₁₆	—	2	2	no	—
111	#362	—	166	4	no	—
112	C₂²×Q₈	—	166	2	no	—
113	C₂×Q₁₆	—	83	4	no	—
114	C₂×Q₁₆	—	166	2	no	—
115	C₂×Q₁₆	—	166	2	no	—
116	C₂×Q₁₆	—	166	2	no	—
117	C₂×Q₁₆	—	166	2	no	—
118	C₂×Q₁₆	—	166	2	no	—
119	C₂×M₁₆	—	166	2	no	—
120	C₂×M₁₆	D₃₃₂	1	1	yes	characteristic
121	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
122	#117	D₃₃₂	1	1	yes	characteristic
123	Q₃₂	—	166	8	no	—
124	M₃₂	—	2	2	no	—
125	C₂×C₁₆	—	2	2	no	—
126	C₂×C₁₆	—	2	1	no	—
127	C₂×C₁₆	C₂²⋊D₈₃	1	2	yes	—
128	#108	—	166	2	no	—
129	#108	—	166	2	no	—
130	#103	—	332	4	no	—
131	#101	—	332	4	no	—
132	#97	—	2	1	no	—
133	C₄×C₈	—	2	1	no	—
134	C₂×Q₈	—	332	2	no	—
135	C₂×Q₈	—	332	2	no	—
136	C₂×Q₈	—	332	2	no	—
137	C₂×Q₈	—	332	2	no	—
138	C₂×Q₈	—	332	2	no	—
139	C₂×Q₈	—	166	4	no	—
140	C₂²×C₄	—	332	2	no	—
141	C₂²×C₄	C₂×D₃₃₂	1	1	yes	characteristic
142	Q₁₆	—	332	2	no	—
143	Q₁₆	—	332	2	no	—
144	Q₁₆	—	166	4	no	—
145	Q₁₆	—	166	4	no	—
146	M₁₆	—	2	2	no	—
147	M₁₆	—	332	2	no	—
148	M₁₆	—	332	2	no	—
149	C₂×C₈	—	2	1	no	—
150	C₂×C₈	—	332	2	no	—
151	C₂×C₈	D₄×D₈₃	1	2	yes	—
152	C₂×C₈	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
153	C₂×C₈	#13050590	1	1	yes	characteristic
154	C₂×C₈	#13050591	1	2	yes	—
155	C₄⋊C₄	D₆₆₄	1	2	yes	—
156	C₂²⋊C₄	—	2	1	no	—
157	C₄²	—	2	1	no	—
158	C₁₆	—	2	2	no	—
159	C₁₆	—	4	1	no	—
160	C₂³	#14581814	1	1	yes	characteristic
161	Q₈	—	332	4	no	—
162	Q₈	—	332	4	no	—
163	Q₈	—	664	2	no	—
164	Q₈	—	664	2	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	332	4	no	—
167	C₂×C₄	—	664	2	no	—
168	C₂×C₄	—	664	2	no	—
169	C₂×C₄	#14581766	1	1	yes	characteristic
170	C₂×C₄	#14581767	1	2	yes	—
171	C₂×C₄	C₂×D₆₆₄	1	1	yes	characteristic
172	C₂×C₄	#14581820	1	2	yes	—
173	C₈	—	4	1	no	—
174	C₈	—	664	2	no	—
175	C₈	#14581760	1	1	yes	characteristic
176	C₈	#14581777	1	2	yes	—
177	C₈	#14581781	1	1	yes	characteristic
178	C₂²	—	2	2	no	—
179	C₂²	#19127917	1	2	yes	—
180	C₂²	#19127937	1	1	yes	characteristic
181	C₄	—	4	2	no	—
182	C₄	—	664	4	no	—
183	C₄	#19127618	1	1	yes	characteristic
184	C₄	#19127897	1	2	yes	—
185	C₄	#19127954	1	1	yes	characteristic
186	C₂	—	2	2	no	—
187	C₂	#24908774	1	1	yes	characteristic
188	C₂	#24909588	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₆₆₄

C₂

yes

maximal, characteristic

C₂×#19127518

C₂

yes

maximal

#24908378

C₂

yes

maximal

#24905557

C₂

yes

maximal, characteristic

C₈₃×#2151

C₂

yes

maximal, characteristic

C₂²×Dic₃₃₂

C₂²

yes

—

C₂×Dic₆₆₄

—

C₂×Dic₆₆₄

—

C₂×#14581728

C₂²

yes

—

#19127709

—

C₂×#14581710

—

#19127518

—

C₂×M₃₂×C₈₃

C₂²

yes

characteristic

C₂²×C₁₆×C₈₃

C₂²

yes

characteristic

#405×C₈₃

C₂²

yes

characteristic

C₄⋊C₁₆×C₈₃

—

C₂²×Dic₁₆₆

—

C₂×Dic₃₃₂

—

C₂×Dic₃₃₂

—

C₂×Dic₃₃₂

—

C₂×Dic₃₃₂

—

C₂×Dic₃₃₂

—

C₂×Dic₃₃₂

D₄

yes

—

C₂×#13050574

—

Dic₆₆₄

—

—

—

—

—

C₂×M₁₆×C₈₃

D₄

yes

characteristic

C₂²×C₈×C₈₃

C₂³

yes

characteristic

#117×C₈₃

D₄

yes

characteristic

M₃₂×C₈₃

—

C₂×C₁₆×C₈₃

—

C₂×C₁₆×C₈₃

—

C₂×C₁₆×C₈₃

D₄

yes

—

#97×C₈₃

—

C₄×C₈×C₈₃

—

C₂×Dic₁₆₆

—

C₂×Dic₁₆₆

—

C₂×Dic₁₆₆

—

C₂×Dic₁₆₆

—

C₂×Dic₁₆₆

—

C₂×Dic₁₆₆

—

C₂²×Dic₈₃

—

Dic₃₃₂

—

Dic₃₃₂

—

Dic₃₃₂

—

Dic₃₃₂

—

#13050574

—

#13050574

—

C₂×C₈₃⋊C₈

—

C₂²×C₄×C₈₃

C₂×D₄

yes

characteristic

M₁₆×C₈₃

—

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

C₂×D₄

yes

—

C₂×C₈×C₈₃

C₂×D₄

yes

C₂×C₈×C₈₃

yes

C₂×C₈×C₈₃

yes

—

C₄⋊C₄×C₈₃

D₈

yes

—

C₂²⋊C₄×C₈₃

—

C₄²×C₈₃

—

C₁₆×C₈₃

—

C₁₆×C₈₃

—

C₂³×C₈₃

#135

yes

characteristic

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

#121

yes

characteristic

C₂×C₄×C₈₃

#121

yes

—

C₂×C₄×C₈₃

C₂×D₈

yes

characteristic

C₂×C₄×C₈₃

AGL(1,ℤ/8)

yes

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

Dic₁₆₆

—

Dic₁₆₆

—

Dic₁₆₆

—

Dic₁₆₆

—

C₈×C₈₃

—

C₈×C₈₃

#120

yes

characteristic

C₈×C₈₃

#123

yes

—

C₈×C₈₃

#124

yes

characteristic

C₈₃⋊C₈

—

C₂²×C₈₃

—

C₂²×C₈₃

#463

yes

—

C₂²×C₈₃

#466

yes

characteristic

C₄×C₈₃

—

C₄×C₈₃

#394

yes

characteristic

C₄×C₈₃

#459

yes

—

C₄×C₈₃

#469

yes

characteristic

Dic₈₃

—

#13441

—

maximal, Sylow

C₂×C₈₃

—

C₂×C₈₃

#2054

yes

characteristic

C₂×C₈₃

#2222

yes

—

C₂²×Q₃₂

—

#2151

D₈₃

yes

—

—

—

100

C₈₃

#13441

yes

characteristic, complemented, Sylow

101

C₂²×Q₁₆

—

102

C₂×Q₃₂

—

166

—

103

C₂×Q₃₂

—

166

—

104

C₂×M₃₂

C₂×D₈₃

yes

characteristic

105

C₂²×C₁₆

C₂×D₈₃

yes

characteristic

106

C₂×#108

—

107

#419

—

166

—

108

C₂×#101

—

166

—

109

#405

C₂×D₈₃

yes

characteristic

110

C₄⋊C₁₆

—

111

#362

—

166

—

112

C₂²×Q₈

—

166

—

113

C₂×Q₁₆

—

114

C₂×Q₁₆

—

166

—

115

C₂×Q₁₆

—

166

—

116

C₂×Q₁₆

—

166

—

117

C₂×Q₁₆

—

166

—

118

C₂×Q₁₆

—

166

—

119

C₂×M₁₆

—

166

—

120

C₂×M₁₆

D₃₃₂

yes

characteristic

121

C₂²×C₈

C₂²×D₈₃

yes

characteristic

122

#117

D₃₃₂

yes

characteristic

123

Q₃₂

—

166

—

124

M₃₂

—

125

C₂×C₁₆

—

126

C₂×C₁₆

—

127

C₂×C₁₆

C₂²⋊D₈₃

yes

—

128

#108

—

166

—

129

#108

—

166

—

130

#103

—

332

—

131

#101

—

332

—

132

#97

—

133

C₄×C₈

—

134

C₂×Q₈

—

332

—

135

C₂×Q₈

—

332

—

136

C₂×Q₈

—

332

—

137

C₂×Q₈

—

332

—

138

C₂×Q₈

—

332

—

139

C₂×Q₈

—

166

—

140

C₂²×C₄

—

332

—

141

C₂²×C₄

C₂×D₃₃₂

yes

characteristic

142

Q₁₆

—

332

—

143

Q₁₆

—

332

—

144

Q₁₆

—

166

—

145

Q₁₆

—

166

—

146

M₁₆

—

147

M₁₆

—

332

—

148

M₁₆

—

332

—

149

C₂×C₈

—

150

C₂×C₈

—

332

—

151

C₂×C₈

D₄×D₈₃

yes

—

152

C₂×C₈

C₂×C₂²⋊D₈₃

yes

characteristic

153

C₂×C₈

#13050590

yes

characteristic

154

C₂×C₈

#13050591

yes

—

155

C₄⋊C₄

D₆₆₄

yes

—

156

C₂²⋊C₄

—

157

C₄²

—

158

C₁₆

—

159

C₁₆

—

160

C₂³

#14581814

yes

characteristic

161

Q₈

—

332

—

162

Q₈

—

332

—

163

Q₈

—

664

—

164

Q₈

—

664

—

165

C₂×C₄

—

166

C₂×C₄

—

332

—

167

C₂×C₄

—

664

—

168

C₂×C₄

—

664

—

169

C₂×C₄

#14581766

yes

characteristic

170

C₂×C₄

#14581767

yes

—

171

C₂×C₄

C₂×D₆₆₄

yes

characteristic

172

C₂×C₄

#14581820

yes

—

173

C₈

—

174

C₈

—

664

—

175

C₈

#14581760

yes

characteristic

176

C₈

#14581777

yes

—

177

C₈

#14581781

yes

characteristic

178

C₂²

—

179

C₂²

#19127917

yes

—

180

C₂²

#19127937

yes

characteristic

181

C₄

—

182

C₄

—

664

—

183

C₄

#19127618

yes

characteristic

184

C₄

#19127897

yes

—

185

C₄

#19127954

yes

characteristic

186

C₂

—

187

C₂

#24908774

yes

characteristic

188

C₂

#24909588

yes

—

	subgroup	G/H
center	C₂²	#19127937
commutator, cosocle	C₂²×C₈×C₈₃	C₂³
Frattini	C₂²×C₈	C₂²×D₈₃
Fitting	C₈₃×#2151	C₂
socle	C₂²×C₈₃	#466
prime radical	C₂³×C₈₃	#135
hypercenter	#2151	D₈₃
hypocenter	C₈₃	#13441
LCS	G ▷ C₂²×C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #405 ◁ #2151

subgroup

G/H

center

C₂²

#19127937

commutator, cosocle

C₂²×C₈×C₈₃

C₂³

Frattini

C₂²×C₈

C₂²×D₈₃

Fitting

C₈₃×#2151

C₂

C₂²×C₈₃

C₂³×C₈₃

D₈₃

C₈₃

G ▷ C₂²×C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #405 ◁ #2151

N	H ≅ G/N	type
100. C₈₃	#13441	semidirect

H ≅ G/N

type

100. C₈₃

#13441

semidirect

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

subgroup	index
C₂²×Dic₆₆₄	2
C₂×#19127518	2
#24908378	2
#24905557	2
C₈₃×#2151	2
#13441	83

subgroup

index

C₂²×Dic₆₆₄

C₈₃×#2151

Subgroups — #35323956

Conjugacy classes of subgroups

Complements

Maximal subgroups