Group #35238352 — subgroups

Subgroups — C₈₃×#10528 #35238352

Order 21248.

342 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 90 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	C₈₃×#3255	C₂	1	1	yes	maximal, characteristic
2	C₈₃×#1849	C₂	1	1	yes	maximal, characteristic
3	C₈₃×#1559	C₂	1	2	yes	maximal
4	C₈₃×#1531	C₂	1	2	yes	maximal
5	C₈₃×#1307	C₂	1	1	yes	maximal, characteristic
6	#533×C₈₃	C₂²	1	2	yes	—
7	#475×C₈₃	—	2	2	no	—
8	#470×C₈₃	—	2	2	no	—
9	#462×C₈₃	—	2	2	no	—
10	#432×C₈₃	C₂²	1	2	yes	—
11	C₂×#108×C₈₃	—	2	2	no	—
12	#423×C₈₃	C₂²	1	1	yes	characteristic
13	#423×C₈₃	C₂²	1	1	yes	characteristic
14	C₂×#97×C₈₃	C₂²	1	1	yes	characteristic
15	#333×C₈₃	—	2	2	no	—
16	#332×C₈₃	—	2	2	no	—
17	#330×C₈₃	—	2	4	no	—
18	C₂×Q₁₆×C₈₃	—	2	2	no	—
19	C₂×M₁₆×C₈₃	—	2	1	no	—
20	C₂×M₁₆×C₈₃	D₄	1	2	yes	—
21	C₂²×C₈×C₈₃	—	2	1	no	—
22	C₄⋊Q₈×C₈₃	D₄	1	2	yes	—
23	C₄⋊Q₈×C₈₃	D₄	1	2	yes	—
24	#125×C₈₃	—	2	2	no	—
25	C₂²⋊Q₈×C₈₃	—	2	2	no	—
26	C₂²⋊Q₈×C₈₃	—	2	2	no	—
27	C₄×Q₈×C₈₃	—	2	2	no	—
28	C₄×Q₈×C₈₃	—	2	2	no	—
29	#117×C₈₃	—	2	2	no	—
30	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
31	#108×C₈₃	—	4	4	no	—
32	#106×C₈₃	—	2	2	no	—
33	C₄⋊C₈×C₈₃	—	2	2	no	—
34	C₄⋊C₈×C₈₃	—	2	2	no	—
35	C₄⋊C₈×C₈₃	—	2	2	no	—
36	#103×C₈₃	—	2	4	no	—
37	C₂²⋊C₈×C₈₃	—	2	2	no	—
38	#97×C₈₃	—	2	2	no	—
39	#97×C₈₃	—	2	1	no	—
40	#97×C₈₃	D₄	1	2	yes	—
41	C₂×Q₈×C₈₃	—	2	2	no	—
42	C₂×Q₈×C₈₃	—	2	2	no	—
43	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
44	C₂²×C₄×C₈₃	#41	1	2	yes	—
45	Q₁₆×C₈₃	—	8	2	no	—
46	M₁₆×C₈₃	—	4	2	no	—
47	M₁₆×C₈₃	—	4	2	no	—
48	C₂×C₈×C₈₃	—	2	2	no	—
49	C₂×C₈×C₈₃	—	2	2	no	—
50	C₂×C₈×C₈₃	—	2	2	no	—
51	C₂×C₈×C₈₃	—	2	2	no	—
52	C₂×C₈×C₈₃	—	4	1	no	—
53	C₂×C₈×C₈₃	—	4	2	no	—
54	C₄⋊C₄×C₈₃	—	2	2	no	—
55	C₄⋊C₄×C₈₃	—	2	2	no	—
56	C₄⋊C₄×C₈₃	—	2	2	no	—
57	C₄⋊C₄×C₈₃	—	2	2	no	—
58	C₄⋊C₄×C₈₃	—	4	2	no	—
59	C₄⋊C₄×C₈₃	—	4	2	no	—
60	C₂²⋊C₄×C₈₃	—	4	2	no	—
61	C₄²×C₈₃	—	4	2	no	—
62	C₄²×C₈₃	C₂×D₄	1	2	yes	—
63	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
64	C₄²×C₈₃	#41	1	1	yes	characteristic
65	C₂³×C₈₃	#124	1	1	yes	characteristic
66	Q₈×C₈₃	—	4	2	no	—
67	Q₈×C₈₃	—	4	2	no	—
68	Q₈×C₈₃	—	8	2	no	—
69	C₂×C₄×C₈₃	—	2	1	no	—
70	C₂×C₄×C₈₃	—	2	1	no	—
71	C₂×C₄×C₈₃	—	4	2	no	—
72	C₂×C₄×C₈₃	—	4	2	no	—
73	C₂×C₄×C₈₃	—	4	2	no	—
74	C₂×C₄×C₈₃	#120	1	1	yes	characteristic
75	C₂×C₄×C₈₃	#121	1	2	yes	—
76	C₂×C₄×C₈₃	#121	1	1	yes	characteristic
77	C₂×C₄×C₈₃	#123	1	2	yes	—
78	C₈×C₈₃	—	2	4	no	—
79	C₈×C₈₃	—	4	2	no	—
80	C₈×C₈₃	—	8	2	no	—
81	C₈×C₈₃	—	8	1	no	—
82	C₂²×C₈₃	—	2	2	no	—
83	C₂²×C₈₃	—	2	1	no	—
84	C₂²×C₈₃	#394	1	1	yes	characteristic
85	C₄×C₈₃	—	2	2	no	—
86	C₄×C₈₃	—	2	2	no	—
87	C₄×C₈₃	—	2	1	no	—
88	C₄×C₈₃	—	8	2	no	—
89	C₄×C₈₃	—	8	2	no	—
90	C₄×C₈₃	#453	1	1	yes	characteristic
91	C₄×C₈₃	#456	1	1	yes	characteristic
92	#10528	C₈₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
93	C₂×C₈₃	—	4	1	no	—
94	C₂×C₈₃	#2039	1	1	yes	characteristic
95	C₂×C₈₃	#2049	1	2	yes	—
96	#3255	C₂×C₈₃	1	1	yes	characteristic
97	#1849	C₂×C₈₃	1	1	yes	characteristic
98	#1559	C₂×C₈₃	1	2	yes	—
99	#1531	C₂×C₈₃	1	2	yes	—
100	#1307	C₂×C₈₃	1	1	yes	characteristic
101	C₈₃	#10528	1	1	yes	characteristic, direct product factor, Sylow
102	#533	C₂²×C₈₃	1	2	yes	—
103	#475	—	2	2	no	—
104	#470	—	2	2	no	—
105	#462	—	2	2	no	—
106	#432	C₂²×C₈₃	1	2	yes	—
107	C₂×#108	—	2	2	no	—
108	#423	C₂²×C₈₃	1	1	yes	characteristic
109	#423	C₂²×C₈₃	1	1	yes	characteristic
110	C₂×#97	C₂²×C₈₃	1	1	yes	characteristic
111	#333	—	2	2	no	—
112	#332	—	2	2	no	—
113	#330	—	2	4	no	—
114	C₂×Q₁₆	—	2	2	no	—
115	C₂×M₁₆	—	2	1	no	—
116	C₂×M₁₆	D₄×C₈₃	1	2	yes	—
117	C₂²×C₈	—	2	1	no	—
118	C₄⋊Q₈	D₄×C₈₃	1	2	yes	—
119	C₄⋊Q₈	D₄×C₈₃	1	2	yes	—
120	#125	—	2	2	no	—
121	C₂²⋊Q₈	—	2	2	no	—
122	C₂²⋊Q₈	—	2	2	no	—
123	C₄×Q₈	—	2	2	no	—
124	C₄×Q₈	—	2	2	no	—
125	#117	—	2	2	no	—
126	C₂×C₄²	C₂³×C₈₃	1	1	yes	characteristic
127	#108	—	4	4	no	—
128	#106	—	2	2	no	—
129	C₄⋊C₈	—	2	2	no	—
130	C₄⋊C₈	—	2	2	no	—
131	C₄⋊C₈	—	2	2	no	—
132	#103	—	2	4	no	—
133	C₂²⋊C₈	—	2	2	no	—
134	#97	—	2	2	no	—
135	#97	—	2	1	no	—
136	#97	D₄×C₈₃	1	2	yes	—
137	C₂×Q₈	—	2	2	no	—
138	C₂×Q₈	—	2	2	no	—
139	C₂²×C₄	C₂×D₄×C₈₃	1	1	yes	characteristic
140	C₂²×C₄	#41×C₈₃	1	2	yes	—
141	Q₁₆	—	8	2	no	—
142	M₁₆	—	4	2	no	—
143	M₁₆	—	4	2	no	—
144	C₂×C₈	—	2	2	no	—
145	C₂×C₈	—	2	2	no	—
146	C₂×C₈	—	2	2	no	—
147	C₂×C₈	—	2	2	no	—
148	C₂×C₈	—	4	1	no	—
149	C₂×C₈	—	4	2	no	—
150	C₄⋊C₄	—	2	2	no	—
151	C₄⋊C₄	—	2	2	no	—
152	C₄⋊C₄	—	2	2	no	—
153	C₄⋊C₄	—	2	2	no	—
154	C₄⋊C₄	—	4	2	no	—
155	C₄⋊C₄	—	4	2	no	—
156	C₂²⋊C₄	—	4	2	no	—
157	C₄²	—	4	2	no	—
158	C₄²	C₂×D₄×C₈₃	1	2	yes	—
159	C₄²	C₂×D₄×C₈₃	1	1	yes	characteristic
160	C₄²	#41×C₈₃	1	1	yes	characteristic
161	C₂³	#124×C₈₃	1	1	yes	characteristic
162	Q₈	—	4	2	no	—
163	Q₈	—	4	2	no	—
164	Q₈	—	8	2	no	—
165	C₂×C₄	—	2	1	no	—
166	C₂×C₄	—	2	1	no	—
167	C₂×C₄	—	4	2	no	—
168	C₂×C₄	—	4	2	no	—
169	C₂×C₄	—	4	2	no	—
170	C₂×C₄	#120×C₈₃	1	1	yes	characteristic
171	C₂×C₄	#121×C₈₃	1	2	yes	—
172	C₂×C₄	#121×C₈₃	1	1	yes	characteristic
173	C₂×C₄	#123×C₈₃	1	2	yes	—
174	C₈	—	2	4	no	—
175	C₈	—	4	2	no	—
176	C₈	—	8	2	no	—
177	C₈	—	8	1	no	—
178	C₂²	—	2	1	no	—
179	C₂²	—	2	2	no	—
180	C₂²	#394×C₈₃	1	1	yes	characteristic
181	C₄	—	2	1	no	—
182	C₄	—	2	2	no	—
183	C₄	—	2	2	no	—
184	C₄	—	8	2	no	—
185	C₄	—	8	2	no	—
186	C₄	#453×C₈₃	1	1	yes	characteristic
187	C₄	#456×C₈₃	1	1	yes	characteristic
188	C₂	—	4	1	no	—
189	C₂	C₈₃×#2039	1	1	yes	characteristic
190	C₂	C₈₃×#2049	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₈₃×#3255

C₂

yes

maximal, characteristic

C₈₃×#1849

C₂

yes

maximal, characteristic

C₈₃×#1559

C₂

yes

maximal

C₈₃×#1531

C₂

yes

maximal

C₈₃×#1307

C₂

yes

maximal, characteristic

#533×C₈₃

C₂²

yes

—

#475×C₈₃

—

#470×C₈₃

—

#462×C₈₃

—

#432×C₈₃

C₂²

yes

—

C₂×#108×C₈₃

—

#423×C₈₃

C₂²

yes

characteristic

#423×C₈₃

C₂²

yes

characteristic

C₂×#97×C₈₃

C₂²

yes

characteristic

#333×C₈₃

—

#332×C₈₃

—

#330×C₈₃

—

C₂×Q₁₆×C₈₃

—

C₂×M₁₆×C₈₃

—

C₂×M₁₆×C₈₃

D₄

yes

—

C₂²×C₈×C₈₃

—

C₄⋊Q₈×C₈₃

D₄

yes

—

C₄⋊Q₈×C₈₃

D₄

yes

—

#125×C₈₃

—

C₂²⋊Q₈×C₈₃

—

C₂²⋊Q₈×C₈₃

—

C₄×Q₈×C₈₃

—

C₄×Q₈×C₈₃

—

#117×C₈₃

—

C₂×C₄²×C₈₃

C₂³

yes

characteristic

#108×C₈₃

—

#106×C₈₃

—

C₄⋊C₈×C₈₃

—

C₄⋊C₈×C₈₃

—

C₄⋊C₈×C₈₃

—

#103×C₈₃

—

C₂²⋊C₈×C₈₃

—

#97×C₈₃

—

#97×C₈₃

—

#97×C₈₃

D₄

yes

—

C₂×Q₈×C₈₃

—

C₂×Q₈×C₈₃

—

C₂²×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₈₃

#41

yes

—

Q₁₆×C₈₃

—

M₁₆×C₈₃

—

M₁₆×C₈₃

—

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

—

C₄⋊C₄×C₈₃

—

C₄⋊C₄×C₈₃

—

C₄⋊C₄×C₈₃

—

C₄⋊C₄×C₈₃

—

C₄⋊C₄×C₈₃

—

C₄⋊C₄×C₈₃

—

C₂²⋊C₄×C₈₃

—

C₄²×C₈₃

—

C₄²×C₈₃

C₂×D₄

yes

—

C₄²×C₈₃

C₂×D₄

yes

C₄²×C₈₃

yes

C₂³×C₈₃

yes

Q₈×C₈₃

—

Q₈×C₈₃

—

Q₈×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

#120

yes

characteristic

C₂×C₄×C₈₃

#121

yes

—

C₂×C₄×C₈₃

#121

yes

characteristic

C₂×C₄×C₈₃

#123

yes

—

C₈×C₈₃

—

C₈×C₈₃

—

C₈×C₈₃

—

C₈×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

#394

yes

characteristic

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

yes

C₄×C₈₃

yes

C₈₃

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₈₃

—

C₂×C₈₃

#2039

yes

characteristic

C₂×C₈₃

#2049

yes

—

#3255

C₂×C₈₃

yes

characteristic

#1849

C₂×C₈₃

yes

characteristic

#1559

C₂×C₈₃

yes

—

#1531

C₂×C₈₃

yes

—

100

#1307

C₂×C₈₃

yes

characteristic

101

C₈₃

#10528

yes

characteristic, direct product factor, Sylow

102

#533

C₂²×C₈₃

yes

—

103

#475

—

104

#470

—

105

#462

—

106

#432

C₂²×C₈₃

yes

—

107

C₂×#108

—

108

#423

C₂²×C₈₃

yes

characteristic

109

#423

C₂²×C₈₃

yes

characteristic

110

C₂×#97

C₂²×C₈₃

yes

characteristic

111

#333

—

112

#332

—

113

#330

—

114

C₂×Q₁₆

—

115

C₂×M₁₆

—

116

C₂×M₁₆

D₄×C₈₃

yes

—

117

C₂²×C₈

—

118

C₄⋊Q₈

D₄×C₈₃

yes

—

119

C₄⋊Q₈

D₄×C₈₃

yes

—

120

#125

—

121

C₂²⋊Q₈

—

122

C₂²⋊Q₈

—

123

C₄×Q₈

—

124

C₄×Q₈

—

125

#117

—

126

C₂×C₄²

C₂³×C₈₃

yes

characteristic

127

#108

—

128

#106

—

129

C₄⋊C₈

—

130

C₄⋊C₈

—

131

C₄⋊C₈

—

132

#103

—

133

C₂²⋊C₈

—

134

#97

—

135

#97

—

136

#97

D₄×C₈₃

yes

—

137

C₂×Q₈

—

138

C₂×Q₈

—

139

C₂²×C₄

C₂×D₄×C₈₃

yes

characteristic

140

C₂²×C₄

#41×C₈₃

yes

—

141

Q₁₆

—

142

M₁₆

—

143

M₁₆

—

144

C₂×C₈

—

145

C₂×C₈

—

146

C₂×C₈

—

147

C₂×C₈

—

148

C₂×C₈

—

149

C₂×C₈

—

150

C₄⋊C₄

—

151

C₄⋊C₄

—

152

C₄⋊C₄

—

153

C₄⋊C₄

—

154

C₄⋊C₄

—

155

C₄⋊C₄

—

156

C₂²⋊C₄

—

157

C₄²

—

158

C₄²

C₂×D₄×C₈₃

yes

—

159

C₄²

C₂×D₄×C₈₃

yes

characteristic

160

C₄²

#41×C₈₃

yes

characteristic

161

C₂³

#124×C₈₃

yes

characteristic

162

Q₈

—

163

Q₈

—

164

Q₈

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

#120×C₈₃

yes

characteristic

171

C₂×C₄

#121×C₈₃

yes

—

172

C₂×C₄

#121×C₈₃

yes

characteristic

173

C₂×C₄

#123×C₈₃

yes

—

174

C₈

—

175

C₈

—

176

C₈

—

177

C₈

—

178

C₂²

—

179

C₂²

—

180

C₂²

#394×C₈₃

yes

characteristic

181

C₄

—

182

C₄

—

183

C₄

—

184

C₄

—

185

C₄

—

186

C₄

#453×C₈₃

yes

characteristic

187

C₄

#456×C₈₃

yes

characteristic

188

C₂

—

189

C₂

C₈₃×#2039

yes

characteristic

190

C₂

C₈₃×#2049

yes

—

	subgroup	G/H
center, socle	C₂²×C₈₃	#394
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₈₃
prime radical	C₂³×C₈₃	#124
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₈₃ ◁ C₂×C₄²×C₈₃ ◁ G

subgroup

G/H

center, socle

C₂²×C₈₃

#394

commutator, Frattini, cosocle

C₂×C₄²

C₂³×C₈₃

prime radical

C₂³×C₈₃

#124

LCS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂²×C₈₃ ◁ C₂×C₄²×C₈₃ ◁ G

N	H ≅ G/N	type
92. #10528	C₈₃	direct (N × H)
101. C₈₃	#10528	direct (N × H)

H ≅ G/N

type

92. #10528

C₈₃

direct (N × H)

101. C₈₃

#10528

direct (N × H)

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

subgroup	index
C₈₃×#3255	2
C₈₃×#1849	2
C₈₃×#1559	2
C₈₃×#1531	2
C₈₃×#1307	2
#10528	83

subgroup

index

C₈₃×#3255

C₈₃×#1849

C₈₃×#1559

C₈₃×#1531

C₈₃×#1307

#10528

Subgroups — C₈₃×#10528 #35238352

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C83×#10528 #35238352

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₃×#10528 #35238352