Group #35238123 — subgroups

Subgroups — C₈₃×#10299 #35238123

Order 21248.

576 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 576 conjugacy classes of subgroups — too many to draw. Only its 130 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×#3229	C₂	1	2	yes	maximal
2	C₈₃×#2868	C₂	1	1	yes	maximal, characteristic
3	C₈₃×#1994	C₂	1	1	yes	maximal, characteristic
4	C₈₃×#1546	C₂	1	2	yes	maximal
5	C₈₃×#1276	C₂	1	1	yes	maximal, characteristic
6	#533×C₈₃	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊Q₈×C₈₃	—	2	2	no	—
8	C₂×C₄⋊Q₈×C₈₃	C₂²	1	2	yes	—
9	#498×C₈₃	—	2	2	no	—
10	#487×C₈₃	—	2	2	no	—
11	#483×C₈₃	—	2	4	no	—
12	#475×C₈₃	—	2	4	no	—
13	#431×C₈₃	C₂²	1	2	yes	—
14	#423×C₈₃	C₂²	1	1	yes	characteristic
15	#384×C₈₃	—	2	4	no	—
16	C₄³×C₈₃	C₂²	1	1	yes	characteristic
17	#339×C₈₃	—	2	2	no	—
18	#332×C₈₃	—	2	4	no	—
19	C₂²×Q₈×C₈₃	—	2	2	no	—
20	C₂×M₁₆×C₈₃	—	2	1	no	—
21	C₄⋊Q₈×C₈₃	—	2	2	no	—
22	C₄⋊Q₈×C₈₃	—	2	2	no	—
23	C₄⋊Q₈×C₈₃	—	2	2	no	—
24	C₄⋊Q₈×C₈₃	—	2	2	no	—
25	C₄⋊Q₈×C₈₃	—	4	4	no	—
26	C₄⋊Q₈×C₈₃	—	4	4	no	—
27	C₄⋊Q₈×C₈₃	D₄	1	2	yes	—
28	#125×C₈₃	—	2	1	no	—
29	C₂²⋊Q₈×C₈₃	—	2	2	no	—
30	C₄×Q₈×C₈₃	—	2	2	no	—
31	#117×C₈₃	—	2	1	no	—
32	C₂×C₄⋊C₄×C₈₃	—	2	8	no	—
33	C₂×C₄⋊C₄×C₈₃	—	2	2	no	—
34	C₂×C₄⋊C₄×C₈₃	D₄	1	4	yes	—
35	C₂×C₄²×C₈₃	—	2	2	no	—
36	C₂×C₄²×C₈₃	D₄	1	2	yes	—
37	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
38	#106×C₈₃	—	2	8	no	—
39	C₄⋊C₈×C₈₃	—	2	2	no	—
40	C₄⋊C₈×C₈₃	D₄	1	4	yes	—
41	#103×C₈₃	—	2	8	no	—
42	C₂²⋊C₈×C₈₃	—	2	2	no	—
43	C₄×C₈×C₈₃	—	2	2	no	—
44	C₂×Q₈×C₈₃	—	2	2	no	—
45	C₂×Q₈×C₈₃	—	4	2	no	—
46	C₂×Q₈×C₈₃	—	4	2	no	—
47	C₂×Q₈×C₈₃	—	4	4	no	—
48	C₂×Q₈×C₈₃	—	4	4	no	—
49	C₂²×C₄×C₈₃	—	2	2	no	—
50	C₂²×C₄×C₈₃	—	2	4	no	—
51	C₂²×C₄×C₈₃	C₂×D₄	1	2	yes	—
52	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
53	M₁₆×C₈₃	—	4	2	no	—
54	C₂×C₈×C₈₃	—	2	4	no	—
55	C₂×C₈×C₈₃	—	4	1	no	—
56	C₄⋊C₄×C₈₃	—	2	4	no	—
57	C₄⋊C₄×C₈₃	—	2	2	no	—
58	C₄⋊C₄×C₈₃	—	2	4	no	—
59	C₄⋊C₄×C₈₃	—	2	2	no	—
60	C₄⋊C₄×C₈₃	—	4	16	no	—
61	C₄⋊C₄×C₈₃	—	4	2	no	—
62	C₄⋊C₄×C₈₃	—	4	2	no	—
63	C₄⋊C₄×C₈₃	—	4	2	no	—
64	C₂²⋊C₄×C₈₃	—	4	1	no	—
65	C₄²×C₈₃	—	2	8	no	—
66	C₄²×C₈₃	—	2	2	no	—
67	C₄²×C₈₃	—	4	1	no	—
68	C₄²×C₈₃	SD₁₆	1	4	yes	—
69	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
70	C₄²×C₈₃	C₂×D₄	1	2	yes	—
71	C₄²×C₈₃	#41	1	1	yes	characteristic
72	C₂³×C₈₃	#120	1	1	yes	characteristic
73	Q₈×C₈₃	—	4	4	no	—
74	Q₈×C₈₃	—	8	4	no	—
75	Q₈×C₈₃	—	8	2	no	—
76	Q₈×C₈₃	—	8	2	no	—
77	C₂×C₄×C₈₃	—	2	8	no	—
78	C₂×C₄×C₈₃	—	2	4	no	—
79	C₂×C₄×C₈₃	—	2	4	no	—
80	C₂×C₄×C₈₃	—	4	2	no	—
81	C₂×C₄×C₈₃	—	4	4	no	—
82	C₂×C₄×C₈₃	—	4	8	no	—
83	C₂×C₄×C₈₃	#120	1	2	yes	—
84	C₂×C₄×C₈₃	#121	1	2	yes	—
85	C₂×C₄×C₈₃	#121	1	1	yes	characteristic
86	C₂×C₄×C₈₃	#127	1	1	yes	characteristic
87	C₂×C₄×C₈₃	C₂×SD₁₆	1	2	yes	—
88	C₂×C₄×C₈₃	AGL(1,ℤ/8)	1	2	yes	—
89	C₈×C₈₃	—	4	4	no	—
90	C₈×C₈₃	—	8	1	no	—
91	C₂²×C₈₃	—	2	2	no	—
92	C₂²×C₈₃	#392	1	1	yes	characteristic
93	C₂²×C₈₃	#450	1	2	yes	—
94	C₄×C₈₃	—	2	8	no	—
95	C₄×C₈₃	—	2	2	no	—
96	C₄×C₈₃	—	8	2	no	—
97	C₄×C₈₃	—	8	4	no	—
98	C₄×C₈₃	#456	1	4	yes	—
99	C₄×C₈₃	#460	1	2	yes	—
100	C₄×C₈₃	#492	1	1	yes	characteristic
101	C₄×C₈₃	#496	1	1	yes	characteristic
102	#10299	C₈₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
103	C₂×C₈₃	—	2	2	no	—
104	C₂×C₈₃	#2005	1	2	yes	—
105	C₂×C₈₃	#2014	1	1	yes	characteristic
106	#3229	C₂×C₈₃	1	2	yes	—
107	#2868	C₂×C₈₃	1	1	yes	characteristic
108	#1994	C₂×C₈₃	1	1	yes	characteristic
109	#1546	C₂×C₈₃	1	2	yes	—
110	#1276	C₂×C₈₃	1	1	yes	characteristic
111	C₈₃	#10299	1	1	yes	characteristic, direct product factor, Sylow
112	#533	C₂²×C₈₃	1	1	yes	characteristic
113	C₂×C₄⋊Q₈	—	2	2	no	—
114	C₂×C₄⋊Q₈	C₂²×C₈₃	1	2	yes	—
115	#498	—	2	2	no	—
116	#487	—	2	2	no	—
117	#483	—	2	4	no	—
118	#475	—	2	4	no	—
119	#431	C₂²×C₈₃	1	2	yes	—
120	#423	C₂²×C₈₃	1	1	yes	characteristic
121	#384	—	2	4	no	—
122	C₄³	C₂²×C₈₃	1	1	yes	characteristic
123	#339	—	2	2	no	—
124	#332	—	2	4	no	—
125	C₂²×Q₈	—	2	2	no	—
126	C₂×M₁₆	—	2	1	no	—
127	C₄⋊Q₈	—	2	2	no	—
128	C₄⋊Q₈	—	2	2	no	—
129	C₄⋊Q₈	—	2	2	no	—
130	C₄⋊Q₈	—	2	2	no	—
131	C₄⋊Q₈	—	4	4	no	—
132	C₄⋊Q₈	—	4	4	no	—
133	C₄⋊Q₈	D₄×C₈₃	1	2	yes	—
134	#125	—	2	1	no	—
135	C₂²⋊Q₈	—	2	2	no	—
136	C₄×Q₈	—	2	2	no	—
137	#117	—	2	1	no	—
138	C₂×C₄⋊C₄	—	2	8	no	—
139	C₂×C₄⋊C₄	—	2	2	no	—
140	C₂×C₄⋊C₄	D₄×C₈₃	1	4	yes	—
141	C₂×C₄²	—	2	2	no	—
142	C₂×C₄²	D₄×C₈₃	1	2	yes	—
143	C₂×C₄²	C₂³×C₈₃	1	1	yes	characteristic
144	#106	—	2	8	no	—
145	C₄⋊C₈	—	2	2	no	—
146	C₄⋊C₈	D₄×C₈₃	1	4	yes	—
147	#103	—	2	8	no	—
148	C₂²⋊C₈	—	2	2	no	—
149	C₄×C₈	—	2	2	no	—
150	C₂×Q₈	—	2	2	no	—
151	C₂×Q₈	—	4	2	no	—
152	C₂×Q₈	—	4	2	no	—
153	C₂×Q₈	—	4	4	no	—
154	C₂×Q₈	—	4	4	no	—
155	C₂²×C₄	—	2	2	no	—
156	C₂²×C₄	—	2	4	no	—
157	C₂²×C₄	C₂×D₄×C₈₃	1	2	yes	—
158	C₂²×C₄	C₂×D₄×C₈₃	1	1	yes	characteristic
159	M₁₆	—	4	2	no	—
160	C₂×C₈	—	2	4	no	—
161	C₂×C₈	—	4	1	no	—
162	C₄⋊C₄	—	2	4	no	—
163	C₄⋊C₄	—	2	2	no	—
164	C₄⋊C₄	—	2	4	no	—
165	C₄⋊C₄	—	2	2	no	—
166	C₄⋊C₄	—	4	16	no	—
167	C₄⋊C₄	—	4	2	no	—
168	C₄⋊C₄	—	4	2	no	—
169	C₄⋊C₄	—	4	2	no	—
170	C₂²⋊C₄	—	4	1	no	—
171	C₄²	—	2	8	no	—
172	C₄²	—	2	2	no	—
173	C₄²	—	4	1	no	—
174	C₄²	SD₁₆×C₈₃	1	4	yes	—
175	C₄²	C₂×D₄×C₈₃	1	1	yes	characteristic
176	C₄²	C₂×D₄×C₈₃	1	2	yes	—
177	C₄²	#41×C₈₃	1	1	yes	characteristic
178	C₂³	#120×C₈₃	1	1	yes	characteristic
179	Q₈	—	4	4	no	—
180	Q₈	—	8	2	no	—
181	Q₈	—	8	2	no	—
182	Q₈	—	8	4	no	—
183	C₂×C₄	—	2	8	no	—
184	C₂×C₄	—	2	4	no	—
185	C₂×C₄	—	2	4	no	—
186	C₂×C₄	—	4	2	no	—
187	C₂×C₄	—	4	4	no	—
188	C₂×C₄	—	4	8	no	—
189	C₂×C₄	#120×C₈₃	1	2	yes	—
190	C₂×C₄	#121×C₈₃	1	2	yes	—
191	C₂×C₄	#121×C₈₃	1	1	yes	characteristic
192	C₂×C₄	#127×C₈₃	1	1	yes	characteristic
193	C₂×C₄	C₂×SD₁₆×C₈₃	1	2	yes	—
194	C₂×C₄	AGL(1,ℤ/8)×C₈₃	1	2	yes	—
195	C₈	—	4	4	no	—
196	C₈	—	8	1	no	—
197	C₂²	—	2	2	no	—
198	C₂²	#392×C₈₃	1	1	yes	characteristic
199	C₂²	#450×C₈₃	1	2	yes	—
200	C₄	—	2	8	no	—
201	C₄	—	2	2	no	—
202	C₄	—	8	2	no	—
203	C₄	—	8	4	no	—
204	C₄	#456×C₈₃	1	4	yes	—
205	C₄	#460×C₈₃	1	2	yes	—
206	C₄	#492×C₈₃	1	1	yes	characteristic
207	C₄	#496×C₈₃	1	1	yes	characteristic
208	C₂	—	2	2	no	—
209	C₂	C₈₃×#2005	1	2	yes	—
210	C₂	C₈₃×#2014	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₈₃×#3229

C₂

yes

maximal

C₈₃×#2868

C₂

yes

maximal, characteristic

C₈₃×#1994

C₂

yes

maximal, characteristic

C₈₃×#1546

C₂

yes

maximal

C₈₃×#1276

C₂

yes

maximal, characteristic

#533×C₈₃

C₂²

yes

characteristic

C₂×C₄⋊Q₈×C₈₃

—

C₂×C₄⋊Q₈×C₈₃

C₂²

yes

—

#498×C₈₃

—

#487×C₈₃

—

#483×C₈₃

—

#475×C₈₃

—

#431×C₈₃

C₂²

yes

—

#423×C₈₃

C₂²

yes

characteristic

#384×C₈₃

—

C₄³×C₈₃

C₂²

yes

characteristic

#339×C₈₃

—

#332×C₈₃

—

C₂²×Q₈×C₈₃

—

C₂×M₁₆×C₈₃

—

C₄⋊Q₈×C₈₃

—

C₄⋊Q₈×C₈₃

—

C₄⋊Q₈×C₈₃

—

C₄⋊Q₈×C₈₃

—

C₄⋊Q₈×C₈₃

—

C₄⋊Q₈×C₈₃

—

C₄⋊Q₈×C₈₃

D₄

yes

—

#125×C₈₃

—

C₂²⋊Q₈×C₈₃

—

C₄×Q₈×C₈₃

—

#117×C₈₃

—

C₂×C₄⋊C₄×C₈₃

—

C₂×C₄⋊C₄×C₈₃

—

C₂×C₄⋊C₄×C₈₃

D₄

yes

—

C₂×C₄²×C₈₃

—

C₂×C₄²×C₈₃

D₄

yes

—

C₂×C₄²×C₈₃

C₂³

yes

characteristic

#106×C₈₃

—

C₄⋊C₈×C₈₃

—

C₄⋊C₈×C₈₃

D₄

yes

—

#103×C₈₃

—

C₂²⋊C₈×C₈₃

—

C₄×C₈×C₈₃

—

C₂×Q₈×C₈₃

—

C₂×Q₈×C₈₃

—

C₂×Q₈×C₈₃

—

C₂×Q₈×C₈₃

—

C₂×Q₈×C₈₃

—

C₂²×C₄×C₈₃

—

C₂²×C₄×C₈₃

—

C₂²×C₄×C₈₃

C₂×D₄

yes

—

C₂²×C₄×C₈₃

C₂×D₄

yes

characteristic

M₁₆×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

—

C₄⋊C₄×C₈₃

—

C₄⋊C₄×C₈₃

—

C₂²⋊C₄×C₈₃

—

C₄²×C₈₃

—

C₄²×C₈₃

—

C₄²×C₈₃

—

C₄²×C₈₃

SD₁₆

yes

—

C₄²×C₈₃

C₂×D₄

yes

characteristic

C₄²×C₈₃

C₂×D₄

yes

—

C₄²×C₈₃

yes

C₂³×C₈₃

yes

Q₈×C₈₃

—

Q₈×C₈₃

—

Q₈×C₈₃

—

Q₈×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

#120

yes

—

C₂×C₄×C₈₃

#121

yes

—

C₂×C₄×C₈₃

yes

C₂×C₄×C₈₃

yes

C₂×C₄×C₈₃

C₂×SD₁₆

yes

—

C₂×C₄×C₈₃

AGL(1,ℤ/8)

yes

—

C₈×C₈₃

—

C₈×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

#392

yes

characteristic

C₂²×C₈₃

#450

yes

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

#456

yes

—

C₄×C₈₃

#460

yes

—

100

C₄×C₈₃

#492

yes

characteristic

101

C₄×C₈₃

#496

yes

characteristic

102

#10299

C₈₃

yes

maximal, characteristic, direct product factor, Sylow

103

C₂×C₈₃

—

104

C₂×C₈₃

#2005

yes

—

105

C₂×C₈₃

#2014

yes

characteristic

106

#3229

C₂×C₈₃

yes

—

107

#2868

C₂×C₈₃

yes

characteristic

108

#1994

C₂×C₈₃

yes

characteristic

109

#1546

C₂×C₈₃

yes

—

110

#1276

C₂×C₈₃

yes

characteristic

111

C₈₃

#10299

yes

characteristic, direct product factor, Sylow

112

#533

C₂²×C₈₃

yes

characteristic

113

C₂×C₄⋊Q₈

—

114

C₂×C₄⋊Q₈

C₂²×C₈₃

yes

—

115

#498

—

116

#487

—

117

#483

—

118

#475

—

119

#431

C₂²×C₈₃

yes

—

120

#423

C₂²×C₈₃

yes

characteristic

121

#384

—

122

C₄³

C₂²×C₈₃

yes

characteristic

123

#339

—

124

#332

—

125

C₂²×Q₈

—

126

C₂×M₁₆

—

127

C₄⋊Q₈

—

128

C₄⋊Q₈

—

129

C₄⋊Q₈

—

130

C₄⋊Q₈

—

131

C₄⋊Q₈

—

132

C₄⋊Q₈

—

133

C₄⋊Q₈

D₄×C₈₃

yes

—

134

#125

—

135

C₂²⋊Q₈

—

136

C₄×Q₈

—

137

#117

—

138

C₂×C₄⋊C₄

—

139

C₂×C₄⋊C₄

—

140

C₂×C₄⋊C₄

D₄×C₈₃

yes

—

141

C₂×C₄²

—

142

C₂×C₄²

D₄×C₈₃

yes

—

143

C₂×C₄²

C₂³×C₈₃

yes

characteristic

144

#106

—

145

C₄⋊C₈

—

146

C₄⋊C₈

D₄×C₈₃

yes

—

147

#103

—

148

C₂²⋊C₈

—

149

C₄×C₈

—

150

C₂×Q₈

—

151

C₂×Q₈

—

152

C₂×Q₈

—

153

C₂×Q₈

—

154

C₂×Q₈

—

155

C₂²×C₄

—

156

C₂²×C₄

—

157

C₂²×C₄

C₂×D₄×C₈₃

yes

—

158

C₂²×C₄

C₂×D₄×C₈₃

yes

characteristic

159

M₁₆

—

160

C₂×C₈

—

161

C₂×C₈

—

162

C₄⋊C₄

—

163

C₄⋊C₄

—

164

C₄⋊C₄

—

165

C₄⋊C₄

—

166

C₄⋊C₄

—

167

C₄⋊C₄

—

168

C₄⋊C₄

—

169

C₄⋊C₄

—

170

C₂²⋊C₄

—

171

C₄²

—

172

C₄²

—

173

C₄²

—

174

C₄²

SD₁₆×C₈₃

yes

—

175

C₄²

C₂×D₄×C₈₃

yes

characteristic

176

C₄²

C₂×D₄×C₈₃

yes

—

177

C₄²

#41×C₈₃

yes

characteristic

178

C₂³

#120×C₈₃

yes

characteristic

179

Q₈

—

180

Q₈

—

181

Q₈

—

182

Q₈

—

183

C₂×C₄

—

184

C₂×C₄

—

185

C₂×C₄

—

186

C₂×C₄

—

187

C₂×C₄

—

188

C₂×C₄

—

189

C₂×C₄

#120×C₈₃

yes

—

190

C₂×C₄

#121×C₈₃

yes

—

191

C₂×C₄

#121×C₈₃

yes

characteristic

192

C₂×C₄

#127×C₈₃

yes

characteristic

193

C₂×C₄

C₂×SD₁₆×C₈₃

yes

—

194

C₂×C₄

AGL(1,ℤ/8)×C₈₃

yes

—

195

C₈

—

196

C₈

—

197

C₂²

—

198

C₂²

#392×C₈₃

yes

characteristic

199

C₂²

#450×C₈₃

yes

—

200

C₄

—

201

C₄

—

202

C₄

—

203

C₄

—

204

C₄

#456×C₈₃

yes

—

205

C₄

#460×C₈₃

yes

—

206

C₄

#492×C₈₃

yes

characteristic

207

C₄

#496×C₈₃

yes

characteristic

208

C₂

—

209

C₂

C₈₃×#2005

yes

—

210

C₂

C₈₃×#2014

yes

characteristic

	subgroup	G/H
center, socle	C₂²×C₈₃	#392
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₈₃
prime radical	C₂³×C₈₃	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₈₃ ◁ C₂×C₄²×C₈₃ ◁ G

subgroup

G/H

center, socle

C₂²×C₈₃

#392

commutator, Frattini, cosocle

C₂×C₄²

C₂³×C₈₃

prime radical

C₂³×C₈₃

#120

LCS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂²×C₈₃ ◁ C₂×C₄²×C₈₃ ◁ G

N	H ≅ G/N	type
102. #10299	C₈₃	direct (N × H)
111. C₈₃	#10299	direct (N × H)

H ≅ G/N

type

102. #10299

C₈₃

direct (N × H)

111. C₈₃

#10299

direct (N × H)

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

subgroup	index
C₈₃×#3229	2
C₈₃×#2868	2
C₈₃×#1994	2
C₈₃×#1546	2
C₈₃×#1276	2
#10299	83

subgroup

index

C₈₃×#3229

C₈₃×#2868

C₈₃×#1994

C₈₃×#1546

C₈₃×#1276

#10299

Subgroups — C₈₃×#10299 #35238123

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C83×#10299 #35238123

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₃×#10299 #35238123