Group #35303173 — subgroups

Subgroups — #35303173

Order 21248.

260 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 115 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	#24906252	C₂	1	2	yes	maximal
2	#24906222	C₂	1	1	yes	maximal, characteristic
3	#24905752	C₂	1	1	yes	maximal, characteristic
4	#24905354	C₂	1	2	yes	maximal
5	C₈₃×#1564	C₂	1	1	yes	maximal, characteristic
6	#19127754	C₂²	1	1	yes	characteristic
7	#19127725	C₂²	1	2	yes	—
8	C₄×#13050574	C₂²	1	1	yes	characteristic
9	#19127454	—	2	2	no	—
10	#19127454	—	2	1	no	—
11	#19127454	—	2	1	no	—
12	#19127452	—	2	2	no	—
13	#19127423	—	2	1	no	—
14	#19127422	—	2	1	no	—
15	#431×C₈₃	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
17	#335×C₈₃	C₄	1	2	yes	—
18	#334×C₈₃	C₄	1	2	yes	—
19	C₂×#13050574	—	2	1	no	—
20	C₂×#13050574	—	2	2	no	—
21	C₄⋊(C₈₃⋊C₈)	—	2	2	no	—
22	C₄⋊(C₈₃⋊C₈)	—	2	2	no	—
23	C₄⋊(C₈₃⋊C₈)	D₄	1	1	yes	characteristic
24	C₄⋊(C₈₃⋊C₈)	D₄	1	1	yes	characteristic
25	C₂²⋊(C₈₃⋊C₈)	—	2	1	no	—
26	#14581703	Q₈	1	2	yes	—
27	C₄×C₈₃⋊C₈	—	2	1	no	—
28	C₄×C₈₃⋊C₈	—	2	1	no	—
29	C₂²×C₈×C₈₃	Q₈	1	1	yes	characteristic
30	C₂²×C₈×C₈₃	Q₈	1	1	yes	characteristic
31	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
32	C₄⋊C₈×C₈₃	C₂×C₄	1	4	yes	—
33	C₂²⋊C₈×C₈₃	—	2	2	no	—
34	C₄×C₈×C₈₃	—	2	2	no	—
35	C₄×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
36	C₄×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
37	C₄×C₈×C₈₃	D₄	1	1	yes	characteristic
38	C₄×C₈×C₈₃	D₄	1	1	yes	characteristic
39	#13050574	—	4	2	no	—
40	#13050574	—	8	2	no	—
41	C₂×C₈₃⋊C₈	—	2	1	no	—
42	C₂×C₈₃⋊C₈	—	2	1	no	—
43	C₂×C₈₃⋊C₈	—	2	2	no	—
44	C₂×C₈₃⋊C₈	—	4	2	no	—
45	C₂²×C₄×C₈₃	C₂×Q₈	1	1	yes	characteristic
46	C₂²×C₄×C₈₃	#41	1	1	yes	characteristic
47	C₂²×C₄×C₈₃	#41	1	1	yes	characteristic
48	C₂×C₈×C₈₃	—	2	2	no	—
49	C₂×C₈×C₈₃	—	2	4	no	—
50	C₂×C₈×C₈₃	—	4	1	no	—
51	C₂×C₈×C₈₃	C₄⋊C₄	1	1	yes	characteristic
52	C₂×C₈×C₈₃	C₄⋊C₄	1	1	yes	characteristic
53	C₂×C₈×C₈₃	C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₈×C₈₃	C₄⋊C₄	1	1	yes	characteristic
55	C₄²×C₈₃	C₂²×C₄	1	1	yes	characteristic
56	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
57	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
58	C₄²×C₈₃	C₂×Q₈	1	1	yes	characteristic
59	C₂³×C₈₃	#125	1	1	yes	characteristic
60	C₂×C₄×C₈₃	—	2	1	no	—
61	C₂×C₄×C₈₃	—	2	1	no	—
62	C₂×C₄×C₈₃	—	2	2	no	—
63	C₂×C₄×C₈₃	—	2	1	no	—
64	C₂×C₄×C₈₃	—	2	1	no	—
65	C₂×C₄×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
66	C₂×C₄×C₈₃	C₄×D₄	1	1	yes	characteristic
67	C₂×C₄×C₈₃	C₄×Q₈	1	1	yes	characteristic
68	C₂×C₄×C₈₃	#121	1	1	yes	characteristic
69	C₂×C₄×C₈₃	C₂²⋊Q₈	1	1	yes	characteristic
70	C₂×C₄×C₈₃	C₄⋊Q₈	1	1	yes	characteristic
71	C₈×C₈₃	—	2	1	no	—
72	C₈×C₈₃	—	2	1	no	—
73	C₈×C₈₃	—	2	1	no	—
74	C₈×C₈₃	—	4	4	no	—
75	C₈×C₈₃	#108	1	2	yes	—
76	C₈₃⋊C₈	—	4	2	no	—
77	C₈₃⋊C₈	—	4	1	no	—
78	C₈₃⋊C₈	—	4	1	no	—
79	C₈₃⋊C₈	—	8	2	no	—
80	C₂²×C₈₃	—	2	1	no	—
81	C₂²×C₈₃	—	2	1	no	—
82	C₂²×C₈₃	—	2	1	no	—
83	C₂²×C₈₃	#389	1	1	yes	characteristic
84	C₄×C₈₃	—	2	1	no	—
85	C₄×C₈₃	—	2	1	no	—
86	C₄×C₈₃	—	2	1	no	—
87	C₄×C₈₃	C₂×#108	1	1	yes	characteristic
88	C₄×C₈₃	#430	1	1	yes	characteristic
89	C₄×C₈₃	#444	1	2	yes	—
90	C₄×C₈₃	#472	1	1	yes	characteristic
91	C₄×C₈₃	#473	1	1	yes	characteristic
92	#9636	—	83	1	no	maximal, Sylow
93	C₂×C₈₃	—	4	1	no	—
94	C₂×C₈₃	#1946	1	1	yes	characteristic
95	C₂×C₈₃	#1950	1	1	yes	characteristic
96	C₂×C₈₃	#1952	1	1	yes	characteristic
97	#1574	—	83	2	no	—
98	#1567	—	83	1	no	—
99	#1564	D₈₃	1	1	yes	characteristic
100	#1451	—	83	1	no	—
101	#1299	—	83	2	no	—
102	C₈₃	#9636	1	1	yes	characteristic, complemented, Sylow
103	#431	—	83	1	no	—
104	#431	C₂×D₈₃	1	2	yes	—
105	#423	—	83	2	no	—
106	C₄×M₁₆	—	83	1	no	—
107	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
108	#335	—	166	2	no	—
109	#335	—	166	1	no	—
110	#335	—	166	1	no	—
111	#335	Dic₈₃	1	2	yes	—
112	#334	—	166	2	no	—
113	#334	Dic₈₃	1	2	yes	—
114	#322	—	166	1	no	—
115	#322	—	166	1	no	—
116	C₂×M₁₆	—	166	2	no	—
117	C₂×M₁₆	—	166	1	no	—
118	C₂²×C₈	Dic₁₆₆	1	1	yes	characteristic
119	C₂²×C₈	Dic₁₆₆	1	1	yes	characteristic
120	C₂×C₄²	C₂²×D₈₃	1	1	yes	characteristic
121	C₄⋊C₈	—	83	1	no	—
122	C₄⋊C₈	—	83	1	no	—
123	C₄⋊C₈	—	166	2	no	—
124	C₄⋊C₈	—	166	2	no	—
125	C₄⋊C₈	C₂×Dic₈₃	1	4	yes	—
126	C₂²⋊C₈	—	2	2	no	—
127	C₂²⋊C₈	—	166	1	no	—
128	#97	—	83	2	no	—
129	C₄×C₈	—	2	2	no	—
130	C₄×C₈	—	166	1	no	—
131	C₄×C₈	—	166	1	no	—
132	C₄×C₈	D₃₃₂	1	1	yes	characteristic
133	C₄×C₈	D₃₃₂	1	1	yes	characteristic
134	C₄×C₈	C₂×Dic₈₃	1	1	yes	characteristic
135	C₄×C₈	C₂×Dic₈₃	1	1	yes	characteristic
136	C₂²×C₄	C₂×Dic₁₆₆	1	1	yes	characteristic
137	C₂²×C₄	#13050589	1	1	yes	characteristic
138	C₂²×C₄	#13050591	1	1	yes	characteristic
139	M₁₆	—	332	2	no	—
140	M₁₆	—	664	2	no	—
141	C₂×C₈	—	2	2	no	—
142	C₂×C₈	—	2	4	no	—
143	C₂×C₈	—	4	1	no	—
144	C₂×C₈	—	332	2	no	—
145	C₂×C₈	—	166	1	no	—
146	C₂×C₈	—	166	1	no	—
147	C₂×C₈	—	166	2	no	—
148	C₂×C₈	C₄⋊Dic₈₃	1	1	yes	characteristic
149	C₂×C₈	C₄⋊Dic₈₃	1	1	yes	characteristic
150	C₂×C₈	C₄⋊Dic₈₃	1	1	yes	characteristic
151	C₂×C₈	C₄⋊Dic₈₃	1	1	yes	characteristic
152	C₄²	C₂²×Dic₈₃	1	1	yes	characteristic
153	C₄²	D₄×D₈₃	1	1	yes	characteristic
154	C₄²	C₂×D₃₃₂	1	1	yes	characteristic
155	C₄²	Q₈×D₈₃	1	1	yes	characteristic
156	C₂³	#14581783	1	1	yes	characteristic
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	2	1	no	—
159	C₂×C₄	—	2	1	no	—
160	C₂×C₄	—	2	1	no	—
161	C₂×C₄	—	2	1	no	—
162	C₂×C₄	C₂×C₄⋊Dic₈₃	1	1	yes	characteristic
163	C₂×C₄	D₄×Dic₈₃	1	1	yes	characteristic
164	C₂×C₄	Q₈×Dic₈₃	1	1	yes	characteristic
165	C₂×C₄	#14581767	1	1	yes	characteristic
166	C₂×C₄	#14581772	1	1	yes	characteristic
167	C₂×C₄	#14581791	1	1	yes	characteristic
168	C₈	—	2	1	no	—
169	C₈	—	2	1	no	—
170	C₈	—	2	1	no	—
171	C₈	—	4	4	no	—
172	C₈	—	332	2	no	—
173	C₈	—	332	1	no	—
174	C₈	—	332	1	no	—
175	C₈	—	664	2	no	—
176	C₈	#14581728	1	2	yes	—
177	C₂²	—	2	1	no	—
178	C₂²	—	2	1	no	—
179	C₂²	—	2	1	no	—
180	C₂²	#19127600	1	1	yes	characteristic
181	C₄	—	2	1	no	—
182	C₄	—	2	1	no	—
183	C₄	—	2	1	no	—
184	C₄	C₂×#14581728	1	1	yes	characteristic
185	C₄	#19127750	1	1	yes	characteristic
186	C₄	#19127823	1	2	yes	—
187	C₄	#19127972	1	1	yes	characteristic
188	C₄	#19127977	1	1	yes	characteristic
189	C₂	—	4	1	no	—
190	C₂	#24908217	1	1	yes	characteristic
191	C₂	#24908239	1	1	yes	characteristic
192	C₂	#24908249	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#24906252

C₂

yes

maximal

#24906222

C₂

yes

maximal, characteristic

#24905752

C₂

yes

maximal, characteristic

#24905354

C₂

yes

maximal

C₈₃×#1564

C₂

yes

maximal, characteristic

#19127754

C₂²

yes

characteristic

#19127725

C₂²

yes

—

C₄×#13050574

C₂²

yes

—

—

—

—

—

—

#431×C₈₃

C₂²

yes

—

C₂×C₄×C₈×C₈₃

C₂²

yes

characteristic

#335×C₈₃

C₄

yes

—

#334×C₈₃

C₄

yes

—

C₂×#13050574

—

C₂×#13050574

—

C₄⋊(C₈₃⋊C₈)

—

C₄⋊(C₈₃⋊C₈)

—

C₄⋊(C₈₃⋊C₈)

D₄

yes

characteristic

C₄⋊(C₈₃⋊C₈)

D₄

yes

characteristic

C₂²⋊(C₈₃⋊C₈)

—

#14581703

Q₈

yes

—

C₄×C₈₃⋊C₈

—

C₄×C₈₃⋊C₈

—

C₂²×C₈×C₈₃

Q₈

yes

characteristic

C₂²×C₈×C₈₃

Q₈

yes

characteristic

C₂×C₄²×C₈₃

C₂³

yes

characteristic

C₄⋊C₈×C₈₃

C₂×C₄

yes

—

C₂²⋊C₈×C₈₃

—

C₄×C₈×C₈₃

—

C₄×C₈×C₈₃

C₂×C₄

yes

characteristic

C₄×C₈×C₈₃

C₂×C₄

yes

characteristic

C₄×C₈×C₈₃

D₄

yes

characteristic

C₄×C₈×C₈₃

D₄

yes

characteristic

#13050574

—

#13050574

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

—

C₂²×C₄×C₈₃

C₂×Q₈

yes

C₂²×C₄×C₈₃

yes

C₂²×C₄×C₈₃

yes

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

C₄⋊C₄

yes

characteristic

C₂×C₈×C₈₃

C₄⋊C₄

yes

characteristic

C₂×C₈×C₈₃

C₄⋊C₄

yes

characteristic

C₂×C₈×C₈₃

C₄⋊C₄

yes

characteristic

C₄²×C₈₃

C₂²×C₄

yes

characteristic

C₄²×C₈₃

C₂×D₄

yes

characteristic

C₄²×C₈₃

C₂×D₄

yes

characteristic

C₄²×C₈₃

C₂×Q₈

yes

characteristic

C₂³×C₈₃

#125

yes

characteristic

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₈₃

C₄×D₄

yes

characteristic

C₂×C₄×C₈₃

C₄×Q₈

yes

characteristic

C₂×C₄×C₈₃

#121

yes

characteristic

C₂×C₄×C₈₃

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₈₃

C₄⋊Q₈

yes

characteristic

C₈×C₈₃

—

C₈×C₈₃

—

C₈×C₈₃

—

C₈×C₈₃

—

C₈×C₈₃

#108

yes

—

C₈₃⋊C₈

—

C₈₃⋊C₈

—

C₈₃⋊C₈

—

C₈₃⋊C₈

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

#389

yes

characteristic

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

yes

C₄×C₈₃

yes

C₄×C₈₃

yes

—

C₄×C₈₃

yes

C₄×C₈₃

yes

—

C₂×C₈₃

—

C₂×C₈₃

yes

C₂×C₈₃

yes

C₂×C₈₃

yes

—

—

D₈₃

yes

characteristic

100

#1451

—

101

#1299

—

102

C₈₃

#9636

yes

characteristic, complemented, Sylow

103

#431

—

104

#431

C₂×D₈₃

yes

—

105

#423

—

106

C₄×M₁₆

—

107

C₂×C₄×C₈

C₂×D₈₃

yes

characteristic

108

#335

—

166

—

109

#335

—

166

—

110

#335

—

166

—

111

#335

Dic₈₃

yes

—

112

#334

—

166

—

113

#334

Dic₈₃

yes

—

114

#322

—

166

—

115

#322

—

166

—

116

C₂×M₁₆

—

166

—

117

C₂×M₁₆

—

166

—

118

C₂²×C₈

Dic₁₆₆

yes

characteristic

119

C₂²×C₈

Dic₁₆₆

yes

characteristic

120

C₂×C₄²

C₂²×D₈₃

yes

characteristic

121

C₄⋊C₈

—

122

C₄⋊C₈

—

123

C₄⋊C₈

—

166

—

124

C₄⋊C₈

—

166

—

125

C₄⋊C₈

C₂×Dic₈₃

yes

—

126

C₂²⋊C₈

—

127

C₂²⋊C₈

—

166

—

128

#97

—

129

C₄×C₈

—

130

C₄×C₈

—

166

—

131

C₄×C₈

—

166

—

132

C₄×C₈

D₃₃₂

yes

characteristic

133

C₄×C₈

D₃₃₂

yes

characteristic

134

C₄×C₈

C₂×Dic₈₃

yes

characteristic

135

C₄×C₈

C₂×Dic₈₃

yes

characteristic

136

C₂²×C₄

C₂×Dic₁₆₆

yes

characteristic

137

C₂²×C₄

#13050589

yes

characteristic

138

C₂²×C₄

#13050591

yes

characteristic

139

M₁₆

—

332

—

140

M₁₆

—

664

—

141

C₂×C₈

—

142

C₂×C₈

—

143

C₂×C₈

—

144

C₂×C₈

—

332

—

145

C₂×C₈

—

166

—

146

C₂×C₈

—

166

—

147

C₂×C₈

—

166

—

148

C₂×C₈

C₄⋊Dic₈₃

yes

characteristic

149

C₂×C₈

C₄⋊Dic₈₃

yes

characteristic

150

C₂×C₈

C₄⋊Dic₈₃

yes

characteristic

151

C₂×C₈

C₄⋊Dic₈₃

yes

characteristic

152

C₄²

C₂²×Dic₈₃

yes

characteristic

153

C₄²

D₄×D₈₃

yes

characteristic

154

C₄²

C₂×D₃₃₂

yes

characteristic

155

C₄²

Q₈×D₈₃

yes

characteristic

156

C₂³

#14581783

yes

characteristic

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

C₂×C₄⋊Dic₈₃

yes

characteristic

163

C₂×C₄

D₄×Dic₈₃

yes

characteristic

164

C₂×C₄

Q₈×Dic₈₃

yes

characteristic

165

C₂×C₄

#14581767

yes

characteristic

166

C₂×C₄

#14581772

yes

characteristic

167

C₂×C₄

#14581791

yes

characteristic

168

C₈

—

169

C₈

—

170

C₈

—

171

C₈

—

172

C₈

—

332

—

173

C₈

—

332

—

174

C₈

—

332

—

175

C₈

—

664

—

176

C₈

#14581728

yes

—

177

C₂²

—

178

C₂²

—

179

C₂²

—

180

C₂²

#19127600

yes

characteristic

181

C₄

—

182

C₄

—

183

C₄

—

184

C₄

C₂×#14581728

yes

characteristic

185

C₄

#19127750

yes

characteristic

186

C₄

#19127823

yes

—

187

C₄

#19127972

yes

characteristic

188

C₄

#19127977

yes

characteristic

189

C₂

—

190

C₂

#24908217

yes

characteristic

191

C₂

#24908239

yes

characteristic

192

C₂

#24908249

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#14581767
commutator	C₄²×C₈₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₃
Fitting	C₈₃×#1564	C₂
socle	C₂²×C₈₃	#389
prime radical	C₂³×C₈₃	#125
cosocle	C₂×C₄²×C₈₃	C₂³
hypercenter	#1564	D₈₃
hypocenter	C₈₃	#9636
LCS	G ▷ C₄²×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1564

subgroup

G/H

center

C₂×C₄

#14581767

commutator

C₄²×C₈₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₈₃

Fitting

C₈₃×#1564

C₂

C₂²×C₈₃

C₂³×C₈₃

C₂×C₄²×C₈₃

C₂³

D₈₃

C₈₃

G ▷ C₄²×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1564

N	H ≅ G/N	type
102. C₈₃	#9636	semidirect

H ≅ G/N

type

102. C₈₃

#9636

semidirect

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

subgroup	index
#24906252	2
#24906222	2
#24905752	2
#24905354	2
C₈₃×#1564	2
#9636	83

subgroup

index

C₈₃×#1564

Subgroups — #35303173

Conjugacy classes of subgroups

Complements

Maximal subgroups