Group #35392111 — subgroups

Subgroups — #35392111

Order 21248.

706 conjugacy classes of subgroups; only the 187 normal subgroups are shown.

normal subgroups only

This group has 706 conjugacy classes of subgroups — too many to draw. Only its 187 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	#24918048	C₂	1	4	yes	maximal
2	#24917974	C₂	1	2	yes	maximal
3	#24907080	C₂	1	2	yes	maximal, complemented
4	#24907073	C₂	1	1	yes	maximal, characteristic, complemented
5	#24907059	C₂	1	4	yes	maximal, complemented
6	C₈²×D₈₃	C₂	1	1	yes	maximal, characteristic
7	C₈₃×#1710	C₂	1	1	yes	maximal, characteristic, complemented
8	#19128256	C₂²	1	4	yes	—
9	#19128106	C₂²	1	2	yes	—
10	#19128104	C₂²	1	4	yes	—
11	#19128102	—	2	4	no	—
12	#19128097	C₂²	1	4	yes	—
13	#19128080	C₂²	1	4	yes	—
14	#19128052	C₂²	1	4	yes	—
15	#19127941	—	2	8	no	—
16	#19127793	—	2	8	no	—
17	#19127738	—	2	4	no	—
18	C₄×C₈×D₈₃	C₂²	1	2	yes	—
19	C₄×C₈×D₈₃	C₂²	1	1	yes	characteristic
20	C₈×C₈₃⋊C₈	C₂²	1	2	yes	—
21	C₈×C₈₃⋊C₈	C₂²	1	1	yes	characteristic
22	#500×C₈₃	C₂²	1	2	yes	—
23	#487×C₈₃	C₂²	1	4	yes	—
24	C₈²×C₈₃	C₂²	1	1	yes	characteristic
25	C₂×#13050580	—	2	8	no	—
26	C₂×C₈×D₈₃	D₄	1	4	yes	—
27	C₂×C₈×D₈₃	D₄	1	2	yes	—
28	#14581792	C₂³	1	4	yes	—
29	#14581783	—	2	4	no	—
30	#14581772	—	2	4	no	—
31	#14581768	—	2	4	no	—
32	Q₈×Dic₈₃	—	2	4	no	—
33	#14581757	—	2	4	no	—
34	#14581750	—	2	4	no	—
35	C₄²×D₈₃	C₂³	1	1	yes	characteristic
36	#14581726	—	2	4	no	—
37	#14581725	—	2	8	no	—
38	#14581723	—	2	8	no	—
39	#14581717	—	2	8	no	—
40	C₄×C₈₃⋊C₈	C₂³	1	2	yes	—
41	C₄×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
42	C₈×Dic₈₃	D₄	1	2	yes	—
43	C₈×Dic₈₃	D₄	1	4	yes	—
44	C₄⋊Q₈×C₈₃	C₂³	1	4	yes	—
45	#107×C₈₃	—	2	4	no	—
46	#103×C₈₃	—	2	8	no	—
47	C₄×C₈×C₈₃	C₂³	1	2	yes	—
48	C₄×C₈×C₈₃	C₂³	1	1	yes	characteristic
49	C₂×Dic₁₆₆	—	2	4	no	—
50	C₂×C₄×D₈₃	C₂×D₄	1	2	yes	—
51	C₂×C₄×D₈₃	C₂×D₄	1	1	yes	characteristic
52	#13050580	—	4	16	no	—
53	C₂×C₈₃⋊C₈	C₂×D₄	1	2	yes	—
54	C₂×C₈₃⋊C₈	C₂×D₄	1	4	yes	—
55	C₈×D₈₃	—	2	8	no	—
56	C₈×D₈₃	Q₁₆	1	8	yes	complemented
57	C₄⋊Dic₈₃	—	2	4	no	—
58	C₄⋊Dic₈₃	—	2	4	no	—
59	C₄⋊Dic₈₃	—	4	4	no	—
60	Dic₈₃⋊C₄	—	4	4	no	—
61	(C₂×D₈₃)⋊C₄	—	4	4	no	—
62	C₄×Dic₈₃	—	4	4	no	—
63	C₄×Dic₈₃	C₂×D₄	1	1	yes	characteristic
64	C₄×Dic₈₃	C₂×D₄	1	2	yes	—
65	C₂×Q₈×C₈₃	—	2	4	no	—
66	C₂×C₈×C₈₃	C₂×D₄	1	4	yes	—
67	C₂×C₈×C₈₃	C₂×D₄	1	2	yes	—
68	C₄⋊C₄×C₈₃	—	2	4	no	—
69	C₄⋊C₄×C₈₃	—	2	4	no	—
70	C₄²×C₈₃	C₂⁴	1	1	yes	characteristic
71	C₂²×D₈₃	#127	1	1	yes	characteristic
72	Q₈×C₈₃	—	4	8	no	—
73	C₂×C₄×C₈₃	—	4	4	no	—
74	C₂×C₄×C₈₃	C₂²×D₄	1	1	yes	characteristic
75	C₂×C₄×C₈₃	C₂²×D₄	1	2	yes	—
76	C₂×Dic₈₃	—	4	4	no	—
77	C₂×Dic₈₃	#127	1	2	yes	—
78	C₂×Dic₈₃	#127	1	1	yes	characteristic
79	C₄×D₈₃	—	2	4	no	—
80	C₄×D₈₃	C₂×Q₁₆	1	4	yes	—
81	Dic₁₆₆	—	4	8	no	—
82	C₈×C₈₃	—	2	4	no	—
83	C₈×C₈₃	C₂×Q₁₆	1	4	yes	—
84	C₈₃⋊C₈	C₂×Q₁₆	1	4	yes	—
85	C₈₃⋊C₈	#135	1	8	yes	—
86	C₂²×C₈₃	C₂×#127	1	1	yes	characteristic
87	C₂×D₈₃	—	2	2	no	—
88	C₂×D₈₃	#494	1	2	yes	—
89	C₄×C₈₃	—	8	4	no	—
90	C₄×C₈₃	C₂²×Q₁₆	1	2	yes	—
91	C₄×C₈₃	C₂×#135	1	4	yes	—
92	Dic₈₃	—	2	2	no	—
93	Dic₈₃	—	8	4	no	—
94	Dic₈₃	#494	1	2	yes	—
95	#19604	—	83	1	no	maximal, Sylow
96	C₂×C₈₃	C₂×#494	1	1	yes	characteristic
97	C₂×C₈₃	#3147	1	2	yes	—
98	D₈₃	—	2	2	no	—
99	#3170	—	83	4	no	—
100	#3160	—	83	2	no	—
101	#1713	—	83	2	no	—
102	#1710	—	83	1	no	—
103	#1710	D₈₃	1	1	yes	characteristic, complemented
104	#1707	—	83	4	no	—
105	C₂×C₈²	—	83	1	no	—
106	C₈₃	#19604	1	1	yes	characteristic, complemented, Sylow
107	#533	—	83	4	no	—
108	#500	—	83	4	no	—
109	#500	—	83	2	no	—
110	#500	C₂×D₈₃	1	2	yes	—
111	#499	—	166	4	no	—
112	#498	—	83	4	no	—
113	#494	—	83	4	no	—
114	#487	—	83	4	no	—
115	#487	C₂×D₈₃	1	4	yes	—
116	#467	—	166	8	no	—
117	C₄×Q₁₆	—	166	8	no	—
118	#427	—	166	4	no	—
119	C₂×C₄×C₈	—	83	1	no	—
120	C₂×C₄×C₈	—	83	2	no	—
121	C₈²	—	83	2	no	—
122	C₈²	—	83	1	no	—
123	C₈²	C₂×D₈₃	1	1	yes	characteristic
124	C₂×Q₁₆	—	166	8	no	—
125	C₂²×C₈	—	83	2	no	—
126	C₂²×C₈	—	83	4	no	—
127	C₄⋊Q₈	—	83	4	no	—
128	C₄⋊Q₈	C₂²×D₈₃	1	4	yes	—
129	#125	—	166	4	no	—
130	C₂²⋊Q₈	—	166	4	no	—
131	C₂²⋊Q₈	—	166	4	no	—
132	C₄×Q₈	—	166	4	no	—
133	C₄×Q₈	—	166	4	no	—
134	#117	—	166	4	no	—
135	C₂×C₄²	—	83	1	no	—
136	#107	—	2	4	no	—
137	#107	—	166	8	no	—
138	#107	—	166	4	no	—
139	#106	—	166	8	no	—
140	#103	—	2	8	no	—
141	#103	—	166	8	no	—
142	C₄×C₈	—	83	4	no	—
143	C₄×C₈	—	83	2	no	—
144	C₄×C₈	—	83	2	no	—
145	C₄×C₈	—	83	1	no	—
146	C₄×C₈	C₂²×D₈₃	1	2	yes	—
147	C₄×C₈	C₂²×D₈₃	1	1	yes	characteristic
148	C₂×Q₈	—	2	4	no	—
149	C₂×Q₈	—	166	4	no	—
150	C₂²×C₄	—	83	1	no	—
151	C₂²×C₄	—	83	2	no	—
152	Q₁₆	—	332	16	no	—
153	C₂×C₈	—	83	2	no	—
154	C₂×C₈	—	83	4	no	—
155	C₂×C₈	—	83	8	no	—
156	C₂×C₈	—	166	8	no	—
157	C₂×C₈	D₄×D₈₃	1	4	yes	—
158	C₂×C₈	D₄×D₈₃	1	2	yes	—
159	C₄⋊C₄	—	2	4	no	—
160	C₄⋊C₄	—	2	4	no	—
161	C₄⋊C₄	—	332	4	no	—
162	C₄⋊C₄	—	332	4	no	—
163	C₄⋊C₄	—	166	4	no	—
164	C₄⋊C₄	—	166	4	no	—
165	C₂²⋊C₄	—	332	4	no	—
166	C₄²	—	332	4	no	—
167	C₄²	—	83	2	no	—
168	C₄²	—	83	1	no	—
169	C₄²	C₂³×D₈₃	1	1	yes	characteristic
170	C₂³	—	83	1	no	—
171	Q₈	—	4	8	no	—
172	Q₈	—	332	8	no	—
173	C₂×C₄	—	4	4	no	—
174	C₂×C₄	—	332	4	no	—
175	C₂×C₄	—	83	1	no	—
176	C₂×C₄	—	83	4	no	—
177	C₂×C₄	—	83	2	no	—
178	C₂×C₄	—	166	4	no	—
179	C₂×C₄	C₂×D₄×D₈₃	1	1	yes	characteristic
180	C₂×C₄	C₂×D₄×D₈₃	1	2	yes	—
181	C₈	—	2	4	no	—
182	C₈	—	83	4	no	—
183	C₈	—	83	8	no	—
184	C₈	Q₁₆×D₈₃	1	4	yes	—
185	C₂²	—	83	2	no	—
186	C₂²	—	166	2	no	—
187	C₂²	#127×D₈₃	1	1	yes	characteristic
188	C₄	—	8	4	no	—
189	C₄	—	83	2	no	—
190	C₄	—	664	4	no	—
191	C₄	—	166	2	no	—
192	C₄	C₂×Q₁₆×D₈₃	1	2	yes	—
193	C₄	#135×D₈₃	1	4	yes	—
194	C₂	—	166	2	no	—
195	C₂	#494×D₈₃	1	1	yes	characteristic
196	C₂	#24917888	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#24918048

C₂

yes

maximal

#24917974

C₂

yes

maximal

#24907080

C₂

yes

maximal, complemented

#24907073

C₂

yes

maximal, characteristic, complemented

#24907059

C₂

yes

maximal, complemented

C₈²×D₈₃

C₂

yes

maximal, characteristic

C₈₃×#1710

C₂

yes

maximal, characteristic, complemented

#19128256

C₂²

yes

—

#19128106

C₂²

yes

—

#19128104

C₂²

yes

—

#19128102

—

#19128097

C₂²

yes

—

#19128080

C₂²

yes

—

#19128052

C₂²

yes

—

#19127941

—

#19127793

—

#19127738

—

C₄×C₈×D₈₃

C₂²

yes

—

C₄×C₈×D₈₃

C₂²

yes

characteristic

C₈×C₈₃⋊C₈

C₂²

yes

—

C₈×C₈₃⋊C₈

C₂²

yes

characteristic

#500×C₈₃

C₂²

yes

—

#487×C₈₃

C₂²

yes

—

C₈²×C₈₃

C₂²

yes

characteristic

C₂×#13050580

—

C₂×C₈×D₈₃

D₄

yes

—

C₂×C₈×D₈₃

D₄

yes

—

#14581792

C₂³

yes

—

#14581783

—

#14581772

—

#14581768

—

Q₈×Dic₈₃

—

#14581757

—

#14581750

—

C₄²×D₈₃

C₂³

yes

—

—

—

—

C₄×C₈₃⋊C₈

C₂³

yes

—

C₄×C₈₃⋊C₈

C₂³

yes

characteristic

C₈×Dic₈₃

D₄

yes

—

C₈×Dic₈₃

D₄

yes

—

C₄⋊Q₈×C₈₃

C₂³

yes

—

#107×C₈₃

—

#103×C₈₃

—

C₄×C₈×C₈₃

C₂³

yes

—

C₄×C₈×C₈₃

C₂³

yes

characteristic

C₂×Dic₁₆₆

—

C₂×C₄×D₈₃

C₂×D₄

yes

—

C₂×C₄×D₈₃

C₂×D₄

yes

characteristic

#13050580

—

C₂×C₈₃⋊C₈

C₂×D₄

yes

—

C₂×C₈₃⋊C₈

C₂×D₄

yes

—

C₈×D₈₃

—

C₈×D₈₃

Q₁₆

yes

complemented

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

—

Dic₈₃⋊C₄

—

(C₂×D₈₃)⋊C₄

—

C₄×Dic₈₃

—

C₄×Dic₈₃

C₂×D₄

yes

characteristic

C₄×Dic₈₃

C₂×D₄

yes

—

C₂×Q₈×C₈₃

—

C₂×C₈×C₈₃

C₂×D₄

yes

—

C₂×C₈×C₈₃

C₂×D₄

yes

—

C₄⋊C₄×C₈₃

—

C₄⋊C₄×C₈₃

—

C₄²×C₈₃

C₂⁴

yes

characteristic

C₂²×D₈₃

#127

yes

characteristic

Q₈×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

C₂²×D₄

yes

characteristic

C₂×C₄×C₈₃

C₂²×D₄

yes

—

C₂×Dic₈₃

—

C₂×Dic₈₃

#127

yes

—

C₂×Dic₈₃

#127

yes

characteristic

C₄×D₈₃

—

C₄×D₈₃

C₂×Q₁₆

yes

—

Dic₁₆₆

—

C₈×C₈₃

—

C₈×C₈₃

C₂×Q₁₆

yes

—

C₈₃⋊C₈

C₂×Q₁₆

yes

—

C₈₃⋊C₈

#135

yes

—

C₂²×C₈₃

C₂×#127

yes

characteristic

C₂×D₈₃

—

C₂×D₈₃

#494

yes

—

C₄×C₈₃

—

C₄×C₈₃

C₂²×Q₁₆

yes

—

C₄×C₈₃

C₂×#135

yes

—

Dic₈₃

—

Dic₈₃

—

Dic₈₃

#494

yes

—

—

C₂×C₈₃

yes

C₂×C₈₃

yes

—

D₈₃

—

#3170

—

100

#3160

—

101

#1713

—

102

#1710

—

103

#1710

D₈₃

yes

characteristic, complemented

104

#1707

—

105

C₂×C₈²

—

106

C₈₃

#19604

yes

characteristic, complemented, Sylow

107

#533

—

108

#500

—

109

#500

—

110

#500

C₂×D₈₃

yes

—

111

#499

—

166

—

112

#498

—

113

#494

—

114

#487

—

115

#487

C₂×D₈₃

yes

—

116

#467

—

166

—

117

C₄×Q₁₆

—

166

—

118

#427

—

166

—

119

C₂×C₄×C₈

—

120

C₂×C₄×C₈

—

121

C₈²

—

122

C₈²

—

123

C₈²

C₂×D₈₃

yes

characteristic

124

C₂×Q₁₆

—

166

—

125

C₂²×C₈

—

126

C₂²×C₈

—

127

C₄⋊Q₈

—

128

C₄⋊Q₈

C₂²×D₈₃

yes

—

129

#125

—

166

—

130

C₂²⋊Q₈

—

166

—

131

C₂²⋊Q₈

—

166

—

132

C₄×Q₈

—

166

—

133

C₄×Q₈

—

166

—

134

#117

—

166

—

135

C₂×C₄²

—

136

#107

—

137

#107

—

166

—

138

#107

—

166

—

139

#106

—

166

—

140

#103

—

141

#103

—

166

—

142

C₄×C₈

—

143

C₄×C₈

—

144

C₄×C₈

—

145

C₄×C₈

—

146

C₄×C₈

C₂²×D₈₃

yes

—

147

C₄×C₈

C₂²×D₈₃

yes

characteristic

148

C₂×Q₈

—

149

C₂×Q₈

—

166

—

150

C₂²×C₄

—

151

C₂²×C₄

—

152

Q₁₆

—

332

—

153

C₂×C₈

—

154

C₂×C₈

—

155

C₂×C₈

—

156

C₂×C₈

—

166

—

157

C₂×C₈

D₄×D₈₃

yes

—

158

C₂×C₈

D₄×D₈₃

yes

—

159

C₄⋊C₄

—

160

C₄⋊C₄

—

161

C₄⋊C₄

—

332

—

162

C₄⋊C₄

—

332

—

163

C₄⋊C₄

—

166

—

164

C₄⋊C₄

—

166

—

165

C₂²⋊C₄

—

332

—

166

C₄²

—

332

—

167

C₄²

—

168

C₄²

—

169

C₄²

C₂³×D₈₃

yes

characteristic

170

C₂³

—

171

Q₈

—

172

Q₈

—

332

—

173

C₂×C₄

—

174

C₂×C₄

—

332

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

166

—

179

C₂×C₄

C₂×D₄×D₈₃

yes

characteristic

180

C₂×C₄

C₂×D₄×D₈₃

yes

—

181

C₈

—

182

C₈

—

183

C₈

—

184

C₈

Q₁₆×D₈₃

yes

—

185

C₂²

—

186

C₂²

—

166

—

187

C₂²

#127×D₈₃

yes

characteristic

188

C₄

—

189

C₄

—

190

C₄

—

664

—

191

C₄

—

166

—

192

C₄

C₂×Q₁₆×D₈₃

yes

—

193

C₄

#135×D₈₃

yes

—

194

C₂

—

166

—

195

C₂

#494×D₈₃

yes

characteristic

196

C₂

#24917888

yes

—

	subgroup	G/H
center	C₂²	#127×D₈₃
commutator, cosocle	C₄²×C₈₃	C₂⁴
Frattini	C₄²	C₂³×D₈₃
Fitting	C₈₃×#1710	C₂
socle	C₂²×C₈₃	C₂×#127
prime radical	C₂²×D₈₃	#127
hypercenter	#1710	D₈₃
hypocenter	C₈₃	#19604
LCS	G ▷ C₄²×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #1710

subgroup

G/H

center

C₂²

#127×D₈₃

commutator, cosocle

C₄²×C₈₃

C₂⁴

Frattini

C₄²

C₂³×D₈₃

Fitting

C₈₃×#1710

C₂

C₂²×C₈₃

C₂²×D₈₃

D₈₃

C₈₃

G ▷ C₄²×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #1710

N	H ≅ G/N	type
3. #24907080	C₂	semidirect
4. #24907073	C₂	semidirect
5. #24907059	C₂	semidirect
7. C₈₃×#1710	C₂	semidirect
56. C₈×D₈₃	Q₁₆	semidirect
103. #1710	D₈₃	semidirect
106. C₈₃	#19604	semidirect

H ≅ G/N

type

3. #24907080

C₂

semidirect

4. #24907073

C₂

semidirect

5. #24907059

C₂

semidirect

7. C₈₃×#1710

C₂

semidirect

56. C₈×D₈₃

Q₁₆

semidirect

103. #1710

D₈₃

semidirect

106. C₈₃

#19604

semidirect

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

subgroup	index
#24918048	2
#24917974	2
#24907080	2
#24907073	2
#24907059	2
C₈²×D₈₃	2
C₈₃×#1710	2
#19604	83

subgroup

index

C₈²×D₈₃

C₈₃×#1710

#19604

Subgroups — #35392111

Conjugacy classes of subgroups

Complements

Maximal subgroups