Group #35307522 — subgroups

Subgroups — #35307522

Order 21248.

402 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 91 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	#24915850	C₂	1	1	yes	maximal, characteristic, complemented
2	#24906023	C₂	1	2	yes	maximal
3	#24905323	C₂	1	2	yes	maximal, complemented
4	#24905296	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈₃×#1962	C₂	1	1	yes	maximal, characteristic
6	#19127754	C₂²	1	2	yes	—
7	#19127599	—	2	2	no	—
8	#19127585	—	2	4	no	—
9	#19127567	C₂²	1	2	yes	—
10	#19127435	—	2	4	no	—
11	#532×C₈₃	C₂²	1	1	yes	characteristic
12	#423×C₈₃	C₂²	1	1	yes	characteristic
13	C₂×#104×C₈₃	—	2	2	no	—
14	C₄³×C₈₃	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊Dic₈₃	—	2	2	no	—
16	C₂×Dic₈₃⋊C₄	—	2	4	no	—
17	C₄⋊(C₈₃⋊C₈)	—	2	2	no	—
18	C₂²⋊(C₈₃⋊C₈)	—	2	2	no	—
19	C₄×C₈₃⋊C₈	—	2	2	no	—
20	C₈₃⋊#95	—	2	4	no	—
21	C₈₃⋊#95	—	2	2	no	—
22	C₈₃⋊#95	D₄	1	4	yes	—
23	C₂×#41×C₈₃	—	2	1	no	—
24	C₂×M₁₆×C₈₃	—	2	1	no	—
25	C₄⋊Q₈×C₈₃	Q₈	1	1	yes	characteristic
26	#127×C₈₃	Q₈	1	1	yes	characteristic
27	#124×C₈₃	—	2	1	no	—
28	#121×C₈₃	—	2	2	no	—
29	C₄×D₄×C₈₃	—	2	2	no	—
30	C₂×C₄²×C₈₃	—	2	2	no	—
31	C₂×C₄²×C₈₃	D₄	1	2	yes	—
32	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
33	C₄⋊C₈×C₈₃	—	2	2	no	—
34	#104×C₈₃	—	4	4	no	—
35	C₂²×Dic₈₃	—	2	4	no	—
36	C₂×C₈₃⋊C₈	—	4	2	no	—
37	C₄⋊Dic₈₃	—	4	2	no	—
38	C₄⋊Dic₈₃	—	4	2	no	—
39	Dic₈₃⋊C₄	—	4	8	no	—
40	#41×C₈₃	—	4	2	no	—
41	#41×C₈₃	—	4	2	no	—
42	C₂×Q₈×C₈₃	—	2	1	no	—
43	C₂×D₄×C₈₃	—	2	1	no	—
44	C₂×D₄×C₈₃	—	2	2	no	—
45	C₂×D₄×C₈₃	—	4	1	no	—
46	C₂²×C₄×C₈₃	—	2	2	no	—
47	C₂²×C₄×C₈₃	—	4	1	no	—
48	C₂²×C₄×C₈₃	C₂×D₄	1	2	yes	—
49	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
50	M₁₆×C₈₃	—	4	2	no	—
51	C₂×C₈×C₈₃	—	4	1	no	—
52	C₄⋊C₄×C₈₃	—	2	2	no	—
53	C₂²⋊C₄×C₈₃	—	4	2	no	—
54	C₄²×C₈₃	—	2	2	no	—
55	C₄²×C₈₃	—	2	2	no	—
56	C₄²×C₈₃	—	4	4	no	—
57	C₄²×C₈₃	C₂×Q₈	1	1	yes	characteristic
58	C₄²×C₈₃	#41	1	2	yes	—
59	C₄²×C₈₃	#41	1	1	yes	characteristic
60	C₂³×C₈₃	—	4	1	no	—
61	C₂³×C₈₃	#120	1	1	yes	characteristic
62	Q₈×C₈₃	—	4	2	no	—
63	D₄×C₈₃	—	4	2	no	—
64	D₄×C₈₃	—	8	2	no	—
65	D₄×C₈₃	—	8	2	no	—
66	C₂×C₄×C₈₃	—	2	4	no	—
67	C₂×C₄×C₈₃	—	4	2	no	—
68	C₂×C₄×C₈₃	—	4	2	no	—
69	C₂×C₄×C₈₃	—	4	2	no	—
70	C₂×C₄×C₈₃	—	4	1	no	—
71	C₂×C₄×C₈₃	—	8	1	no	—
72	C₂×C₄×C₈₃	—	8	1	no	—
73	C₂×C₄×C₈₃	C₂²⋊Q₈	1	2	yes	—
74	C₂×C₄×C₈₃	C₂²⋊Q₈	1	1	yes	characteristic
75	C₂×C₄×C₈₃	#124	1	2	yes	—
76	C₂×C₄×C₈₃	#124	1	1	yes	characteristic
77	C₂×C₄×C₈₃	#135	1	2	yes	—
78	C₂×C₄×C₈₃	AGL(1,ℤ/8)	1	2	yes	—
79	C₂×Dic₈₃	—	4	8	no	—
80	C₂×Dic₈₃	—	4	4	no	—
81	C₈×C₈₃	—	8	1	no	—
82	C₈₃⋊C₈	—	8	2	no	—
83	C₂²×C₈₃	—	2	2	no	—
84	C₂²×C₈₃	—	8	1	no	—
85	C₂²×C₈₃	—	8	2	no	—
86	C₂²×C₈₃	#393	1	1	yes	characteristic
87	C₂²×C₈₃	#449	1	2	yes	—
88	C₄×C₈₃	—	2	2	no	—
89	C₄×C₈₃	—	2	2	no	—
90	C₄×C₈₃	—	4	4	no	—
91	C₄×C₈₃	—	8	1	no	—
92	C₄×C₈₃	#478	1	2	yes	—
93	C₄×C₈₃	#488	1	1	yes	characteristic
94	C₄×C₈₃	#490	1	1	yes	characteristic
95	Dic₈₃	—	8	4	no	—
96	#10368	—	83	1	no	maximal, Sylow
97	C₂×C₈₃	—	2	2	no	—
98	C₂×C₈₃	—	16	1	no	—
99	C₂×C₈₃	#2025	1	1	yes	characteristic
100	C₂×C₈₃	#2031	1	2	yes	—
101	#2864	—	83	1	no	—
102	#1962	D₈₃	1	1	yes	characteristic
103	#1517	—	83	2	no	—
104	#1286	—	83	2	no	—
105	#1276	—	83	1	no	—
106	C₈₃	#10368	1	1	yes	characteristic, complemented, Sylow
107	#532	C₂×D₈₃	1	1	yes	characteristic
108	#431	—	83	2	no	—
109	#423	C₂×D₈₃	1	1	yes	characteristic
110	C₂×#104	—	2	2	no	—
111	#389	—	166	2	no	—
112	#387	—	166	4	no	—
113	#382	—	83	2	no	—
114	C₄³	C₂×D₈₃	1	1	yes	characteristic
115	#327	—	166	4	no	—
116	C₂×#41	—	2	1	no	—
117	C₂×M₁₆	—	2	1	no	—
118	C₄⋊Q₈	Dic₁₆₆	1	1	yes	characteristic
119	#127	Dic₁₆₆	1	1	yes	characteristic
120	#124	—	2	1	no	—
121	#121	—	2	2	no	—
122	C₄×D₄	—	2	2	no	—
123	C₂×C₄⋊C₄	—	166	4	no	—
124	C₂×C₄⋊C₄	—	166	2	no	—
125	C₂×C₄²	—	2	2	no	—
126	C₂×C₄²	C₂²⋊D₈₃	1	2	yes	—
127	C₂×C₄²	C₂²×D₈₃	1	1	yes	characteristic
128	C₄⋊C₈	—	2	2	no	—
129	C₄⋊C₈	—	166	2	no	—
130	#104	—	4	4	no	—
131	C₂²⋊C₈	—	166	2	no	—
132	C₄×C₈	—	166	2	no	—
133	#95	—	83	4	no	—
134	#95	—	166	4	no	—
135	#95	—	166	2	no	—
136	#41	—	4	2	no	—
137	#41	—	4	2	no	—
138	C₂×Q₈	—	2	1	no	—
139	C₂×D₄	—	2	2	no	—
140	C₂×D₄	—	2	1	no	—
141	C₂×D₄	—	4	1	no	—
142	C₂²×C₄	—	2	2	no	—
143	C₂²×C₄	—	4	1	no	—
144	C₂²×C₄	—	166	4	no	—
145	C₂²×C₄	D₄×D₈₃	1	2	yes	—
146	C₂²×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
147	M₁₆	—	4	2	no	—
148	C₂×C₈	—	4	1	no	—
149	C₂×C₈	—	332	2	no	—
150	C₄⋊C₄	—	2	2	no	—
151	C₄⋊C₄	—	332	8	no	—
152	C₄⋊C₄	—	332	2	no	—
153	C₄⋊C₄	—	332	2	no	—
154	C₂²⋊C₄	—	4	2	no	—
155	C₄²	—	2	2	no	—
156	C₄²	—	2	2	no	—
157	C₄²	—	4	4	no	—
158	C₄²	C₂×Dic₁₆₆	1	1	yes	characteristic
159	C₄²	#13050589	1	2	yes	—
160	C₄²	#13050590	1	1	yes	characteristic
161	C₂³	—	4	1	no	—
162	C₂³	#14581760	1	1	yes	characteristic
163	Q₈	—	4	2	no	—
164	D₄	—	4	2	no	—
165	D₄	—	8	2	no	—
166	D₄	—	8	2	no	—
167	C₂×C₄	—	2	4	no	—
168	C₂×C₄	—	4	2	no	—
169	C₂×C₄	—	4	2	no	—
170	C₂×C₄	—	4	2	no	—
171	C₂×C₄	—	4	1	no	—
172	C₂×C₄	—	8	1	no	—
173	C₂×C₄	—	8	1	no	—
174	C₂×C₄	—	332	8	no	—
175	C₂×C₄	—	332	4	no	—
176	C₂×C₄	#14581769	1	2	yes	—
177	C₂×C₄	#14581771	1	1	yes	characteristic
178	C₂×C₄	#14581778	1	2	yes	—
179	C₂×C₄	#14581779	1	1	yes	characteristic
180	C₂×C₄	#14581812	1	2	yes	—
181	C₂×C₄	#14581818	1	2	yes	—
182	C₈	—	8	1	no	—
183	C₈	—	664	2	no	—
184	C₂²	—	2	2	no	—
185	C₂²	—	8	1	no	—
186	C₂²	—	8	2	no	—
187	C₂²	#19127613	1	1	yes	characteristic
188	C₂²	#19127844	1	2	yes	—
189	C₄	—	2	2	no	—
190	C₄	—	2	2	no	—
191	C₄	—	4	4	no	—
192	C₄	—	8	1	no	—
193	C₄	—	664	4	no	—
194	C₄	#19128001	1	2	yes	—
195	C₄	#19128055	1	1	yes	characteristic
196	C₄	#19128065	1	1	yes	characteristic
197	C₂	—	2	2	no	—
198	C₂	—	16	1	no	—
199	C₂	#24908609	1	1	yes	characteristic
200	C₂	#24908635	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#24915850

C₂

yes

maximal, characteristic, complemented

#24906023

C₂

yes

maximal

#24905323

C₂

yes

maximal, complemented

#24905296

C₂

yes

maximal, characteristic, complemented

C₈₃×#1962

C₂

yes

maximal, characteristic

#19127754

C₂²

yes

—

#19127599

—

#19127585

—

#19127567

C₂²

yes

—

#19127435

—

#532×C₈₃

C₂²

yes

characteristic

#423×C₈₃

C₂²

yes

characteristic

C₂×#104×C₈₃

—

C₄³×C₈₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₈₃

—

C₂×Dic₈₃⋊C₄

—

C₄⋊(C₈₃⋊C₈)

—

C₂²⋊(C₈₃⋊C₈)

—

C₄×C₈₃⋊C₈

—

C₈₃⋊#95

—

C₈₃⋊#95

—

C₈₃⋊#95

D₄

yes

—

C₂×#41×C₈₃

—

C₂×M₁₆×C₈₃

—

C₄⋊Q₈×C₈₃

Q₈

yes

characteristic

#127×C₈₃

Q₈

yes

characteristic

#124×C₈₃

—

#121×C₈₃

—

C₄×D₄×C₈₃

—

C₂×C₄²×C₈₃

—

C₂×C₄²×C₈₃

D₄

yes

—

C₂×C₄²×C₈₃

C₂³

yes

characteristic

C₄⋊C₈×C₈₃

—

#104×C₈₃

—

C₂²×Dic₈₃

—

C₂×C₈₃⋊C₈

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

—

Dic₈₃⋊C₄

—

#41×C₈₃

—

#41×C₈₃

—

C₂×Q₈×C₈₃

—

C₂×D₄×C₈₃

—

C₂×D₄×C₈₃

—

C₂×D₄×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

—

C₄⋊C₄×C₈₃

—

C₂²⋊C₄×C₈₃

—

C₄²×C₈₃

—

C₄²×C₈₃

—

C₄²×C₈₃

—

C₄²×C₈₃

C₂×Q₈

yes

characteristic

C₄²×C₈₃

#41

yes

—

C₄²×C₈₃

#41

yes

characteristic

C₂³×C₈₃

—

C₂³×C₈₃

#120

yes

characteristic

Q₈×C₈₃

—

D₄×C₈₃

—

D₄×C₈₃

—

D₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

C₂²⋊Q₈

yes

—

C₂×C₄×C₈₃

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₈₃

#124

yes

—

C₂×C₄×C₈₃

#124

yes

characteristic

C₂×C₄×C₈₃

#135

yes

—

C₂×C₄×C₈₃

AGL(1,ℤ/8)

yes

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₈×C₈₃

—

C₈₃⋊C₈

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

#393

yes

characteristic

C₂²×C₈₃

#449

yes

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

#478

yes

—

C₄×C₈₃

yes

C₄×C₈₃

yes

Dic₈₃

—

#10368

—

maximal, Sylow

C₂×C₈₃

—

C₂×C₈₃

—

C₂×C₈₃

#2025

yes

characteristic

100

C₂×C₈₃

#2031

yes

—

101

#2864

—

102

#1962

D₈₃

yes

characteristic

103

#1517

—

104

#1286

—

105

#1276

—

106

C₈₃

#10368

yes

characteristic, complemented, Sylow

107

#532

C₂×D₈₃

yes

characteristic

108

#431

—

109

#423

C₂×D₈₃

yes

characteristic

110

C₂×#104

—

111

#389

—

166

—

112

#387

—

166

—

113

#382

—

114

C₄³

C₂×D₈₃

yes

characteristic

115

#327

—

166

—

116

C₂×#41

—

117

C₂×M₁₆

—

118

C₄⋊Q₈

Dic₁₆₆

yes

characteristic

119

#127

Dic₁₆₆

yes

characteristic

120

#124

—

121

#121

—

122

C₄×D₄

—

123

C₂×C₄⋊C₄

—

166

—

124

C₂×C₄⋊C₄

—

166

—

125

C₂×C₄²

—

126

C₂×C₄²

C₂²⋊D₈₃

yes

—

127

C₂×C₄²

C₂²×D₈₃

yes

characteristic

128

C₄⋊C₈

—

129

C₄⋊C₈

—

166

—

130

#104

—

131

C₂²⋊C₈

—

166

—

132

C₄×C₈

—

166

—

133

#95

—

134

#95

—

166

—

135

#95

—

166

—

136

#41

—

137

#41

—

138

C₂×Q₈

—

139

C₂×D₄

—

140

C₂×D₄

—

141

C₂×D₄

—

142

C₂²×C₄

—

143

C₂²×C₄

—

144

C₂²×C₄

—

166

—

145

C₂²×C₄

D₄×D₈₃

yes

—

146

C₂²×C₄

C₂×C₂²⋊D₈₃

yes

characteristic

147

M₁₆

—

148

C₂×C₈

—

149

C₂×C₈

—

332

—

150

C₄⋊C₄

—

151

C₄⋊C₄

—

332

—

152

C₄⋊C₄

—

332

—

153

C₄⋊C₄

—

332

—

154

C₂²⋊C₄

—

155

C₄²

—

156

C₄²

—

157

C₄²

—

158

C₄²

C₂×Dic₁₆₆

yes

characteristic

159

C₄²

#13050589

yes

—

160

C₄²

#13050590

yes

characteristic

161

C₂³

—

162

C₂³

#14581760

yes

characteristic

163

Q₈

—

164

D₄

—

165

D₄

—

166

D₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

332

—

175

C₂×C₄

—

332

—

176

C₂×C₄

#14581769

yes

—

177

C₂×C₄

#14581771

yes

characteristic

178

C₂×C₄

#14581778

yes

—

179

C₂×C₄

#14581779

yes

characteristic

180

C₂×C₄

#14581812

yes

—

181

C₂×C₄

#14581818

yes

—

182

C₈

—

183

C₈

—

664

—

184

C₂²

—

185

C₂²

—

186

C₂²

—

187

C₂²

#19127613

yes

characteristic

188

C₂²

#19127844

yes

—

189

C₄

—

190

C₄

—

191

C₄

—

192

C₄

—

193

C₄

—

664

—

194

C₄

#19128001

yes

—

195

C₄

#19128055

yes

characteristic

196

C₄

#19128065

yes

characteristic

197

C₂

—

198

C₂

—

199

C₂

#24908609

yes

characteristic

200

C₂

#24908635

yes

—

	subgroup	G/H
center	C₂²	#19127613
commutator, cosocle	C₂×C₄²×C₈₃	C₂³
Frattini	C₂×C₄²	C₂²×D₈₃
Fitting	C₈₃×#1962	C₂
socle	C₂²×C₈₃	#393
prime radical	#532×C₈₃	C₂²
hypercenter	#1962	D₈₃
hypocenter	C₈₃	#10368
LCS	G ▷ C₂×C₄²×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1962

subgroup

G/H

center

C₂²

#19127613

commutator, cosocle

C₂×C₄²×C₈₃

C₂³

Frattini

C₂×C₄²

C₂²×D₈₃

Fitting

C₈₃×#1962

C₂

socle

C₂²×C₈₃

#393

prime radical

#532×C₈₃

C₂²

D₈₃

C₈₃

G ▷ C₂×C₄²×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1962

N	H ≅ G/N	type
1. #24915850	C₂	semidirect
3. #24905323	C₂	semidirect
4. #24905296	C₂	semidirect
106. C₈₃	#10368	semidirect

H ≅ G/N

type

1. #24915850

C₂

semidirect

3. #24905323

C₂

semidirect

4. #24905296

C₂

semidirect

106. C₈₃

#10368

semidirect

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

subgroup	index
#24915850	2
#24906023	2
#24905323	2
#24905296	2
C₈₃×#1962	2
#10368	83

subgroup

index

C₈₃×#1962

Subgroups — #35307522

Conjugacy classes of subgroups

Complements

Maximal subgroups