Group #35309179 — subgroups

Subgroups — #35309179

Order 21248.

292 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 93 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	#24908444	C₂	1	1	yes	maximal, characteristic
2	#24907617	C₂	1	1	yes	maximal, characteristic
3	#24906118	C₂	1	2	yes	maximal
4	#24905359	C₂	1	2	yes	maximal
5	C₈₃×#1274	C₂	1	1	yes	maximal, characteristic
6	#19128258	C₂²	1	1	yes	characteristic
7	#19127725	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊(C₈₃⋊C₈)	—	2	1	no	—
9	C₂×C₄⋊(C₈₃⋊C₈)	—	2	1	no	—
10	C₂×C₄⋊(C₈₃⋊C₈)	C₂²	1	2	yes	—
11	#19127461	—	2	1	no	—
12	#19127461	—	2	1	no	—
13	#19127449	—	2	2	no	—
14	#19127448	—	2	2	no	—
15	#431×C₈₃	C₂²	1	2	yes	—
16	#376×C₈₃	C₂²	1	1	yes	characteristic
17	C₂×#13050574	—	2	1	no	—
18	C₂²×C₈₃⋊C₈	—	2	2	no	—
19	#14581792	D₄	1	2	yes	—
20	#14581784	—	2	1	no	—
21	#14581771	—	2	2	no	—
22	C₄×Dic₁₆₆	—	2	2	no	—
23	#14581751	—	2	1	no	—
24	C₄⋊(C₈₃⋊C₈)	—	2	2	no	—
25	C₄⋊(C₈₃⋊C₈)	—	2	2	no	—
26	C₄⋊(C₈₃⋊C₈)	—	2	2	no	—
27	C₄⋊(C₈₃⋊C₈)	—	4	2	no	—
28	C₄⋊(C₈₃⋊C₈)	—	4	2	no	—
29	C₂×C₄²×C₈₃	D₄	1	1	yes	characteristic
30	C₂×C₄²×C₈₃	D₄	1	1	yes	characteristic
31	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
32	C₄⋊C₈×C₈₃	—	2	2	no	—
33	C₂²⋊C₈×C₈₃	—	2	2	no	—
34	C₄×C₈×C₈₃	—	2	2	no	—
35	#95×C₈₃	—	2	2	no	—
36	C₂×Dic₁₆₆	—	2	2	no	—
37	#13050574	—	4	1	no	—
38	#13050574	—	4	1	no	—
39	C₂×C₈₃⋊C₈	—	4	2	no	—
40	C₂×C₈₃⋊C₈	—	4	1	no	—
41	C₂×C₈₃⋊C₈	—	4	2	no	—
42	C₂×C₈₃⋊C₈	—	4	2	no	—
43	C₄⋊Dic₈₃	—	2	2	no	—
44	C₄⋊Dic₈₃	—	2	2	no	—
45	Dic₈₃⋊C₄	—	4	2	no	—
46	C₂²⋊Dic₈₃	—	4	1	no	—
47	C₄×Dic₈₃	—	4	1	no	—
48	C₂²×C₄×C₈₃	—	2	1	no	—
49	C₂²×C₄×C₈₃	—	2	1	no	—
50	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄×C₈₃	#41	1	2	yes	—
52	C₂×C₈×C₈₃	—	4	2	no	—
53	C₄²×C₈₃	—	2	1	no	—
54	C₄²×C₈₃	—	2	1	no	—
55	C₄²×C₈₃	SD₁₆	1	2	yes	—
56	C₄²×C₈₃	Q₁₆	1	2	yes	—
57	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
58	C₄²×C₈₃	#41	1	2	yes	—
59	C₄²×C₈₃	#41	1	1	yes	characteristic
60	C₂³×C₈₃	#123	1	1	yes	characteristic
61	C₂×C₄×C₈₃	—	2	2	no	—
62	C₂×C₄×C₈₃	—	2	2	no	—
63	C₂×C₄×C₈₃	—	4	2	no	—
64	C₂×C₄×C₈₃	—	4	2	no	—
65	C₂×C₄×C₈₃	—	4	2	no	—
66	C₂×C₄×C₈₃	#121	1	1	yes	characteristic
67	C₂×C₄×C₈₃	#123	1	2	yes	—
68	C₂×C₄×C₈₃	#124	1	1	yes	characteristic
69	C₂×C₄×C₈₃	#126	1	2	yes	—
70	C₂×C₄×C₈₃	C₂×SD₁₆	1	1	yes	characteristic
71	C₂×C₄×C₈₃	C₂×Q₁₆	1	1	yes	characteristic
72	C₂×C₄×C₈₃	AGL(1,ℤ/8)	1	1	yes	characteristic
73	C₂×C₄×C₈₃	#137	1	1	yes	characteristic
74	C₂×Dic₈₃	—	4	2	no	—
75	Dic₁₆₆	—	8	2	no	—
76	C₈×C₈₃	—	8	2	no	—
77	C₈₃⋊C₈	—	8	2	no	—
78	C₈₃⋊C₈	—	8	1	no	—
79	C₂²×C₈₃	—	2	2	no	—
80	C₂²×C₈₃	#397	1	1	yes	characteristic
81	C₂²×C₈₃	#483	1	1	yes	characteristic
82	C₂²×C₈₃	#484	1	1	yes	characteristic
83	C₄×C₈₃	—	2	2	no	—
84	C₄×C₈₃	—	2	2	no	—
85	C₄×C₈₃	—	4	2	no	—
86	C₄×C₈₃	—	4	2	no	—
87	C₄×C₈₃	#460	1	1	yes	characteristic
88	C₄×C₈₃	#462	1	1	yes	characteristic
89	C₄×C₈₃	#487	1	1	yes	characteristic
90	C₄×C₈₃	#489	1	1	yes	characteristic
91	Dic₈₃	—	8	2	no	—
92	#10628	—	83	1	no	maximal, Sylow
93	C₂×C₈₃	—	2	1	no	—
94	C₂×C₈₃	—	2	1	no	—
95	C₂×C₈₃	#2072	1	1	yes	characteristic
96	C₂×C₈₃	#2080	1	2	yes	—
97	#1995	—	83	1	no	—
98	#1843	—	83	1	no	—
99	#1542	—	83	2	no	—
100	#1301	—	83	2	no	—
101	#1274	D₈₃	1	1	yes	characteristic
102	C₈₃	#10628	1	1	yes	characteristic, complemented, Sylow
103	#533	—	83	1	no	—
104	#431	C₂×D₈₃	1	2	yes	—
105	#423	—	83	1	no	—
106	C₂×C₄⋊C₈	—	83	2	no	—
107	C₂×C₄⋊C₈	—	166	1	no	—
108	C₂×C₄⋊C₈	—	166	1	no	—
109	#376	C₂×D₈₃	1	1	yes	characteristic
110	#339	—	166	1	no	—
111	#339	—	166	1	no	—
112	#333	—	166	2	no	—
113	#332	—	166	2	no	—
114	C₂×M₁₆	—	166	1	no	—
115	C₂²×C₈	—	166	2	no	—
116	C₄⋊Q₈	—	83	2	no	—
117	#125	—	166	1	no	—
118	C₂²⋊Q₈	—	166	2	no	—
119	C₄×Q₈	—	166	2	no	—
120	#117	—	166	1	no	—
121	C₂×C₄²	D₃₃₂	1	1	yes	characteristic
122	C₂×C₄²	D₃₃₂	1	1	yes	characteristic
123	C₂×C₄²	C₂²×D₈₃	1	1	yes	characteristic
124	C₄⋊C₈	—	2	2	no	—
125	C₄⋊C₈	—	332	2	no	—
126	C₄⋊C₈	—	332	2	no	—
127	C₄⋊C₈	—	166	2	no	—
128	C₄⋊C₈	—	166	2	no	—
129	C₄⋊C₈	—	166	2	no	—
130	C₂²⋊C₈	—	2	2	no	—
131	C₄×C₈	—	2	2	no	—
132	#95	—	2	2	no	—
133	C₂×Q₈	—	166	2	no	—
134	C₂²×C₄	—	2	1	no	—
135	C₂²×C₄	—	2	1	no	—
136	C₂²×C₄	C₂×D₃₃₂	1	1	yes	characteristic
137	C₂²×C₄	#13050589	1	2	yes	—
138	M₁₆	—	332	1	no	—
139	M₁₆	—	332	1	no	—
140	C₂×C₈	—	4	2	no	—
141	C₂×C₈	—	332	2	no	—
142	C₂×C₈	—	332	1	no	—
143	C₂×C₈	—	332	2	no	—
144	C₂×C₈	—	332	2	no	—
145	C₄⋊C₄	—	332	2	no	—
146	C₄⋊C₄	—	166	2	no	—
147	C₄⋊C₄	—	166	2	no	—
148	C₂²⋊C₄	—	332	1	no	—
149	C₄²	—	2	1	no	—
150	C₄²	—	2	1	no	—
151	C₄²	—	332	1	no	—
152	C₄²	#13050579	1	2	yes	—
153	C₄²	Dic₃₃₂	1	2	yes	—
154	C₄²	D₄×D₈₃	1	1	yes	characteristic
155	C₄²	#13050590	1	2	yes	—
156	C₄²	#13050591	1	1	yes	characteristic
157	C₂³	#14581776	1	1	yes	characteristic
158	Q₈	—	664	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	4	2	no	—
162	C₂×C₄	—	4	2	no	—
163	C₂×C₄	—	4	2	no	—
164	C₂×C₄	—	332	2	no	—
165	C₂×C₄	#14581767	1	1	yes	characteristic
166	C₂×C₄	#14581774	1	2	yes	—
167	C₂×C₄	#14581780	1	1	yes	characteristic
168	C₂×C₄	#14581787	1	2	yes	—
169	C₂×C₄	C₂×#13050579	1	1	yes	characteristic
170	C₂×C₄	C₂×Dic₃₃₂	1	1	yes	characteristic
171	C₂×C₄	#14581820	1	1	yes	characteristic
172	C₂×C₄	#14581825	1	1	yes	characteristic
173	C₈	—	8	2	no	—
174	C₈	—	664	2	no	—
175	C₈	—	664	1	no	—
176	C₂²	—	2	2	no	—
177	C₂²	#19127628	1	1	yes	characteristic
178	C₂²	#19128032	1	1	yes	characteristic
179	C₂²	#19128037	1	1	yes	characteristic
180	C₄	—	2	2	no	—
181	C₄	—	2	2	no	—
182	C₄	—	4	2	no	—
183	C₄	—	4	2	no	—
184	C₄	—	664	2	no	—
185	C₄	#19127902	1	1	yes	characteristic
186	C₄	#19127912	1	1	yes	characteristic
187	C₄	#19128052	1	1	yes	characteristic
188	C₄	#19128061	1	1	yes	characteristic
189	C₂	—	2	1	no	—
190	C₂	—	2	1	no	—
191	C₂	#24908873	1	1	yes	characteristic
192	C₂	#24908915	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#24908444

C₂

yes

maximal, characteristic

#24907617

C₂

yes

maximal, characteristic

#24906118

C₂

yes

maximal

#24905359

C₂

yes

maximal

C₈₃×#1274

C₂

yes

maximal, characteristic

#19128258

C₂²

yes

characteristic

#19127725

C₂²

yes

characteristic

C₂×C₄⋊(C₈₃⋊C₈)

—

C₂×C₄⋊(C₈₃⋊C₈)

—

C₂×C₄⋊(C₈₃⋊C₈)

C₂²

yes

—

—

—

—

—

#431×C₈₃

C₂²

yes

—

#376×C₈₃

C₂²

yes

characteristic

C₂×#13050574

—

C₂²×C₈₃⋊C₈

—

#14581792

D₄

yes

—

#14581784

—

#14581771

—

C₄×Dic₁₆₆

—

#14581751

—

C₄⋊(C₈₃⋊C₈)

—

C₄⋊(C₈₃⋊C₈)

—

C₄⋊(C₈₃⋊C₈)

—

C₄⋊(C₈₃⋊C₈)

—

C₄⋊(C₈₃⋊C₈)

—

C₂×C₄²×C₈₃

D₄

yes

characteristic

C₂×C₄²×C₈₃

D₄

yes

characteristic

C₂×C₄²×C₈₃

C₂³

yes

characteristic

C₄⋊C₈×C₈₃

—

C₂²⋊C₈×C₈₃

—

C₄×C₈×C₈₃

—

#95×C₈₃

—

C₂×Dic₁₆₆

—

#13050574

—

#13050574

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

—

Dic₈₃⋊C₄

—

C₂²⋊Dic₈₃

—

C₄×Dic₈₃

—

C₂²×C₄×C₈₃

—

C₂²×C₄×C₈₃

—

C₂²×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₈₃

#41

yes

—

C₂×C₈×C₈₃

—

C₄²×C₈₃

—

C₄²×C₈₃

—

C₄²×C₈₃

SD₁₆

yes

—

C₄²×C₈₃

Q₁₆

yes

—

C₄²×C₈₃

C₂×D₄

yes

characteristic

C₄²×C₈₃

#41

yes

—

C₄²×C₈₃

yes

C₂³×C₈₃

yes

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

#121

yes

characteristic

C₂×C₄×C₈₃

#123

yes

—

C₂×C₄×C₈₃

#124

yes

characteristic

C₂×C₄×C₈₃

#126

yes

—

C₂×C₄×C₈₃

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₈₃

C₂×Q₁₆

yes

C₂×C₄×C₈₃

yes

C₂×C₄×C₈₃

yes

C₂×Dic₈₃

—

Dic₁₆₆

—

C₈×C₈₃

—

C₈₃⋊C₈

—

C₈₃⋊C₈

—

C₂²×C₈₃

—

C₂²×C₈₃

yes

C₂²×C₈₃

yes

C₂²×C₈₃

yes

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

yes

C₄×C₈₃

yes

C₄×C₈₃

yes

C₄×C₈₃

yes

Dic₈₃

—

#10628

—

maximal, Sylow

C₂×C₈₃

—

C₂×C₈₃

—

C₂×C₈₃

#2072

yes

characteristic

C₂×C₈₃

#2080

yes

—

#1995

—

#1843

—

#1542

—

100

#1301

—

101

#1274

D₈₃

yes

characteristic

102

C₈₃

#10628

yes

characteristic, complemented, Sylow

103

#533

—

104

#431

C₂×D₈₃

yes

—

105

#423

—

106

C₂×C₄⋊C₈

—

107

C₂×C₄⋊C₈

—

166

—

108

C₂×C₄⋊C₈

—

166

—

109

#376

C₂×D₈₃

yes

characteristic

110

#339

—

166

—

111

#339

—

166

—

112

#333

—

166

—

113

#332

—

166

—

114

C₂×M₁₆

—

166

—

115

C₂²×C₈

—

166

—

116

C₄⋊Q₈

—

117

#125

—

166

—

118

C₂²⋊Q₈

—

166

—

119

C₄×Q₈

—

166

—

120

#117

—

166

—

121

C₂×C₄²

D₃₃₂

yes

characteristic

122

C₂×C₄²

D₃₃₂

yes

characteristic

123

C₂×C₄²

C₂²×D₈₃

yes

characteristic

124

C₄⋊C₈

—

125

C₄⋊C₈

—

332

—

126

C₄⋊C₈

—

332

—

127

C₄⋊C₈

—

166

—

128

C₄⋊C₈

—

166

—

129

C₄⋊C₈

—

166

—

130

C₂²⋊C₈

—

131

C₄×C₈

—

132

#95

—

133

C₂×Q₈

—

166

—

134

C₂²×C₄

—

135

C₂²×C₄

—

136

C₂²×C₄

C₂×D₃₃₂

yes

characteristic

137

C₂²×C₄

#13050589

yes

—

138

M₁₆

—

332

—

139

M₁₆

—

332

—

140

C₂×C₈

—

141

C₂×C₈

—

332

—

142

C₂×C₈

—

332

—

143

C₂×C₈

—

332

—

144

C₂×C₈

—

332

—

145

C₄⋊C₄

—

332

—

146

C₄⋊C₄

—

166

—

147

C₄⋊C₄

—

166

—

148

C₂²⋊C₄

—

332

—

149

C₄²

—

150

C₄²

—

151

C₄²

—

332

—

152

C₄²

#13050579

yes

—

153

C₄²

Dic₃₃₂

yes

—

154

C₄²

D₄×D₈₃

yes

characteristic

155

C₄²

#13050590

yes

—

156

C₄²

#13050591

yes

characteristic

157

C₂³

#14581776

yes

characteristic

158

Q₈

—

664

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

332

—

165

C₂×C₄

#14581767

yes

characteristic

166

C₂×C₄

#14581774

yes

—

167

C₂×C₄

#14581780

yes

characteristic

168

C₂×C₄

#14581787

yes

—

169

C₂×C₄

C₂×#13050579

yes

characteristic

170

C₂×C₄

C₂×Dic₃₃₂

yes

characteristic

171

C₂×C₄

#14581820

yes

characteristic

172

C₂×C₄

#14581825

yes

characteristic

173

C₈

—

174

C₈

—

664

—

175

C₈

—

664

—

176

C₂²

—

177

C₂²

#19127628

yes

characteristic

178

C₂²

#19128032

yes

characteristic

179

C₂²

#19128037

yes

characteristic

180

C₄

—

181

C₄

—

182

C₄

—

183

C₄

—

184

C₄

—

664

—

185

C₄

#19127902

yes

characteristic

186

C₄

#19127912

yes

characteristic

187

C₄

#19128052

yes

characteristic

188

C₄

#19128061

yes

characteristic

189

C₂

—

190

C₂

—

191

C₂

#24908873

yes

characteristic

192

C₂

#24908915

yes

—

	subgroup	G/H
center	C₂×C₄	#14581780
commutator, cosocle	C₂×C₄²×C₈₃	C₂³
Frattini	C₂×C₄²	C₂²×D₈₃
Fitting	C₈₃×#1274	C₂
socle	C₂²×C₈₃	#397
prime radical	C₂³×C₈₃	#123
hypercenter	#1274	D₈₃
hypocenter	C₈₃	#10628
LCS	G ▷ C₂×C₄²×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1274

subgroup

G/H

center

C₂×C₄

#14581780

commutator, cosocle

C₂×C₄²×C₈₃

C₂³

Frattini

C₂×C₄²

C₂²×D₈₃

Fitting

C₈₃×#1274

C₂

C₂²×C₈₃

C₂³×C₈₃

D₈₃

C₈₃

G ▷ C₂×C₄²×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1274

N	H ≅ G/N	type
102. C₈₃	#10628	semidirect

H ≅ G/N

type

102. C₈₃

#10628

semidirect

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

subgroup	index
#24908444	2
#24907617	2
#24906118	2
#24905359	2
C₈₃×#1274	2
#10628	83

subgroup

index

C₈₃×#1274

Subgroups — #35309179

Conjugacy classes of subgroups

Complements

Maximal subgroups