Group #35318553 — subgroups

Subgroups — #35318553

Order 21248.

218 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 69 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	#24909615	C₂	1	1	yes	maximal, characteristic
2	#24907057	C₂	1	1	yes	maximal, characteristic
3	Q₁₆×C₈₃⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#24905526	C₂	1	1	yes	maximal, characteristic
5	#24905520	C₂	1	1	yes	maximal, characteristic
6	#24905512	C₂	1	1	yes	maximal, characteristic
7	C₈₃×#2209	C₂	1	1	yes	maximal, characteristic
8	#19128106	C₂²	1	1	yes	characteristic
9	#19128097	—	2	1	no	—
10	#19128053	—	2	1	no	—
11	Q₈×C₈₃⋊C₈	—	2	1	no	—
12	#19127526	—	2	1	no	—
13	C₄⋊(C₈₃⋊C₁₆)	C₂²	1	1	yes	characteristic
14	#19127505	—	2	1	no	—
15	#19127454	C₂²	1	1	yes	characteristic
16	Q₈⋊(C₈₃⋊C₈)	—	2	1	no	—
17	C₈×C₈₃⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×Q₃₂×C₈₃	—	2	1	no	—
19	#494×C₈₃	C₂²	1	1	yes	characteristic
20	C₄×Q₁₆×C₈₃	C₂²	1	1	yes	characteristic
21	C₄⋊C₁₆×C₈₃	C₂²	1	1	yes	characteristic
22	#358×C₈₃	—	2	1	no	—
23	#14581792	—	2	1	no	—
24	C₂×C₈₃⋊C₁₆	—	2	1	no	—
25	#14581726	—	2	1	no	—
26	#14581726	—	2	1	no	—
27	#14581723	—	4	1	no	—
28	#14581723	—	4	1	no	—
29	C₄⋊(C₈₃⋊C₈)	—	2	1	no	—
30	C₄⋊(C₈₃⋊C₈)	—	4	1	no	—
31	#14581715	—	4	2	no	—
32	C₄×C₈₃⋊C₈	—	4	1	no	—
33	C₄×C₈₃⋊C₈	D₄	1	1	yes	characteristic
34	C₄×C₈₃⋊C₈	D₄	1	1	yes	characteristic
35	C₂×Q₁₆×C₈₃	—	2	1	no	—
36	C₂×Q₁₆×C₈₃	—	2	1	no	—
37	C₂×Q₁₆×C₈₃	D₄	1	1	yes	characteristic
38	C₄⋊Q₈×C₈₃	—	2	1	no	—
39	C₄×Q₈×C₈₃	—	2	1	no	—
40	Q₃₂×C₈₃	—	4	2	no	—
41	C₂×C₁₆×C₈₃	—	2	1	no	—
42	#107×C₈₃	D₄	1	1	yes	characteristic
43	#103×C₈₃	—	2	1	no	—
44	C₄×C₈×C₈₃	C₂³	1	1	yes	characteristic
45	C₂×Dic₁₆₆	—	4	1	no	—
46	C₂×C₈₃⋊C₈	—	2	1	no	—
47	C₂×C₈₃⋊C₈	—	4	1	no	—
48	C₂×C₈₃⋊C₈	D₈	1	1	yes	characteristic
49	C₂×C₈₃⋊C₈	D₈	1	1	yes	characteristic
50	C₄⋊Dic₈₃	—	4	1	no	—
51	C₄⋊Dic₈₃	—	4	1	no	—
52	C₈₃⋊C₁₆	—	4	1	no	—
53	C₂×Q₈×C₈₃	—	2	1	no	—
54	C₂×Q₈×C₈₃	—	4	1	no	—
55	Q₁₆×C₈₃	—	2	1	no	—
56	Q₁₆×C₈₃	—	4	1	no	—
57	Q₁₆×C₈₃	—	4	1	no	—
58	Q₁₆×C₈₃	—	8	1	no	—
59	Q₁₆×C₈₃	SD₁₆	1	2	yes	—
60	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
61	C₂×C₈×C₈₃	#41	1	1	yes	characteristic
62	C₄⋊C₄×C₈₃	—	2	1	no	—
63	C₄⋊C₄×C₈₃	—	4	1	no	—
64	C₄⋊C₄×C₈₃	—	4	1	no	—
65	C₄⋊C₄×C₈₃	—	4	1	no	—
66	C₄²×C₈₃	—	4	1	no	—
67	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
68	C₁₆×C₈₃	—	4	1	no	—
69	Q₈×C₈₃	—	2	2	no	—
70	Q₈×C₈₃	—	4	1	no	—
71	Q₈×C₈₃	—	8	1	no	—
72	Q₈×C₈₃	—	8	1	no	—
73	C₂×C₄×C₈₃	—	4	1	no	—
74	C₂×C₄×C₈₃	—	4	1	no	—
75	C₂×C₄×C₈₃	—	8	1	no	—
76	C₂×C₄×C₈₃	#121	1	1	yes	characteristic
77	C₂×C₄×C₈₃	C₂×D₈	1	1	yes	characteristic
78	C₂×C₄×C₈₃	AGL(1,ℤ/8)	1	1	yes	characteristic
79	C₂×Dic₈₃	—	8	1	no	—
80	Dic₁₆₆	—	8	2	no	—
81	C₈×C₈₃	—	2	1	no	—
82	C₈×C₈₃	C₂×SD₁₆	1	1	yes	characteristic
83	C₈×C₈₃	#135	1	1	yes	characteristic
84	C₈₃⋊C₈	—	2	1	no	—
85	C₈₃⋊C₈	—	4	1	no	—
86	C₈₃⋊C₈	—	8	1	no	—
87	C₈₃⋊C₈	Q₃₂	1	2	yes	complemented
88	C₂²×C₈₃	#459	1	1	yes	characteristic
89	C₄×C₈₃	—	2	1	no	—
90	C₄×C₈₃	—	4	2	no	—
91	C₄×C₈₃	—	8	1	no	—
92	C₄×C₈₃	—	16	1	no	—
93	C₄×C₈₃	#465	1	1	yes	characteristic
94	C₄×C₈₃	#471	1	1	yes	characteristic
95	C₄×C₈₃	C₂×Q₃₂	1	1	yes	characteristic
96	C₄×C₈₃	#510	1	1	yes	characteristic
97	Dic₈₃	—	16	1	no	—
98	#12421	—	83	1	no	maximal, Sylow
99	C₂×C₈₃	#1665	1	1	yes	characteristic
100	C₂×C₈₃	#2209	1	1	yes	characteristic
101	C₂×C₈₃	#2212	1	1	yes	characteristic
102	#2227	—	83	1	no	—
103	#2209	D₈₃	1	1	yes	characteristic
104	#1707	—	83	1	no	—
105	C₈×Q₁₆	—	83	1	no	—
106	#1367	—	83	1	no	—
107	#1365	—	83	1	no	—
108	#1362	—	83	1	no	—
109	C₈₃	#12421	1	1	yes	characteristic, complemented, Sylow
110	C₂×Q₃₂	—	2	1	no	—
111	#500	—	83	1	no	—
112	#498	—	166	1	no	—
113	#494	C₂×D₈₃	1	1	yes	characteristic
114	#487	—	166	1	no	—
115	C₈×Q₈	—	166	1	no	—
116	C₄×Q₁₆	C₂×D₈₃	1	1	yes	characteristic
117	#366	—	166	1	no	—
118	C₄⋊C₁₆	—	83	1	no	—
119	C₄⋊C₁₆	C₂×D₈₃	1	1	yes	characteristic
120	#358	—	2	1	no	—
121	#358	—	166	1	no	—
122	#335	—	83	1	no	—
123	Q₈⋊C₈	—	166	1	no	—
124	C₈²	—	83	1	no	—
125	C₂×Q₁₆	—	2	1	no	—
126	C₂×Q₁₆	—	2	1	no	—
127	C₂×Q₁₆	C₂²⋊D₈₃	1	1	yes	characteristic
128	C₄⋊Q₈	—	2	1	no	—
129	C₄⋊Q₈	—	166	1	no	—
130	C₄×Q₈	—	2	1	no	—
131	Q₃₂	—	4	2	no	—
132	C₂×C₁₆	—	2	1	no	—
133	C₂×C₁₆	—	166	1	no	—
134	#107	—	166	1	no	—
135	#107	—	166	1	no	—
136	#107	C₂²⋊D₈₃	1	1	yes	characteristic
137	#106	—	332	1	no	—
138	#106	—	332	1	no	—
139	C₄⋊C₈	—	332	1	no	—
140	C₄⋊C₈	—	166	1	no	—
141	#103	—	2	1	no	—
142	#103	—	332	2	no	—
143	C₄×C₈	—	332	1	no	—
144	C₄×C₈	—	83	1	no	—
145	C₄×C₈	—	83	1	no	—
146	C₄×C₈	C₂²×D₈₃	1	1	yes	characteristic
147	C₂×Q₈	—	2	1	no	—
148	C₂×Q₈	—	4	1	no	—
149	C₂×Q₈	—	332	1	no	—
150	Q₁₆	—	2	1	no	—
151	Q₁₆	—	4	1	no	—
152	Q₁₆	—	4	1	no	—
153	Q₁₆	—	8	1	no	—
154	Q₁₆	#13050578	1	2	yes	—
155	C₂×C₈	—	332	1	no	—
156	C₂×C₈	—	83	1	no	—
157	C₂×C₈	—	83	1	no	—
158	C₂×C₈	—	166	1	no	—
159	C₂×C₈	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
160	C₂×C₈	#13050589	1	1	yes	characteristic
161	C₄⋊C₄	—	2	1	no	—
162	C₄⋊C₄	—	4	1	no	—
163	C₄⋊C₄	—	4	1	no	—
164	C₄⋊C₄	—	4	1	no	—
165	C₄⋊C₄	—	332	1	no	—
166	C₄⋊C₄	—	332	1	no	—
167	C₄²	—	4	1	no	—
168	C₄²	D₄×D₈₃	1	1	yes	characteristic
169	C₁₆	—	4	1	no	—
170	C₁₆	—	332	1	no	—
171	Q₈	—	2	2	no	—
172	Q₈	—	4	1	no	—
173	Q₈	—	8	1	no	—
174	Q₈	—	8	1	no	—
175	Q₈	—	664	2	no	—
176	C₂×C₄	—	4	1	no	—
177	C₂×C₄	—	4	1	no	—
178	C₂×C₄	—	8	1	no	—
179	C₂×C₄	—	664	1	no	—
180	C₂×C₄	#14581764	1	1	yes	characteristic
181	C₂×C₄	D₈×D₈₃	1	1	yes	characteristic
182	C₂×C₄	#14581816	1	1	yes	characteristic
183	C₈	—	2	1	no	—
184	C₈	—	332	1	no	—
185	C₈	—	83	2	no	—
186	C₈	—	664	1	no	—
187	C₈	—	166	1	no	—
188	C₈	C₂×#13050578	1	1	yes	characteristic
189	C₈	#14581812	1	1	yes	characteristic
190	C₂²	#19127894	1	1	yes	characteristic
191	C₄	—	2	1	no	—
192	C₄	—	4	2	no	—
193	C₄	—	8	1	no	—
194	C₄	—	16	1	no	—
195	C₄	—	1328	1	no	—
196	C₄	#19127928	1	1	yes	characteristic
197	C₄	#19127962	1	1	yes	characteristic
198	C₄	Q₃₂×D₈₃	1	1	yes	characteristic
199	C₄	#19128143	1	1	yes	characteristic
200	C₂	#24906802	1	1	yes	characteristic
201	C₂	#24909511	1	1	yes	characteristic
202	C₂	#24909534	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#24909615

C₂

yes

maximal, characteristic

#24907057

C₂

yes

maximal, characteristic

Q₁₆×C₈₃⋊C₈

C₂

yes

maximal, characteristic

#24905526

C₂

yes

maximal, characteristic

#24905520

C₂

yes

maximal, characteristic

#24905512

C₂

yes

maximal, characteristic

C₈₃×#2209

C₂

yes

maximal, characteristic

#19128106

C₂²

yes

characteristic

#19128097

—

#19128053

—

Q₈×C₈₃⋊C₈

—

#19127526

—

C₄⋊(C₈₃⋊C₁₆)

C₂²

yes

characteristic

#19127505

—

#19127454

C₂²

yes

characteristic

Q₈⋊(C₈₃⋊C₈)

—

C₈×C₈₃⋊C₈

C₂²

yes

characteristic

C₂×Q₃₂×C₈₃

—

#494×C₈₃

C₂²

yes

characteristic

C₄×Q₁₆×C₈₃

C₂²

yes

characteristic

C₄⋊C₁₆×C₈₃

C₂²

yes

characteristic

#358×C₈₃

—

#14581792

—

C₂×C₈₃⋊C₁₆

—

—

—

—

—

C₄⋊(C₈₃⋊C₈)

—

C₄⋊(C₈₃⋊C₈)

—

#14581715

—

C₄×C₈₃⋊C₈

—

C₄×C₈₃⋊C₈

D₄

yes

characteristic

C₄×C₈₃⋊C₈

D₄

yes

characteristic

C₂×Q₁₆×C₈₃

—

C₂×Q₁₆×C₈₃

—

C₂×Q₁₆×C₈₃

D₄

yes

characteristic

C₄⋊Q₈×C₈₃

—

C₄×Q₈×C₈₃

—

Q₃₂×C₈₃

—

C₂×C₁₆×C₈₃

—

#107×C₈₃

D₄

yes

characteristic

#103×C₈₃

—

C₄×C₈×C₈₃

C₂³

yes

characteristic

C₂×Dic₁₆₆

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

D₈

yes

characteristic

C₂×C₈₃⋊C₈

D₈

yes

characteristic

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

—

C₈₃⋊C₁₆

—

C₂×Q₈×C₈₃

—

C₂×Q₈×C₈₃

—

Q₁₆×C₈₃

—

Q₁₆×C₈₃

—

Q₁₆×C₈₃

—

Q₁₆×C₈₃

—

Q₁₆×C₈₃

SD₁₆

yes

—

C₂×C₈×C₈₃

C₂×D₄

yes

characteristic

C₂×C₈×C₈₃

#41

yes

characteristic

C₄⋊C₄×C₈₃

—

C₄⋊C₄×C₈₃

—

C₄⋊C₄×C₈₃

—

C₄⋊C₄×C₈₃

—

C₄²×C₈₃

—

C₄²×C₈₃

C₂×D₄

yes

characteristic

C₁₆×C₈₃

—

Q₈×C₈₃

—

Q₈×C₈₃

—

Q₈×C₈₃

—

Q₈×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

#121

yes

characteristic

C₂×C₄×C₈₃

C₂×D₈

yes

characteristic

C₂×C₄×C₈₃

AGL(1,ℤ/8)

yes

characteristic

C₂×Dic₈₃

—

Dic₁₆₆

—

C₈×C₈₃

—

C₈×C₈₃

C₂×SD₁₆

yes

characteristic

C₈×C₈₃

#135

yes

characteristic

C₈₃⋊C₈

—

C₈₃⋊C₈

—

C₈₃⋊C₈

—

C₈₃⋊C₈

Q₃₂

yes

complemented

C₂²×C₈₃

#459

yes

characteristic

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

yes

C₄×C₈₃

yes

C₄×C₈₃

C₂×Q₃₂

yes

characteristic

C₄×C₈₃

#510

yes

characteristic

Dic₈₃

—

—

C₂×C₈₃

yes

100

C₂×C₈₃

#2209

yes

characteristic

101

C₂×C₈₃

#2212

yes

characteristic

102

#2227

—

103

#2209

D₈₃

yes

characteristic

104

#1707

—

105

C₈×Q₁₆

—

106

#1367

—

107

#1365

—

108

#1362

—

109

C₈₃

#12421

yes

characteristic, complemented, Sylow

110

C₂×Q₃₂

—

111

#500

—

112

#498

—

166

—

113

#494

C₂×D₈₃

yes

characteristic

114

#487

—

166

—

115

C₈×Q₈

—

166

—

116

C₄×Q₁₆

C₂×D₈₃

yes

characteristic

117

#366

—

166

—

118

C₄⋊C₁₆

—

119

C₄⋊C₁₆

C₂×D₈₃

yes

characteristic

120

#358

—

121

#358

—

166

—

122

#335

—

123

Q₈⋊C₈

—

166

—

124

C₈²

—

125

C₂×Q₁₆

—

126

C₂×Q₁₆

—

127

C₂×Q₁₆

C₂²⋊D₈₃

yes

characteristic

128

C₄⋊Q₈

—

129

C₄⋊Q₈

—

166

—

130

C₄×Q₈

—

131

Q₃₂

—

132

C₂×C₁₆

—

133

C₂×C₁₆

—

166

—

134

#107

—

166

—

135

#107

—

166

—

136

#107

C₂²⋊D₈₃

yes

characteristic

137

#106

—

332

—

138

#106

—

332

—

139

C₄⋊C₈

—

332

—

140

C₄⋊C₈

—

166

—

141

#103

—

142

#103

—

332

—

143

C₄×C₈

—

332

—

144

C₄×C₈

—

145

C₄×C₈

—

146

C₄×C₈

C₂²×D₈₃

yes

characteristic

147

C₂×Q₈

—

148

C₂×Q₈

—

149

C₂×Q₈

—

332

—

150

Q₁₆

—

151

Q₁₆

—

152

Q₁₆

—

153

Q₁₆

—

154

Q₁₆

#13050578

yes

—

155

C₂×C₈

—

332

—

156

C₂×C₈

—

157

C₂×C₈

—

158

C₂×C₈

—

166

—

159

C₂×C₈

C₂×C₂²⋊D₈₃

yes

characteristic

160

C₂×C₈

#13050589

yes

characteristic

161

C₄⋊C₄

—

162

C₄⋊C₄

—

163

C₄⋊C₄

—

164

C₄⋊C₄

—

165

C₄⋊C₄

—

332

—

166

C₄⋊C₄

—

332

—

167

C₄²

—

168

C₄²

D₄×D₈₃

yes

characteristic

169

C₁₆

—

170

C₁₆

—

332

—

171

Q₈

—

172

Q₈

—

173

Q₈

—

174

Q₈

—

175

Q₈

—

664

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

—

664

—

180

C₂×C₄

#14581764

yes

characteristic

181

C₂×C₄

D₈×D₈₃

yes

characteristic

182

C₂×C₄

#14581816

yes

characteristic

183

C₈

—

184

C₈

—

332

—

185

C₈

—

186

C₈

—

664

—

187

C₈

—

166

—

188

C₈

C₂×#13050578

yes

characteristic

189

C₈

#14581812

yes

characteristic

190

C₂²

#19127894

yes

characteristic

191

C₄

—

192

C₄

—

193

C₄

—

194

C₄

—

195

C₄

—

1328

—

196

C₄

#19127928

yes

characteristic

197

C₄

#19127962

yes

characteristic

198

C₄

Q₃₂×D₈₃

yes

characteristic

199

C₄

#19128143

yes

characteristic

200

C₂

#24906802

yes

characteristic

201

C₂

#24909511

yes

characteristic

202

C₂

#24909534

yes

characteristic

	subgroup	G/H
center	C₂²	#19127894
commutator, cosocle	C₄×C₈×C₈₃	C₂³
Frattini	C₄×C₈	C₂²×D₈₃
Fitting	C₈₃×#2209	C₂
socle, prime radical	C₂²×C₈₃	#459
hypercenter	#2209	D₈₃
hypocenter	C₈₃	#12421
LCS	G ▷ C₄×C₈×C₈₃ ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2209

subgroup

G/H

center

C₂²

#19127894

commutator, cosocle

C₄×C₈×C₈₃

C₂³

Frattini

C₄×C₈

C₂²×D₈₃

Fitting

C₈₃×#2209

C₂

C₂²×C₈₃

D₈₃

C₈₃

G ▷ C₄×C₈×C₈₃ ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2209

N	H ≅ G/N	type
87. C₈₃⋊C₈	Q₃₂	semidirect
109. C₈₃	#12421	semidirect

H ≅ G/N

type

87. C₈₃⋊C₈

Q₃₂

semidirect

109. C₈₃

#12421

semidirect

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

subgroup	index
#24909615	2
#24907057	2
Q₁₆×C₈₃⋊C₈	2
#24905526	2
#24905520	2
#24905512	2
C₈₃×#2209	2
#12421	83

subgroup

index

Q₁₆×C₈₃⋊C₈

C₈₃×#2209

Subgroups — #35318553

Conjugacy classes of subgroups

Complements

Maximal subgroups