Group #35323957 — subgroups

Subgroups — #35323957

Order 21248.

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

normal subgroups only

This group has 378 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	C₂²×Dic₆₆₄	C₂	1	1	yes	maximal, characteristic
2	#24909221	C₂	1	1	yes	maximal, characteristic
3	C₂×#19127516	C₂	1	2	yes	maximal
4	#24908376	C₂	1	2	yes	maximal
5	C₈₃×#1381	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₃₃₂	C₂²	1	2	yes	—
7	C₂×Dic₆₆₄	—	2	2	no	—
8	C₂×Dic₆₆₄	—	2	4	no	—
9	C₂×#14581732	C₂²	1	1	yes	characteristic
10	#19127711	—	2	2	no	—
11	C₂×#14581710	—	2	2	no	—
12	#19127656	C₂²	1	1	yes	characteristic
13	Dic₈₃⋊C₁₆	—	2	2	no	—
14	#19127516	—	2	4	no	—
15	C₂²×C₁₆×C₈₃	C₂²	1	1	yes	characteristic
16	C₂×#108×C₈₃	C₂²	1	2	yes	—
17	C₂²×Dic₁₆₆	—	2	2	no	—
18	C₂×Dic₃₃₂	—	2	2	no	—
19	C₂×Dic₃₃₂	—	2	2	no	—
20	C₂×Dic₃₃₂	—	2	2	no	—
21	C₂×Dic₃₃₂	—	2	2	no	—
22	C₂×Dic₃₃₂	—	2	2	no	—
23	C₂×Dic₃₃₂	D₄	1	4	yes	—
24	C₂×#13050574	D₄	1	1	yes	characteristic
25	#14581751	D₄	1	1	yes	characteristic
26	Dic₆₆₄	—	2	8	no	—
27	#14581732	—	2	2	no	—
28	C₂×C₈₃⋊C₁₆	—	2	1	no	—
29	#14581717	—	4	4	no	—
30	#14581710	—	4	4	no	—
31	#14581702	—	2	1	no	—
32	C₈×Dic₈₃	—	2	1	no	—
33	C₂×M₁₆×C₈₃	—	2	2	no	—
34	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
35	C₂×C₁₆×C₈₃	—	2	2	no	—
36	C₂×C₁₆×C₈₃	D₄	1	2	yes	—
37	#108×C₈₃	—	2	2	no	—
38	#108×C₈₃	—	2	2	no	—
39	C₂×Dic₁₆₆	—	2	4	no	—
40	C₂×Dic₁₆₆	—	4	2	no	—
41	C₂×Dic₁₆₆	—	4	2	no	—
42	C₂×Dic₁₆₆	—	4	2	no	—
43	C₂×Dic₁₆₆	—	4	2	no	—
44	C₂×Dic₁₆₆	—	4	2	no	—
45	C₂²×Dic₈₃	—	4	2	no	—
46	Dic₃₃₂	—	2	4	no	—
47	Dic₃₃₂	—	2	4	no	—
48	Dic₃₃₂	—	4	2	no	—
49	Dic₃₃₂	—	4	2	no	—
50	#13050574	—	2	2	no	—
51	C₂×C₈₃⋊C₈	—	2	1	no	—
52	C₄⋊Dic₈₃	D₈	1	2	yes	—
53	C₂²⋊Dic₈₃	—	2	1	no	—
54	C₄×Dic₈₃	—	2	1	no	—
55	C₈₃⋊C₁₆	—	4	1	no	—
56	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
57	M₁₆×C₈₃	—	4	2	no	—
58	M₁₆×C₈₃	—	4	2	no	—
59	C₂×C₈×C₈₃	—	4	2	no	—
60	C₂×C₈×C₈₃	C₂×D₄	1	2	yes	—
61	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
62	C₂×C₈×C₈₃	#41	1	2	yes	—
63	C₂×C₈×C₈₃	#41	1	1	yes	characteristic
64	C₁₆×C₈₃	—	2	2	no	—
65	C₂³×C₈₃	#135	1	1	yes	characteristic
66	C₂×C₄×C₈₃	#121	1	2	yes	—
67	C₂×C₄×C₈₃	#121	1	1	yes	characteristic
68	C₂×C₄×C₈₃	C₂×D₈	1	1	yes	characteristic
69	C₂×C₄×C₈₃	AGL(1,ℤ/8)	1	2	yes	—
70	C₂×Dic₈₃	—	2	2	no	—
71	C₂×Dic₈₃	—	4	4	no	—
72	C₂×Dic₈₃	—	8	2	no	—
73	C₂×Dic₈₃	—	8	2	no	—
74	Dic₁₆₆	—	4	4	no	—
75	Dic₁₆₆	—	4	4	no	—
76	Dic₁₆₆	—	8	2	no	—
77	Dic₁₆₆	—	8	2	no	—
78	C₈×C₈₃	—	8	2	no	—
79	C₈×C₈₃	#120	1	1	yes	characteristic
80	C₈×C₈₃	#123	1	2	yes	—
81	C₈×C₈₃	#124	1	1	yes	characteristic
82	C₈₃⋊C₈	—	4	1	no	—
83	C₂²×C₈₃	—	2	2	no	—
84	C₂²×C₈₃	#463	1	2	yes	—
85	C₂²×C₈₃	#466	1	1	yes	characteristic
86	C₄×C₈₃	#394	1	1	yes	characteristic
87	C₄×C₈₃	#459	1	2	yes	—
88	C₄×C₈₃	#469	1	1	yes	characteristic
89	Dic₈₃	—	4	2	no	—
90	Dic₈₃	—	8	4	no	—
91	#13441	—	83	1	no	maximal, Sylow
92	C₂×C₈₃	—	2	2	no	—
93	C₂×C₈₃	#2054	1	1	yes	characteristic
94	C₂×C₈₃	#2222	1	2	yes	—
95	C₂²×Q₃₂	—	83	1	no	—
96	#2151	—	83	1	no	—
97	C₂×#362	—	83	2	no	—
98	#1979	—	83	2	no	—
99	#1381	D₈₃	1	1	yes	characteristic
100	C₈₃	#13441	1	1	yes	characteristic, complemented, Sylow
101	C₂²×Q₁₆	—	83	2	no	—
102	C₂×Q₃₂	—	166	4	no	—
103	C₂×Q₃₂	—	166	2	no	—
104	C₂×M₃₂	—	83	1	no	—
105	C₂²×C₁₆	C₂×D₈₃	1	1	yes	characteristic
106	C₂×#108	C₂×D₈₃	1	2	yes	—
107	#419	—	166	2	no	—
108	C₂×#101	—	166	2	no	—
109	#405	—	83	1	no	—
110	C₄⋊C₁₆	—	166	2	no	—
111	#362	—	166	4	no	—
112	C₂²×Q₈	—	166	2	no	—
113	C₂×Q₁₆	—	83	4	no	—
114	C₂×Q₁₆	—	166	2	no	—
115	C₂×Q₁₆	—	166	2	no	—
116	C₂×Q₁₆	—	166	2	no	—
117	C₂×Q₁₆	—	166	2	no	—
118	C₂×Q₁₆	—	166	2	no	—
119	C₂×M₁₆	—	2	2	no	—
120	C₂×M₁₆	—	83	1	no	—
121	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
122	#117	—	83	1	no	—
123	Q₃₂	—	166	8	no	—
124	M₃₂	—	166	2	no	—
125	C₂×C₁₆	—	2	2	no	—
126	C₂×C₁₆	—	166	1	no	—
127	C₂×C₁₆	D₃₃₂	1	2	yes	—
128	#108	—	2	2	no	—
129	#108	—	2	2	no	—
130	#103	—	332	4	no	—
131	#101	—	332	4	no	—
132	#97	—	166	1	no	—
133	C₄×C₈	—	166	1	no	—
134	C₂×Q₈	—	332	2	no	—
135	C₂×Q₈	—	332	2	no	—
136	C₂×Q₈	—	332	2	no	—
137	C₂×Q₈	—	332	2	no	—
138	C₂×Q₈	—	332	2	no	—
139	C₂×Q₈	—	166	4	no	—
140	C₂²×C₄	—	332	2	no	—
141	C₂²×C₄	D₄×D₈₃	1	1	yes	characteristic
142	Q₁₆	—	332	2	no	—
143	Q₁₆	—	332	2	no	—
144	Q₁₆	—	166	4	no	—
145	Q₁₆	—	166	4	no	—
146	M₁₆	—	4	2	no	—
147	M₁₆	—	4	2	no	—
148	M₁₆	—	166	2	no	—
149	C₂×C₈	—	4	2	no	—
150	C₂×C₈	—	166	1	no	—
151	C₂×C₈	D₄×D₈₃	1	2	yes	—
152	C₂×C₈	C₂×D₃₃₂	1	1	yes	characteristic
153	C₂×C₈	#13050590	1	2	yes	—
154	C₂×C₈	#13050591	1	1	yes	characteristic
155	C₄⋊C₄	—	83	2	no	—
156	C₂²⋊C₄	—	166	1	no	—
157	C₄²	—	166	1	no	—
158	C₁₆	—	2	2	no	—
159	C₁₆	—	332	1	no	—
160	C₂³	#14581815	1	1	yes	characteristic
161	Q₈	—	332	4	no	—
162	Q₈	—	332	4	no	—
163	Q₈	—	664	2	no	—
164	Q₈	—	664	2	no	—
165	C₂×C₄	—	332	4	no	—
166	C₂×C₄	—	664	2	no	—
167	C₂×C₄	—	664	2	no	—
168	C₂×C₄	—	166	2	no	—
169	C₂×C₄	#14581765	1	2	yes	—
170	C₂×C₄	#14581767	1	1	yes	characteristic
171	C₂×C₄	D₈×D₈₃	1	1	yes	characteristic
172	C₂×C₄	#14581819	1	2	yes	—
173	C₈	—	8	2	no	—
174	C₈	—	332	1	no	—
175	C₈	#14581762	1	1	yes	characteristic
176	C₈	#14581777	1	2	yes	—
177	C₈	#14581780	1	1	yes	characteristic
178	C₂²	—	2	2	no	—
179	C₂²	#19127919	1	2	yes	—
180	C₂²	#19127938	1	1	yes	characteristic
181	C₄	—	332	2	no	—
182	C₄	—	664	4	no	—
183	C₄	#19127619	1	1	yes	characteristic
184	C₄	#19127895	1	2	yes	—
185	C₄	#19127955	1	1	yes	characteristic
186	C₂	—	2	2	no	—
187	C₂	#24908775	1	1	yes	characteristic
188	C₂	#24909589	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₆₆₄

C₂

yes

maximal, characteristic

#24909221

C₂

yes

maximal, characteristic

C₂×#19127516

C₂

yes

maximal

#24908376

C₂

yes

maximal

C₈₃×#1381

C₂

yes

maximal, characteristic

C₂²×Dic₃₃₂

C₂²

yes

—

C₂×Dic₆₆₄

—

C₂×Dic₆₆₄

—

C₂×#14581732

C₂²

yes

—

—

C₂²

yes

characteristic

Dic₈₃⋊C₁₆

—

#19127516

—

C₂²×C₁₆×C₈₃

C₂²

yes

characteristic

C₂×#108×C₈₃

C₂²

yes

—

C₂²×Dic₁₆₆

—

C₂×Dic₃₃₂

—

C₂×Dic₃₃₂

—

C₂×Dic₃₃₂

—

C₂×Dic₃₃₂

—

C₂×Dic₃₃₂

—

C₂×Dic₃₃₂

D₄

yes

—

C₂×#13050574

D₄

yes

characteristic

#14581751

D₄

yes

characteristic

Dic₆₆₄

—

#14581732

—

C₂×C₈₃⋊C₁₆

—

#14581717

—

#14581710

—

#14581702

—

C₈×Dic₈₃

—

C₂×M₁₆×C₈₃

—

C₂²×C₈×C₈₃

C₂³

yes

characteristic

C₂×C₁₆×C₈₃

—

C₂×C₁₆×C₈₃

D₄

yes

—

#108×C₈₃

—

#108×C₈₃

—

C₂×Dic₁₆₆

—

C₂×Dic₁₆₆

—

C₂×Dic₁₆₆

—

C₂×Dic₁₆₆

—

C₂×Dic₁₆₆

—

C₂×Dic₁₆₆

—

C₂²×Dic₈₃

—

Dic₃₃₂

—

Dic₃₃₂

—

Dic₃₃₂

—

Dic₃₃₂

—

#13050574

—

C₂×C₈₃⋊C₈

—

C₄⋊Dic₈₃

D₈

yes

—

C₂²⋊Dic₈₃

—

C₄×Dic₈₃

—

C₈₃⋊C₁₆

—

C₂²×C₄×C₈₃

C₂×D₄

yes

characteristic

M₁₆×C₈₃

—

M₁₆×C₈₃

—

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

C₂×D₄

yes

—

C₂×C₈×C₈₃

C₂×D₄

yes

characteristic

C₂×C₈×C₈₃

#41

yes

—

C₂×C₈×C₈₃

#41

yes

characteristic

C₁₆×C₈₃

—

C₂³×C₈₃

#135

yes

characteristic

C₂×C₄×C₈₃

#121

yes

—

C₂×C₄×C₈₃

#121

yes

characteristic

C₂×C₄×C₈₃

C₂×D₈

yes

characteristic

C₂×C₄×C₈₃

AGL(1,ℤ/8)

yes

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

Dic₁₆₆

—

Dic₁₆₆

—

Dic₁₆₆

—

Dic₁₆₆

—

C₈×C₈₃

—

C₈×C₈₃

#120

yes

characteristic

C₈×C₈₃

#123

yes

—

C₈×C₈₃

#124

yes

characteristic

C₈₃⋊C₈

—

C₂²×C₈₃

—

C₂²×C₈₃

#463

yes

—

C₂²×C₈₃

yes

C₄×C₈₃

yes

C₄×C₈₃

yes

—

C₄×C₈₃

#469

yes

characteristic

Dic₈₃

—

Dic₈₃

—

#13441

—

maximal, Sylow

C₂×C₈₃

—

C₂×C₈₃

#2054

yes

characteristic

C₂×C₈₃

#2222

yes

—

C₂²×Q₃₂

—

—

—

—

D₈₃

yes

characteristic

100

C₈₃

#13441

yes

characteristic, complemented, Sylow

101

C₂²×Q₁₆

—

102

C₂×Q₃₂

—

166

—

103

C₂×Q₃₂

—

166

—

104

C₂×M₃₂

—

105

C₂²×C₁₆

C₂×D₈₃

yes

characteristic

106

C₂×#108

C₂×D₈₃

yes

—

107

#419

—

166

—

108

C₂×#101

—

166

—

109

#405

—

110

C₄⋊C₁₆

—

166

—

111

#362

—

166

—

112

C₂²×Q₈

—

166

—

113

C₂×Q₁₆

—

114

C₂×Q₁₆

—

166

—

115

C₂×Q₁₆

—

166

—

116

C₂×Q₁₆

—

166

—

117

C₂×Q₁₆

—

166

—

118

C₂×Q₁₆

—

166

—

119

C₂×M₁₆

—

120

C₂×M₁₆

—

121

C₂²×C₈

C₂²×D₈₃

yes

characteristic

122

#117

—

123

Q₃₂

—

166

—

124

M₃₂

—

166

—

125

C₂×C₁₆

—

126

C₂×C₁₆

—

166

—

127

C₂×C₁₆

D₃₃₂

yes

—

128

#108

—

129

#108

—

130

#103

—

332

—

131

#101

—

332

—

132

#97

—

166

—

133

C₄×C₈

—

166

—

134

C₂×Q₈

—

332

—

135

C₂×Q₈

—

332

—

136

C₂×Q₈

—

332

—

137

C₂×Q₈

—

332

—

138

C₂×Q₈

—

332

—

139

C₂×Q₈

—

166

—

140

C₂²×C₄

—

332

—

141

C₂²×C₄

D₄×D₈₃

yes

characteristic

142

Q₁₆

—

332

—

143

Q₁₆

—

332

—

144

Q₁₆

—

166

—

145

Q₁₆

—

166

—

146

M₁₆

—

147

M₁₆

—

148

M₁₆

—

166

—

149

C₂×C₈

—

150

C₂×C₈

—

166

—

151

C₂×C₈

D₄×D₈₃

yes

—

152

C₂×C₈

C₂×D₃₃₂

yes

characteristic

153

C₂×C₈

#13050590

yes

—

154

C₂×C₈

#13050591

yes

characteristic

155

C₄⋊C₄

—

156

C₂²⋊C₄

—

166

—

157

C₄²

—

166

—

158

C₁₆

—

159

C₁₆

—

332

—

160

C₂³

#14581815

yes

characteristic

161

Q₈

—

332

—

162

Q₈

—

332

—

163

Q₈

—

664

—

164

Q₈

—

664

—

165

C₂×C₄

—

332

—

166

C₂×C₄

—

664

—

167

C₂×C₄

—

664

—

168

C₂×C₄

—

166

—

169

C₂×C₄

#14581765

yes

—

170

C₂×C₄

#14581767

yes

characteristic

171

C₂×C₄

D₈×D₈₃

yes

characteristic

172

C₂×C₄

#14581819

yes

—

173

C₈

—

174

C₈

—

332

—

175

C₈

#14581762

yes

characteristic

176

C₈

#14581777

yes

—

177

C₈

#14581780

yes

characteristic

178

C₂²

—

179

C₂²

#19127919

yes

—

180

C₂²

#19127938

yes

characteristic

181

C₄

—

332

—

182

C₄

—

664

—

183

C₄

#19127619

yes

characteristic

184

C₄

#19127895

yes

—

185

C₄

#19127955

yes

characteristic

186

C₂

—

187

C₂

#24908775

yes

characteristic

188

C₂

#24909589

yes

—

	subgroup	G/H
center	C₂²	#19127938
commutator, cosocle	C₂²×C₈×C₈₃	C₂³
Frattini	C₂²×C₈	C₂²×D₈₃
Fitting	C₈₃×#1381	C₂
socle	C₂²×C₈₃	#466
prime radical	C₂³×C₈₃	#135
hypercenter	#1381	D₈₃
hypocenter	C₈₃	#13441
LCS	G ▷ C₂²×C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1381

subgroup

G/H

center

C₂²

#19127938

commutator, cosocle

C₂²×C₈×C₈₃

C₂³

Frattini

C₂²×C₈

C₂²×D₈₃

Fitting

C₈₃×#1381

C₂

C₂²×C₈₃

C₂³×C₈₃

D₈₃

C₈₃

G ▷ C₂²×C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1381

N	H ≅ G/N	type
100. C₈₃	#13441	semidirect

H ≅ G/N

type

100. C₈₃

#13441

semidirect

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

subgroup	index
C₂²×Dic₆₆₄	2
#24909221	2
C₂×#19127516	2
#24908376	2
C₈₃×#1381	2
#13441	83

subgroup

index

C₂²×Dic₆₆₄

C₈₃×#1381

Subgroups — #35323957

Conjugacy classes of subgroups

Complements

Maximal subgroups