Group #35324296 — subgroups

Subgroups — #35324296

Order 21248.

292 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 83 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	#24909258	C₂	1	1	yes	maximal, characteristic, complemented
2	#24909073	C₂	1	1	yes	maximal, characteristic
3	#24907498	C₂	1	2	yes	maximal
4	#24905576	C₂	1	2	yes	maximal, complemented
5	C₈₃×#1385	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₈×D₈₃	C₂²	1	1	yes	characteristic
7	C₂×#14581732	C₂²	1	1	yes	characteristic
8	C₂×#14581728	C₂²	1	2	yes	—
9	#19127664	—	2	2	no	—
10	#19127531	—	2	2	no	—
11	(C₂×D₈₃)⋊C₁₆	—	2	2	no	—
12	#19127465	—	2	2	no	—
13	C₂×M₃₂×C₈₃	C₂²	1	1	yes	characteristic
14	C₂×#108×C₈₃	C₂²	1	2	yes	—
15	C₂²×C₄×D₈₃	D₄	1	1	yes	characteristic
16	C₂×#13050574	—	2	2	no	—
17	C₂²×C₈₃⋊C₈	D₄	1	1	yes	characteristic
18	C₂×C₈×D₈₃	—	2	1	no	—
19	C₂×C₈×D₈₃	—	2	2	no	—
20	C₂×C₈×D₈₃	—	2	2	no	—
21	C₂×C₈×D₈₃	—	2	1	no	—
22	#14581732	—	2	2	no	—
23	C₂×C₈₃⋊C₁₆	—	2	1	no	—
24	#14581728	—	2	2	no	—
25	#14581728	—	2	2	no	—
26	(C₂×D₈₃)⋊C₈	—	4	4	no	—
27	C₂×M₁₆×C₈₃	—	2	2	no	—
28	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
29	M₃₂×C₈₃	—	2	2	no	—
30	C₂×C₁₆×C₈₃	D₄	1	2	yes	—
31	#108×C₈₃	—	2	2	no	—
32	#108×C₈₃	—	2	2	no	—
33	C₂³×D₈₃	Q₁₆	1	1	yes	characteristic
34	C₂²×Dic₈₃	Q₁₆	1	1	yes	characteristic
35	C₂×C₄×D₈₃	—	2	1	no	—
36	C₂×C₄×D₈₃	—	2	2	no	—
37	C₂×C₄×D₈₃	—	2	1	no	—
38	C₂×C₄×D₈₃	—	2	2	no	—
39	#13050574	—	4	2	no	—
40	#13050574	—	4	2	no	—
41	C₂×C₈₃⋊C₈	—	2	1	no	—
42	C₂×C₈₃⋊C₈	—	2	2	no	—
43	C₂×C₈₃⋊C₈	—	4	2	no	—
44	C₈×D₈₃	—	4	1	no	—
45	C₈×D₈₃	—	4	1	no	—
46	C₈×D₈₃	—	4	2	no	—
47	C₈₃⋊C₁₆	—	4	1	no	—
48	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
49	M₁₆×C₈₃	—	4	2	no	—
50	M₁₆×C₈₃	—	4	2	no	—
51	C₂×C₈×C₈₃	—	4	2	no	—
52	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
53	C₂×C₈×C₈₃	#41	1	2	yes	—
54	C₂×C₈×C₈₃	#41	1	2	yes	—
55	C₂×C₈×C₈₃	#41	1	1	yes	characteristic
56	C₁₆×C₈₃	—	2	2	no	—
57	C₂³×C₈₃	C₂×Q₁₆	1	1	yes	characteristic
58	C₂²×D₈₃	—	2	1	no	—
59	C₂²×D₈₃	—	2	2	no	—
60	C₂²×D₈₃	—	4	2	no	—
61	C₂×C₄×C₈₃	#121	1	1	yes	characteristic
62	C₂×C₄×C₈₃	#123	1	2	yes	—
63	C₂×C₄×C₈₃	#135	1	1	yes	characteristic
64	C₂×C₄×C₈₃	AGL(1,ℤ/8)	1	2	yes	—
65	C₂×Dic₈₃	—	2	1	no	—
66	C₂×Dic₈₃	—	2	2	no	—
67	C₄×D₈₃	—	4	1	no	—
68	C₄×D₈₃	—	4	1	no	—
69	C₄×D₈₃	—	4	2	no	—
70	C₈×C₈₃	—	8	2	no	—
71	C₈×C₈₃	#123	1	1	yes	characteristic
72	C₈×C₈₃	#124	1	1	yes	characteristic
73	C₈×C₈₃	#126	1	2	yes	—
74	C₈₃⋊C₈	—	4	1	no	—
75	C₈₃⋊C₈	—	8	2	no	—
76	C₂²×C₈₃	—	4	1	no	—
77	C₂²×C₈₃	#467	1	1	yes	characteristic
78	C₂²×C₈₃	#484	1	2	yes	—
79	C₂×D₈₃	—	2	2	no	—
80	C₂×D₈₃	—	2	4	no	—
81	C₂×D₈₃	—	4	1	no	—
82	C₂×D₈₃	—	4	2	no	—
83	C₂×D₈₃	—	4	1	no	—
84	C₄×C₈₃	#397	1	1	yes	characteristic
85	C₄×C₈₃	#469	1	1	yes	characteristic
86	C₄×C₈₃	#482	1	2	yes	—
87	Dic₈₃	—	4	1	no	—
88	#13498	—	83	1	no	maximal, Sylow
89	C₂×C₈₃	—	2	1	no	—
90	C₂×C₈₃	—	2	2	no	—
91	C₂×C₈₃	#2076	1	1	yes	characteristic
92	D₈₃	—	4	2	no	—
93	#2161	—	83	1	no	—
94	#2112	—	83	1	no	—
95	#1810	—	83	2	no	—
96	#1388	—	83	2	no	—
97	#1385	D₈₃	1	1	yes	characteristic, complemented
98	C₈₃	#13498	1	1	yes	characteristic, complemented, Sylow
99	C₂³×C₈	—	83	1	no	—
100	C₂×M₃₂	—	83	1	no	—
101	C₂×M₃₂	C₂×D₈₃	1	1	yes	characteristic
102	C₂×#108	—	83	2	no	—
103	C₂×#108	C₂×D₈₃	1	2	yes	—
104	#407	—	166	2	no	—
105	#368	—	166	2	no	—
106	C₂²⋊C₁₆	—	166	2	no	—
107	#341	—	166	2	no	—
108	C₂³×C₄	—	83	1	no	—
109	C₂×M₁₆	—	2	2	no	—
110	C₂×M₁₆	—	166	2	no	—
111	C₂²×C₈	—	83	1	no	—
112	C₂²×C₈	—	166	1	no	—
113	C₂²×C₈	—	166	2	no	—
114	C₂²×C₈	—	166	2	no	—
115	C₂²×C₈	—	166	1	no	—
116	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
117	M₃₂	—	2	2	no	—
118	M₃₂	—	166	2	no	—
119	C₂×C₁₆	—	166	1	no	—
120	C₂×C₁₆	D₃₃₂	1	2	yes	—
121	#108	—	2	2	no	—
122	#108	—	2	2	no	—
123	#108	—	166	2	no	—
124	#108	—	166	2	no	—
125	C₂²⋊C₈	—	332	4	no	—
126	C₂⁴	—	83	1	no	—
127	C₂²×C₄	—	83	1	no	—
128	C₂²×C₄	—	166	1	no	—
129	C₂²×C₄	—	166	2	no	—
130	C₂²×C₄	—	166	1	no	—
131	C₂²×C₄	—	166	2	no	—
132	C₂²×C₄	D₄×D₈₃	1	1	yes	characteristic
133	M₁₆	—	4	2	no	—
134	M₁₆	—	4	2	no	—
135	M₁₆	—	332	2	no	—
136	M₁₆	—	332	2	no	—
137	C₂×C₈	—	4	2	no	—
138	C₂×C₈	—	332	1	no	—
139	C₂×C₈	—	332	1	no	—
140	C₂×C₈	—	332	2	no	—
141	C₂×C₈	—	332	2	no	—
142	C₂×C₈	—	166	1	no	—
143	C₂×C₈	—	166	2	no	—
144	C₂×C₈	C₂×D₃₃₂	1	1	yes	characteristic
145	C₂×C₈	#13050589	1	2	yes	—
146	C₂×C₈	#13050590	1	2	yes	—
147	C₂×C₈	#13050591	1	1	yes	characteristic
148	C₁₆	—	2	2	no	—
149	C₁₆	—	332	1	no	—
150	C₂³	—	332	2	no	—
151	C₂³	—	166	1	no	—
152	C₂³	—	166	2	no	—
153	C₂³	Q₁₆×D₈₃	1	1	yes	characteristic
154	C₂×C₄	—	332	1	no	—
155	C₂×C₄	—	332	1	no	—
156	C₂×C₄	—	332	2	no	—
157	C₂×C₄	—	166	1	no	—
158	C₂×C₄	—	166	2	no	—
159	C₂×C₄	#14581767	1	1	yes	characteristic
160	C₂×C₄	#14581774	1	2	yes	—
161	C₂×C₄	#14581811	1	1	yes	characteristic
162	C₂×C₄	#14581819	1	2	yes	—
163	C₈	—	8	2	no	—
164	C₈	—	332	1	no	—
165	C₈	—	664	2	no	—
166	C₈	#14581776	1	1	yes	characteristic
167	C₈	#14581780	1	1	yes	characteristic
168	C₈	#14581787	1	2	yes	—
169	C₂²	—	4	1	no	—
170	C₂²	—	332	1	no	—
171	C₂²	—	332	2	no	—
172	C₂²	—	332	1	no	—
173	C₂²	—	166	2	no	—
174	C₂²	—	166	4	no	—
175	C₂²	#19127943	1	1	yes	characteristic
176	C₂²	#19128035	1	2	yes	—
177	C₄	—	332	1	no	—
178	C₄	#19127628	1	1	yes	characteristic
179	C₄	#19127955	1	1	yes	characteristic
180	C₄	#19128023	1	2	yes	—
181	C₂	—	2	1	no	—
182	C₂	—	2	2	no	—
183	C₂	—	332	2	no	—
184	C₂	#24908893	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#24909258

C₂

yes

maximal, characteristic, complemented

#24909073

C₂

yes

maximal, characteristic

#24907498

C₂

yes

maximal

#24905576

C₂

yes

maximal, complemented

C₈₃×#1385

C₂

yes

maximal, characteristic, complemented

C₂²×C₈×D₈₃

C₂²

yes

characteristic

C₂×#14581732

C₂²

yes

characteristic

C₂×#14581728

C₂²

yes

—

#19127664

—

#19127531

—

(C₂×D₈₃)⋊C₁₆

—

#19127465

—

C₂×M₃₂×C₈₃

C₂²

yes

characteristic

C₂×#108×C₈₃

C₂²

yes

—

C₂²×C₄×D₈₃

D₄

yes

characteristic

C₂×#13050574

—

C₂²×C₈₃⋊C₈

D₄

yes

characteristic

C₂×C₈×D₈₃

—

C₂×C₈×D₈₃

—

C₂×C₈×D₈₃

—

C₂×C₈×D₈₃

—

#14581732

—

C₂×C₈₃⋊C₁₆

—

#14581728

—

#14581728

—

(C₂×D₈₃)⋊C₈

—

C₂×M₁₆×C₈₃

—

C₂²×C₈×C₈₃

C₂³

yes

characteristic

M₃₂×C₈₃

—

C₂×C₁₆×C₈₃

D₄

yes

—

#108×C₈₃

—

#108×C₈₃

—

C₂³×D₈₃

Q₁₆

yes

characteristic

C₂²×Dic₈₃

Q₁₆

yes

characteristic

C₂×C₄×D₈₃

—

C₂×C₄×D₈₃

—

C₂×C₄×D₈₃

—

C₂×C₄×D₈₃

—

#13050574

—

#13050574

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

—

C₈×D₈₃

—

C₈×D₈₃

—

C₈×D₈₃

—

C₈₃⋊C₁₆

—

C₂²×C₄×C₈₃

C₂×D₄

yes

characteristic

M₁₆×C₈₃

—

M₁₆×C₈₃

—

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

C₂×D₄

yes

characteristic

C₂×C₈×C₈₃

#41

yes

—

C₂×C₈×C₈₃

#41

yes

—

C₂×C₈×C₈₃

#41

yes

characteristic

C₁₆×C₈₃

—

C₂³×C₈₃

C₂×Q₁₆

yes

characteristic

C₂²×D₈₃

—

C₂²×D₈₃

—

C₂²×D₈₃

—

C₂×C₄×C₈₃

#121

yes

characteristic

C₂×C₄×C₈₃

#123

yes

—

C₂×C₄×C₈₃

#135

yes

characteristic

C₂×C₄×C₈₃

AGL(1,ℤ/8)

yes

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₄×D₈₃

—

C₄×D₈₃

—

C₄×D₈₃

—

C₈×C₈₃

—

C₈×C₈₃

yes

C₈×C₈₃

yes

C₈×C₈₃

yes

—

C₈₃⋊C₈

—

C₈₃⋊C₈

—

C₂²×C₈₃

—

C₂²×C₈₃

#467

yes

characteristic

C₂²×C₈₃

#484

yes

—

C₂×D₈₃

—

C₂×D₈₃

—

C₂×D₈₃

—

C₂×D₈₃

—

C₂×D₈₃

—

C₄×C₈₃

yes

C₄×C₈₃

yes

C₄×C₈₃

yes

—

Dic₈₃

—

#13498

—

maximal, Sylow

C₂×C₈₃

—

C₂×C₈₃

—

C₂×C₈₃

#2076

yes

characteristic

D₈₃

—

—

—

—

—

D₈₃

yes

characteristic, complemented

C₈₃

#13498

yes

characteristic, complemented, Sylow

C₂³×C₈

—

100

C₂×M₃₂

—

101

C₂×M₃₂

C₂×D₈₃

yes

characteristic

102

C₂×#108

—

103

C₂×#108

C₂×D₈₃

yes

—

104

#407

—

166

—

105

#368

—

166

—

106

C₂²⋊C₁₆

—

166

—

107

#341

—

166

—

108

C₂³×C₄

—

109

C₂×M₁₆

—

110

C₂×M₁₆

—

166

—

111

C₂²×C₈

—

112

C₂²×C₈

—

166

—

113

C₂²×C₈

—

166

—

114

C₂²×C₈

—

166

—

115

C₂²×C₈

—

166

—

116

C₂²×C₈

C₂²×D₈₃

yes

characteristic

117

M₃₂

—

118

M₃₂

—

166

—

119

C₂×C₁₆

—

166

—

120

C₂×C₁₆

D₃₃₂

yes

—

121

#108

—

122

#108

—

123

#108

—

166

—

124

#108

—

166

—

125

C₂²⋊C₈

—

332

—

126

C₂⁴

—

127

C₂²×C₄

—

128

C₂²×C₄

—

166

—

129

C₂²×C₄

—

166

—

130

C₂²×C₄

—

166

—

131

C₂²×C₄

—

166

—

132

C₂²×C₄

D₄×D₈₃

yes

characteristic

133

M₁₆

—

134

M₁₆

—

135

M₁₆

—

332

—

136

M₁₆

—

332

—

137

C₂×C₈

—

138

C₂×C₈

—

332

—

139

C₂×C₈

—

332

—

140

C₂×C₈

—

332

—

141

C₂×C₈

—

332

—

142

C₂×C₈

—

166

—

143

C₂×C₈

—

166

—

144

C₂×C₈

C₂×D₃₃₂

yes

characteristic

145

C₂×C₈

#13050589

yes

—

146

C₂×C₈

#13050590

yes

—

147

C₂×C₈

#13050591

yes

characteristic

148

C₁₆

—

149

C₁₆

—

332

—

150

C₂³

—

332

—

151

C₂³

—

166

—

152

C₂³

—

166

—

153

C₂³

Q₁₆×D₈₃

yes

characteristic

154

C₂×C₄

—

332

—

155

C₂×C₄

—

332

—

156

C₂×C₄

—

332

—

157

C₂×C₄

—

166

—

158

C₂×C₄

—

166

—

159

C₂×C₄

#14581767

yes

characteristic

160

C₂×C₄

#14581774

yes

—

161

C₂×C₄

#14581811

yes

characteristic

162

C₂×C₄

#14581819

yes

—

163

C₈

—

164

C₈

—

332

—

165

C₈

—

664

—

166

C₈

#14581776

yes

characteristic

167

C₈

#14581780

yes

characteristic

168

C₈

#14581787

yes

—

169

C₂²

—

170

C₂²

—

332

—

171

C₂²

—

332

—

172

C₂²

—

332

—

173

C₂²

—

166

—

174

C₂²

—

166

—

175

C₂²

#19127943

yes

characteristic

176

C₂²

#19128035

yes

—

177

C₄

—

332

—

178

C₄

#19127628

yes

characteristic

179

C₄

#19127955

yes

characteristic

180

C₄

#19128023

yes

—

181

C₂

—

182

C₂

—

183

C₂

—

332

—

184

C₂

#24908893

yes

characteristic

	subgroup	G/H
center	C₄	#19127955
commutator, cosocle	C₂²×C₈×C₈₃	C₂³
Frattini	C₂²×C₈	C₂²×D₈₃
Fitting	C₈₃×#1385	C₂
socle	C₂×C₈₃	#2076
prime radical	C₂³×D₈₃	Q₁₆
hypercenter	#1385	D₈₃
hypocenter	C₈₃	#13498
LCS	G ▷ C₂²×C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1385

subgroup

G/H

center

C₄

#19127955

commutator, cosocle

C₂²×C₈×C₈₃

C₂³

Frattini

C₂²×C₈

C₂²×D₈₃

Fitting

C₈₃×#1385

C₂

socle

C₂×C₈₃

#2076

prime radical

C₂³×D₈₃

Q₁₆

D₈₃

C₈₃

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

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1385

N	H ≅ G/N	type
1. #24909258	C₂	semidirect
4. #24905576	C₂	semidirect
5. C₈₃×#1385	C₂	semidirect
97. #1385	D₈₃	semidirect
98. C₈₃	#13498	semidirect

H ≅ G/N

type

1. #24909258

C₂

semidirect

4. #24905576

C₂

semidirect

5. C₈₃×#1385

C₂

semidirect

97. #1385

D₈₃

semidirect

98. C₈₃

#13498

semidirect

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

subgroup	index
#24909258	2
#24909073	2
#24907498	2
#24905576	2
C₈₃×#1385	2
#13498	83

subgroup

index

C₈₃×#1385

Subgroups — #35324296

Conjugacy classes of subgroups

Complements

Maximal subgroups