Group #35303872 — subgroups

Subgroups — #35303872

Order 21248.

582 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 582 conjugacy classes of subgroups — too many to draw. Only its 159 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	#24907635	C₂	1	1	yes	maximal, characteristic, complemented
2	#24906131	C₂	1	4	yes	maximal, complemented
3	#24905746	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#493×C₈₃	C₂	1	1	yes	maximal, characteristic
5	#19127754	C₂²	1	2	yes	—
6	C₂×#14581728	—	2	2	no	—
7	#19127725	C₂²	1	2	yes	—
8	#19127445	—	2	8	no	—
9	#19127422	—	2	1	no	—
10	C₈×C₈₃⋊C₈	—	2	1	no	—
11	C₂²×D₈×C₈₃	—	2	2	no	—
12	C₂×#127×C₈₃	C₂²	1	2	yes	—
13	#493×C₈₃	—	2	1	no	—
14	#493×C₈₃	—	2	1	no	—
15	#493×C₈₃	C₄	1	4	yes	—
16	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
17	C₂×#13050574	—	2	2	no	—
18	#14581728	—	4	4	no	—
19	C₄⋊(C₈₃⋊C₈)	—	2	4	no	—
20	C₄⋊(C₈₃⋊C₈)	D₄	1	4	yes	—
21	C₂²⋊(C₈₃⋊C₈)	—	2	2	no	—
22	C₄×C₈₃⋊C₈	—	2	2	no	—
23	C₂²×D₄×C₈₃	—	2	2	no	—
24	C₂²×D₄×C₈₃	—	2	4	no	—
25	C₂×D₈×C₈₃	—	2	4	no	—
26	C₂×D₈×C₈₃	—	2	4	no	—
27	C₂×D₈×C₈₃	—	4	4	no	—
28	C₂×D₈×C₈₃	—	4	4	no	—
29	C₂²×C₈×C₈₃	D₄	1	2	yes	—
30	#127×C₈₃	—	2	2	no	—
31	#127×C₈₃	—	2	4	no	—
32	#127×C₈₃	C₂×C₄	1	4	yes	—
33	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
34	C₄×C₈×C₈₃	C₂×C₄	1	2	yes	—
35	C₄×C₈×C₈₃	D₄	1	1	yes	characteristic
36	C₄×C₈×C₈₃	D₄	1	1	yes	characteristic
37	#13050574	—	4	4	no	—
38	C₂×C₈₃⋊C₈	—	2	4	no	—
39	C₂×C₈₃⋊C₈	—	4	2	no	—
40	C₂⁴×C₈₃	—	4	2	no	—
41	C₂×D₄×C₈₃	—	2	4	no	—
42	C₂×D₄×C₈₃	—	2	8	no	—
43	C₂×D₄×C₈₃	—	4	4	no	—
44	C₂×D₄×C₈₃	—	4	2	no	—
45	C₂×D₄×C₈₃	—	4	4	no	—
46	C₂×D₄×C₈₃	—	4	4	no	—
47	C₂×D₄×C₈₃	—	8	8	no	—
48	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₈₃	#41	1	2	yes	—
50	D₈×C₈₃	—	4	8	no	—
51	D₈×C₈₃	—	4	8	no	—
52	C₂×C₈×C₈₃	C₂²⋊C₄	1	2	yes	—
53	C₂×C₈×C₈₃	C₂²⋊C₄	1	2	yes	—
54	C₂×C₈×C₈₃	D₈	1	4	yes	—
55	C₂×C₈×C₈₃	SD₁₆	1	4	yes	—
56	C₄²×C₈₃	C₂²×C₄	1	1	yes	characteristic
57	C₄²×C₈₃	C₂×D₄	1	2	yes	—
58	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
59	C₂³×C₈₃	—	4	4	no	—
60	C₂³×C₈₃	—	8	8	no	—
61	C₂³×C₈₃	—	8	4	no	—
62	C₂³×C₈₃	#124	1	1	yes	characteristic
63	D₄×C₈₃	—	4	8	no	—
64	D₄×C₈₃	—	8	8	no	—
65	D₄×C₈₃	—	8	8	no	—
66	D₄×C₈₃	—	8	2	no	—
67	D₄×C₈₃	—	8	2	no	—
68	C₂×C₄×C₈₃	—	2	4	no	—
69	C₂×C₄×C₈₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₄×C₈₃	C₄×D₄	1	2	yes	—
71	C₂×C₄×C₈₃	#121	1	2	yes	—
72	C₂×C₄×C₈₃	#127	1	1	yes	characteristic
73	C₂×C₄×C₈₃	C₂×D₈	1	2	yes	—
74	C₂×C₄×C₈₃	C₂×SD₁₆	1	2	yes	—
75	C₈×C₈₃	#102	1	4	yes	—
76	C₈×C₈₃	#102	1	4	yes	—
77	C₈₃⋊C₈	—	4	4	no	—
78	C₈₃⋊C₈	—	8	2	no	—
79	C₂²×C₈₃	—	2	2	no	—
80	C₂²×C₈₃	—	8	4	no	—
81	C₂²×C₈₃	—	8	8	no	—
82	C₂²×C₈₃	—	16	8	no	—
83	C₂²×C₈₃	#390	1	1	yes	characteristic
84	C₂²×C₈₃	#486	1	2	yes	—
85	C₄×C₈₃	—	2	2	no	—
86	C₄×C₈₃	C₂×#102	1	2	yes	—
87	C₄×C₈₃	#472	1	4	yes	—
88	C₄×C₈₃	#492	1	1	yes	characteristic
89	C₄×C₈₃	#493	1	1	yes	characteristic
90	#9773	—	83	1	no	maximal, Sylow
91	C₂×C₈₃	—	2	2	no	—
92	C₂×C₈₃	—	16	4	no	—
93	C₂×C₈₃	#1957	1	1	yes	characteristic
94	C₂×C₈₃	#1978	1	2	yes	—
95	C₂×#493	D₈₃	1	1	yes	characteristic
96	#1848	—	83	1	no	—
97	#1545	—	83	4	no	—
98	#1450	—	83	1	no	—
99	C₈₃	#9773	1	1	yes	characteristic, complemented, Sylow
100	C₂²×D₈	—	2	2	no	—
101	C₂×#127	C₂×D₈₃	1	2	yes	—
102	#493	—	2	1	no	—
103	#493	—	2	1	no	—
104	#493	Dic₈₃	1	4	yes	—
105	#431	—	83	2	no	—
106	C₂×#108	—	166	2	no	—
107	#423	—	83	2	no	—
108	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
109	#331	—	166	8	no	—
110	#322	—	166	1	no	—
111	C₈²	—	166	1	no	—
112	C₂²×D₄	—	2	2	no	—
113	C₂²×D₄	—	2	4	no	—
114	C₂×D₈	—	2	4	no	—
115	C₂×D₈	—	2	4	no	—
116	C₂×D₈	—	4	4	no	—
117	C₂×D₈	—	4	4	no	—
118	C₂×M₁₆	—	166	2	no	—
119	C₂²×C₈	C₂²⋊D₈₃	1	2	yes	—
120	#127	—	2	4	no	—
121	#127	—	2	2	no	—
122	#127	C₂×Dic₈₃	1	4	yes	—
123	C₂×C₄²	C₂²×D₈₃	1	1	yes	characteristic
124	#108	—	332	4	no	—
125	C₄⋊C₈	—	83	4	no	—
126	C₄⋊C₈	—	166	4	no	—
127	C₂²⋊C₈	—	166	2	no	—
128	C₄×C₈	—	166	2	no	—
129	C₄×C₈	C₂×Dic₈₃	1	2	yes	—
130	C₄×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
131	C₄×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
132	C₂⁴	—	4	2	no	—
133	C₂×D₄	—	2	4	no	—
134	C₂×D₄	—	2	8	no	—
135	C₂×D₄	—	4	4	no	—
136	C₂×D₄	—	4	2	no	—
137	C₂×D₄	—	4	4	no	—
138	C₂×D₄	—	4	4	no	—
139	C₂×D₄	—	8	8	no	—
140	C₂²×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
141	C₂²×C₄	#13050589	1	2	yes	—
142	D₈	—	4	8	no	—
143	D₈	—	4	8	no	—
144	M₁₆	—	332	4	no	—
145	C₂×C₈	—	332	2	no	—
146	C₂×C₈	—	166	4	no	—
147	C₂×C₈	C₂²⋊Dic₈₃	1	2	yes	—
148	C₂×C₈	C₂²⋊Dic₈₃	1	2	yes	—
149	C₂×C₈	#13050575	1	4	yes	—
150	C₂×C₈	#13050578	1	4	yes	—
151	C₄²	C₂²×Dic₈₃	1	1	yes	characteristic
152	C₄²	D₄×D₈₃	1	2	yes	—
153	C₄²	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
154	C₂³	—	4	4	no	—
155	C₂³	—	8	8	no	—
156	C₂³	—	8	4	no	—
157	C₂³	#14581779	1	1	yes	characteristic
158	D₄	—	4	8	no	—
159	D₄	—	8	8	no	—
160	D₄	—	8	2	no	—
161	D₄	—	8	2	no	—
162	D₄	—	8	8	no	—
163	C₂×C₄	—	2	4	no	—
164	C₂×C₄	C₂×C₂²⋊Dic₈₃	1	1	yes	characteristic
165	C₂×C₄	D₄×Dic₈₃	1	2	yes	—
166	C₂×C₄	#14581763	1	2	yes	—
167	C₂×C₄	#14581788	1	1	yes	characteristic
168	C₂×C₄	C₂×#13050575	1	2	yes	—
169	C₂×C₄	C₂×#13050578	1	2	yes	—
170	C₈	—	332	4	no	—
171	C₈	—	664	2	no	—
172	C₈	#14581713	1	4	yes	—
173	C₈	#14581713	1	4	yes	—
174	C₂²	—	2	2	no	—
175	C₂²	—	8	8	no	—
176	C₂²	—	8	4	no	—
177	C₂²	—	16	8	no	—
178	C₂²	#19127604	1	1	yes	characteristic
179	C₂²	#19128047	1	2	yes	—
180	C₄	—	2	2	no	—
181	C₄	C₂×#14581713	1	2	yes	—
182	C₄	#19127968	1	4	yes	—
183	C₄	#19128073	1	1	yes	characteristic
184	C₄	#19128077	1	1	yes	characteristic
185	C₂	—	2	2	no	—
186	C₂	—	16	4	no	—
187	C₂	#24908270	1	1	yes	characteristic
188	C₂	#24908374	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#24907635

C₂

yes

maximal, characteristic, complemented

#24906131

C₂

yes

maximal, complemented

#24905746

C₂

yes

maximal, characteristic, complemented

C₂×#493×C₈₃

C₂

yes

maximal, characteristic

#19127754

C₂²

yes

—

C₂×#14581728

—

#19127725

C₂²

yes

—

#19127445

—

#19127422

—

C₈×C₈₃⋊C₈

—

C₂²×D₈×C₈₃

—

C₂×#127×C₈₃

C₂²

yes

—

#493×C₈₃

—

#493×C₈₃

—

#493×C₈₃

C₄

yes

—

C₂×C₄×C₈×C₈₃

C₂²

yes

characteristic

C₂×#13050574

—

#14581728

—

C₄⋊(C₈₃⋊C₈)

—

C₄⋊(C₈₃⋊C₈)

D₄

yes

—

C₂²⋊(C₈₃⋊C₈)

—

C₄×C₈₃⋊C₈

—

C₂²×D₄×C₈₃

—

C₂²×D₄×C₈₃

—

C₂×D₈×C₈₃

—

C₂×D₈×C₈₃

—

C₂×D₈×C₈₃

—

C₂×D₈×C₈₃

—

C₂²×C₈×C₈₃

D₄

yes

—

#127×C₈₃

—

#127×C₈₃

—

#127×C₈₃

C₂×C₄

yes

—

C₂×C₄²×C₈₃

C₂³

yes

characteristic

C₄×C₈×C₈₃

C₂×C₄

yes

—

C₄×C₈×C₈₃

D₄

yes

characteristic

C₄×C₈×C₈₃

D₄

yes

characteristic

#13050574

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

—

C₂⁴×C₈₃

—

C₂×D₄×C₈₃

—

C₂×D₄×C₈₃

—

C₂×D₄×C₈₃

—

C₂×D₄×C₈₃

—

C₂×D₄×C₈₃

—

C₂×D₄×C₈₃

—

C₂×D₄×C₈₃

—

C₂²×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₈₃

#41

yes

—

D₈×C₈₃

—

D₈×C₈₃

—

C₂×C₈×C₈₃

C₂²⋊C₄

yes

—

C₂×C₈×C₈₃

C₂²⋊C₄

yes

—

C₂×C₈×C₈₃

D₈

yes

—

C₂×C₈×C₈₃

SD₁₆

yes

—

C₄²×C₈₃

C₂²×C₄

yes

characteristic

C₄²×C₈₃

C₂×D₄

yes

—

C₄²×C₈₃

C₂×D₄

yes

characteristic

C₂³×C₈₃

—

C₂³×C₈₃

—

C₂³×C₈₃

—

C₂³×C₈₃

#124

yes

characteristic

D₄×C₈₃

—

D₄×C₈₃

—

D₄×C₈₃

—

D₄×C₈₃

—

D₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₈₃

C₄×D₄

yes

—

C₂×C₄×C₈₃

#121

yes

—

C₂×C₄×C₈₃

#127

yes

characteristic

C₂×C₄×C₈₃

C₂×D₈

yes

—

C₂×C₄×C₈₃

C₂×SD₁₆

yes

—

C₈×C₈₃

#102

yes

—

C₈×C₈₃

#102

yes

—

C₈₃⋊C₈

—

C₈₃⋊C₈

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

#390

yes

characteristic

C₂²×C₈₃

#486

yes

—

C₄×C₈₃

—

C₄×C₈₃

C₂×#102

yes

—

C₄×C₈₃

#472

yes

—

C₄×C₈₃

yes

C₄×C₈₃

yes

—

C₂×C₈₃

—

C₂×C₈₃

—

C₂×C₈₃

#1957

yes

characteristic

C₂×C₈₃

#1978

yes

—

C₂×#493

D₈₃

yes

—

—

—

C₈₃

#9773

yes

characteristic, complemented, Sylow

100

C₂²×D₈

—

101

C₂×#127

C₂×D₈₃

yes

—

102

#493

—

103

#493

—

104

#493

Dic₈₃

yes

—

105

#431

—

106

C₂×#108

—

166

—

107

#423

—

108

C₂×C₄×C₈

C₂×D₈₃

yes

characteristic

109

#331

—

166

—

110

#322

—

166

—

111

C₈²

—

166

—

112

C₂²×D₄

—

113

C₂²×D₄

—

114

C₂×D₈

—

115

C₂×D₈

—

116

C₂×D₈

—

117

C₂×D₈

—

118

C₂×M₁₆

—

166

—

119

C₂²×C₈

C₂²⋊D₈₃

yes

—

120

#127

—

121

#127

—

122

#127

C₂×Dic₈₃

yes

—

123

C₂×C₄²

C₂²×D₈₃

yes

characteristic

124

#108

—

332

—

125

C₄⋊C₈

—

126

C₄⋊C₈

—

166

—

127

C₂²⋊C₈

—

166

—

128

C₄×C₈

—

166

—

129

C₄×C₈

C₂×Dic₈₃

yes

—

130

C₄×C₈

C₂²⋊D₈₃

yes

characteristic

131

C₄×C₈

C₂²⋊D₈₃

yes

characteristic

132

C₂⁴

—

133

C₂×D₄

—

134

C₂×D₄

—

135

C₂×D₄

—

136

C₂×D₄

—

137

C₂×D₄

—

138

C₂×D₄

—

139

C₂×D₄

—

140

C₂²×C₄

C₂×C₂²⋊D₈₃

yes

characteristic

141

C₂²×C₄

#13050589

yes

—

142

D₈

—

143

D₈

—

144

M₁₆

—

332

—

145

C₂×C₈

—

332

—

146

C₂×C₈

—

166

—

147

C₂×C₈

C₂²⋊Dic₈₃

yes

—

148

C₂×C₈

C₂²⋊Dic₈₃

yes

—

149

C₂×C₈

#13050575

yes

—

150

C₂×C₈

#13050578

yes

—

151

C₄²

C₂²×Dic₈₃

yes

characteristic

152

C₄²

D₄×D₈₃

yes

—

153

C₄²

C₂×C₂²⋊D₈₃

yes

characteristic

154

C₂³

—

155

C₂³

—

156

C₂³

—

157

C₂³

#14581779

yes

characteristic

158

D₄

—

159

D₄

—

160

D₄

—

161

D₄

—

162

D₄

—

163

C₂×C₄

—

164

C₂×C₄

C₂×C₂²⋊Dic₈₃

yes

characteristic

165

C₂×C₄

D₄×Dic₈₃

yes

—

166

C₂×C₄

#14581763

yes

—

167

C₂×C₄

#14581788

yes

characteristic

168

C₂×C₄

C₂×#13050575

yes

—

169

C₂×C₄

C₂×#13050578

yes

—

170

C₈

—

332

—

171

C₈

—

664

—

172

C₈

#14581713

yes

—

173

C₈

#14581713

yes

—

174

C₂²

—

175

C₂²

—

176

C₂²

—

177

C₂²

—

178

C₂²

#19127604

yes

characteristic

179

C₂²

#19128047

yes

—

180

C₄

—

181

C₄

C₂×#14581713

yes

—

182

C₄

#19127968

yes

—

183

C₄

#19128073

yes

characteristic

184

C₄

#19128077

yes

characteristic

185

C₂

—

186

C₂

—

187

C₂

#24908270

yes

characteristic

188

C₂

#24908374

yes

—

	subgroup	G/H
center	C₂²	#19127604
commutator	C₄²×C₈₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₃
Fitting, prime radical	C₂×#493×C₈₃	C₂
socle	C₂²×C₈₃	#390
cosocle	C₂×C₄²×C₈₃	C₂³
hypercenter	C₂×#493	D₈₃
hypocenter	C₈₃	#9773
LCS	G ▷ C₄²×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#493

subgroup

G/H

center

C₂²

#19127604

commutator

C₄²×C₈₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₈₃

Fitting, prime radical

C₂×#493×C₈₃

C₂

socle

C₂²×C₈₃

#390

cosocle

C₂×C₄²×C₈₃

C₂³

D₈₃

C₈₃

G ▷ C₄²×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#493

N	H ≅ G/N	type
1. #24907635	C₂	semidirect
2. #24906131	C₂	semidirect
3. #24905746	C₂	semidirect
99. C₈₃	#9773	semidirect

H ≅ G/N

type

1. #24907635

C₂

semidirect

2. #24906131

C₂

semidirect

3. #24905746

C₂

semidirect

99. C₈₃

#9773

semidirect

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

subgroup	index
#24907635	2
#24906131	2
#24905746	2
C₂×#493×C₈₃	2
#9773	83

subgroup

index

C₂×#493×C₈₃

Subgroups — #35303872

Conjugacy classes of subgroups

Complements

Maximal subgroups