Group #35309469 — subgroups

Subgroups — #35309469

Order 21248.

410 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 149 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	#24908424	C₂	1	2	yes	maximal
2	#24908265	C₂	1	1	yes	maximal, characteristic
3	#24907623	C₂	1	1	yes	maximal, characteristic
4	#24907591	C₂	1	2	yes	maximal
5	C₈₃×#1273	C₂	1	1	yes	maximal, characteristic
6	C₂×#14581792	C₂²	1	1	yes	characteristic
7	C₂×#14581726	—	2	2	no	—
8	C₂×#14581717	—	2	2	no	—
9	#19127571	C₂²	1	1	yes	characteristic
10	#19127571	C₂²	1	2	yes	—
11	#19127464	—	2	4	no	—
12	#19127464	—	2	4	no	—
13	C₂×C₄×C₈×C₈₃	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
15	C₂²×Dic₁₆₆	—	2	1	no	—
16	#14581792	—	2	2	no	—
17	#14581792	—	2	1	no	—
18	#14581792	Q₈	1	2	yes	—
19	C₂×C₄⋊Dic₈₃	—	2	2	no	—
20	C₂×C₄⋊Dic₈₃	—	2	1	no	—
21	C₂×C₄⋊Dic₈₃	—	2	2	no	—
22	C₂×C₄⋊Dic₈₃	—	2	2	no	—
23	C₂×C₄⋊Dic₈₃	—	2	2	no	—
24	C₂×C₄⋊Dic₈₃	—	2	2	no	—
25	#14581726	—	4	4	no	—
26	#14581717	—	4	4	no	—
27	C₂²×C₈×C₈₃	D₄	1	4	yes	—
28	C₂²×C₈×C₈₃	D₄	1	2	yes	—
29	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
30	C₄×C₈×C₈₃	—	2	2	no	—
31	C₄×C₈×C₈₃	—	2	2	no	—
32	C₄×C₈×C₈₃	—	2	2	no	—
33	C₂×Dic₁₆₆	—	2	2	no	—
34	C₂×Dic₁₆₆	—	4	2	no	—
35	C₂×Dic₁₆₆	—	4	2	no	—
36	C₂×Dic₁₆₆	—	4	1	no	—
37	C₂²×Dic₈₃	—	4	1	no	—
38	C₂²×Dic₈₃	—	4	2	no	—
39	C₂²×Dic₈₃	—	4	1	no	—
40	C₄⋊Dic₈₃	—	2	4	no	—
41	C₄⋊Dic₈₃	—	4	2	no	—
42	C₄⋊Dic₈₃	—	4	2	no	—
43	C₄⋊Dic₈₃	—	4	4	no	—
44	C₄⋊Dic₈₃	—	4	4	no	—
45	C₄⋊Dic₈₃	—	4	4	no	—
46	C₄⋊Dic₈₃	—	4	4	no	—
47	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₄×C₈₃	C₂×D₄	1	2	yes	—
49	C₂×C₈×C₈₃	—	2	2	no	—
50	C₂×C₈×C₈₃	—	2	2	no	—
51	C₂×C₈×C₈₃	—	2	4	no	—
52	C₂×C₈×C₈₃	—	2	4	no	—
53	C₂×C₈×C₈₃	—	2	4	no	—
54	C₂×C₈×C₈₃	D₈	1	4	yes	—
55	C₄²×C₈₃	C₂×Q₈	1	1	yes	characteristic
56	C₄²×C₈₃	#41	1	1	yes	characteristic
57	C₄²×C₈₃	#41	1	2	yes	—
58	C₂³×C₈₃	#127	1	1	yes	characteristic
59	C₂×C₄×C₈₃	C₂²⋊Q₈	1	1	yes	characteristic
60	C₂×C₄×C₈₃	C₂²⋊Q₈	1	2	yes	—
61	C₂×C₄×C₈₃	#123	1	1	yes	characteristic
62	C₂×C₄×C₈₃	#123	1	2	yes	—
63	C₂×C₄×C₈₃	C₂×D₈	1	2	yes	—
64	C₂×C₄×C₈₃	#135	1	4	yes	—
65	C₂×C₄×C₈₃	#137	1	2	yes	—
66	C₂×C₄×C₈₃	#137	1	4	yes	—
67	C₂×Dic₈₃	—	4	2	no	—
68	C₂×Dic₈₃	—	8	1	no	—
69	C₂×Dic₈₃	—	8	1	no	—
70	C₂×Dic₈₃	—	8	2	no	—
71	C₂×Dic₈₃	—	8	1	no	—
72	C₂×Dic₈₃	—	8	2	no	—
73	C₂×Dic₈₃	—	8	2	no	—
74	C₂×Dic₈₃	—	8	2	no	—
75	Dic₁₆₆	—	4	4	no	—
76	Dic₁₆₆	—	8	2	no	—
77	C₈×C₈₃	—	2	4	no	—
78	C₈×C₈₃	—	4	4	no	—
79	C₂²×C₈₃	#399	1	1	yes	characteristic
80	C₂²×C₈₃	#493	1	1	yes	characteristic
81	C₂²×C₈₃	#495	1	2	yes	—
82	C₂²×C₈₃	#497	1	1	yes	characteristic
83	C₂²×C₈₃	#497	1	2	yes	—
84	C₄×C₈₃	#474	1	2	yes	—
85	C₄×C₈₃	#479	1	4	yes	—
86	C₄×C₈₃	#480	1	2	yes	—
87	C₄×C₈₃	#485	1	4	yes	—
88	Dic₈₃	—	8	2	no	—
89	Dic₈₃	—	16	2	no	—
90	Dic₈₃	—	16	1	no	—
91	#10676	—	83	1	no	maximal, Sylow
92	C₂×C₈₃	#1709	1	1	yes	characteristic
93	C₂×C₈₃	#1718	1	2	yes	—
94	C₂×C₈₃	#1722	1	1	yes	characteristic
95	C₂×C₈₃	#2093	1	1	yes	characteristic
96	C₂×C₈₃	#2097	1	2	yes	—
97	#1990	—	83	2	no	—
98	#1956	—	83	1	no	—
99	#1845	—	83	1	no	—
100	#1834	—	83	2	no	—
101	#1273	D₈₃	1	1	yes	characteristic
102	C₈₃	#10676	1	1	yes	characteristic, complemented, Sylow
103	C₂×C₄⋊Q₈	—	83	1	no	—
104	C₂×#107	—	166	2	no	—
105	C₂×#103	—	166	2	no	—
106	C₂×C₄×C₈	C₂×D₈₃	1	2	yes	—
107	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
108	#384	—	83	1	no	—
109	#384	—	83	2	no	—
110	#340	—	166	4	no	—
111	#340	—	166	4	no	—
112	C₂²×Q₈	—	166	1	no	—
113	C₂²×C₈	D₃₃₂	1	4	yes	—
114	C₂²×C₈	D₃₃₂	1	2	yes	—
115	C₄⋊Q₈	—	83	2	no	—
116	C₄⋊Q₈	—	166	1	no	—
117	C₄⋊Q₈	—	166	2	no	—
118	C₂×C₄⋊C₄	—	166	2	no	—
119	C₂×C₄⋊C₄	—	166	2	no	—
120	C₂×C₄⋊C₄	—	166	2	no	—
121	C₂×C₄⋊C₄	—	166	1	no	—
122	C₂×C₄⋊C₄	—	166	2	no	—
123	C₂×C₄⋊C₄	—	166	2	no	—
124	C₂×C₄²	C₂²×D₈₃	1	1	yes	characteristic
125	#107	—	332	4	no	—
126	#103	—	332	4	no	—
127	C₄×C₈	—	2	2	no	—
128	C₄×C₈	—	2	2	no	—
129	C₄×C₈	—	2	2	no	—
130	C₂×Q₈	—	332	1	no	—
131	C₂×Q₈	—	332	2	no	—
132	C₂×Q₈	—	332	2	no	—
133	C₂×Q₈	—	166	2	no	—
134	C₂²×C₄	—	332	1	no	—
135	C₂²×C₄	—	332	2	no	—
136	C₂²×C₄	—	332	1	no	—
137	C₂²×C₄	C₂×D₃₃₂	1	1	yes	characteristic
138	C₂²×C₄	C₂×D₃₃₂	1	2	yes	—
139	C₂×C₈	—	2	2	no	—
140	C₂×C₈	—	2	2	no	—
141	C₂×C₈	—	2	4	no	—
142	C₂×C₈	—	2	4	no	—
143	C₂×C₈	—	2	4	no	—
144	C₂×C₈	D₆₆₄	1	4	yes	—
145	C₄⋊C₄	—	332	2	no	—
146	C₄⋊C₄	—	332	2	no	—
147	C₄⋊C₄	—	332	4	no	—
148	C₄⋊C₄	—	332	4	no	—
149	C₄⋊C₄	—	332	4	no	—
150	C₄⋊C₄	—	332	4	no	—
151	C₄⋊C₄	—	166	4	no	—
152	C₄²	Q₈×D₈₃	1	1	yes	characteristic
153	C₄²	#13050589	1	1	yes	characteristic
154	C₄²	#13050589	1	2	yes	—
155	C₂³	#14581789	1	1	yes	characteristic
156	Q₈	—	332	4	no	—
157	Q₈	—	664	2	no	—
158	C₂×C₄	—	332	2	no	—
159	C₂×C₄	—	664	2	no	—
160	C₂×C₄	—	664	1	no	—
161	C₂×C₄	—	664	2	no	—
162	C₂×C₄	—	664	2	no	—
163	C₂×C₄	—	664	2	no	—
164	C₂×C₄	—	664	1	no	—
165	C₂×C₄	—	664	1	no	—
166	C₂×C₄	#14581772	1	1	yes	characteristic
167	C₂×C₄	#14581772	1	2	yes	—
168	C₂×C₄	#14581776	1	1	yes	characteristic
169	C₂×C₄	#14581776	1	2	yes	—
170	C₂×C₄	C₂×D₆₆₄	1	2	yes	—
171	C₂×C₄	#14581814	1	4	yes	—
172	C₂×C₄	#14581825	1	2	yes	—
173	C₂×C₄	#14581825	1	4	yes	—
174	C₈	—	2	4	no	—
175	C₈	—	4	4	no	—
176	C₂²	#19127638	1	1	yes	characteristic
177	C₂²	#19128078	1	1	yes	characteristic
178	C₂²	#19128085	1	2	yes	—
179	C₂²	#19128095	1	1	yes	characteristic
180	C₂²	#19128095	1	2	yes	—
181	C₄	—	1328	2	no	—
182	C₄	—	1328	1	no	—
183	C₄	—	664	2	no	—
184	C₄	#19127982	1	2	yes	—
185	C₄	#19128009	1	4	yes	—
186	C₄	#19128015	1	2	yes	—
187	C₄	#19128043	1	4	yes	—
188	C₂	#24907069	1	1	yes	characteristic
189	C₂	#24907109	1	2	yes	—
190	C₂	#24907134	1	1	yes	characteristic
191	C₂	#24908990	1	1	yes	characteristic
192	C₂	#24909010	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#24908424

C₂

yes

maximal

#24908265

C₂

yes

maximal, characteristic

#24907623

C₂

yes

maximal, characteristic

#24907591

C₂

yes

maximal

C₈₃×#1273

C₂

yes

maximal, characteristic

C₂×#14581792

C₂²

yes

—

—

C₂²

yes

characteristic

#19127571

C₂²

yes

—

#19127464

—

#19127464

—

C₂×C₄×C₈×C₈₃

C₂²

yes

—

C₂×C₄×C₈×C₈₃

C₂²

yes

characteristic

C₂²×Dic₁₆₆

—

#14581792

—

#14581792

—

#14581792

Q₈

yes

—

C₂×C₄⋊Dic₈₃

—

C₂×C₄⋊Dic₈₃

—

C₂×C₄⋊Dic₈₃

—

C₂×C₄⋊Dic₈₃

—

C₂×C₄⋊Dic₈₃

—

C₂×C₄⋊Dic₈₃

—

#14581726

—

#14581717

—

C₂²×C₈×C₈₃

D₄

yes

—

C₂²×C₈×C₈₃

D₄

yes

—

C₂×C₄²×C₈₃

C₂³

yes

characteristic

C₄×C₈×C₈₃

—

C₄×C₈×C₈₃

—

C₄×C₈×C₈₃

—

C₂×Dic₁₆₆

—

C₂×Dic₁₆₆

—

C₂×Dic₁₆₆

—

C₂×Dic₁₆₆

—

C₂²×Dic₈₃

—

C₂²×Dic₈₃

—

C₂²×Dic₈₃

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

—

C₂²×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₈₃

C₂×D₄

yes

—

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

D₈

yes

—

C₄²×C₈₃

C₂×Q₈

yes

C₄²×C₈₃

yes

C₄²×C₈₃

yes

—

C₂³×C₈₃

#127

yes

characteristic

C₂×C₄×C₈₃

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₈₃

C₂²⋊Q₈

yes

—

C₂×C₄×C₈₃

#123

yes

characteristic

C₂×C₄×C₈₃

#123

yes

—

C₂×C₄×C₈₃

C₂×D₈

yes

—

C₂×C₄×C₈₃

#135

yes

—

C₂×C₄×C₈₃

#137

yes

—

C₂×C₄×C₈₃

#137

yes

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

Dic₁₆₆

—

Dic₁₆₆

—

C₈×C₈₃

—

C₈×C₈₃

—

C₂²×C₈₃

yes

C₂²×C₈₃

yes

C₂²×C₈₃

yes

—

C₂²×C₈₃

#497

yes

characteristic

C₂²×C₈₃

#497

yes

—

C₄×C₈₃

#474

yes

—

C₄×C₈₃

#479

yes

—

C₄×C₈₃

#480

yes

—

C₄×C₈₃

#485

yes

—

Dic₈₃

—

Dic₈₃

—

Dic₈₃

—

—

C₂×C₈₃

yes

C₂×C₈₃

yes

—

C₂×C₈₃

yes

C₂×C₈₃

yes

C₂×C₈₃

yes

—

#1990

—

#1956

—

#1845

—

100

#1834

—

101

#1273

D₈₃

yes

characteristic

102

C₈₃

#10676

yes

characteristic, complemented, Sylow

103

C₂×C₄⋊Q₈

—

104

C₂×#107

—

166

—

105

C₂×#103

—

166

—

106

C₂×C₄×C₈

C₂×D₈₃

yes

—

107

C₂×C₄×C₈

C₂×D₈₃

yes

characteristic

108

#384

—

109

#384

—

110

#340

—

166

—

111

#340

—

166

—

112

C₂²×Q₈

—

166

—

113

C₂²×C₈

D₃₃₂

yes

—

114

C₂²×C₈

D₃₃₂

yes

—

115

C₄⋊Q₈

—

116

C₄⋊Q₈

—

166

—

117

C₄⋊Q₈

—

166

—

118

C₂×C₄⋊C₄

—

166

—

119

C₂×C₄⋊C₄

—

166

—

120

C₂×C₄⋊C₄

—

166

—

121

C₂×C₄⋊C₄

—

166

—

122

C₂×C₄⋊C₄

—

166

—

123

C₂×C₄⋊C₄

—

166

—

124

C₂×C₄²

C₂²×D₈₃

yes

characteristic

125

#107

—

332

—

126

#103

—

332

—

127

C₄×C₈

—

128

C₄×C₈

—

129

C₄×C₈

—

130

C₂×Q₈

—

332

—

131

C₂×Q₈

—

332

—

132

C₂×Q₈

—

332

—

133

C₂×Q₈

—

166

—

134

C₂²×C₄

—

332

—

135

C₂²×C₄

—

332

—

136

C₂²×C₄

—

332

—

137

C₂²×C₄

C₂×D₃₃₂

yes

characteristic

138

C₂²×C₄

C₂×D₃₃₂

yes

—

139

C₂×C₈

—

140

C₂×C₈

—

141

C₂×C₈

—

142

C₂×C₈

—

143

C₂×C₈

—

144

C₂×C₈

D₆₆₄

yes

—

145

C₄⋊C₄

—

332

—

146

C₄⋊C₄

—

332

—

147

C₄⋊C₄

—

332

—

148

C₄⋊C₄

—

332

—

149

C₄⋊C₄

—

332

—

150

C₄⋊C₄

—

332

—

151

C₄⋊C₄

—

166

—

152

C₄²

Q₈×D₈₃

yes

characteristic

153

C₄²

#13050589

yes

characteristic

154

C₄²

#13050589

yes

—

155

C₂³

#14581789

yes

characteristic

156

Q₈

—

332

—

157

Q₈

—

664

—

158

C₂×C₄

—

332

—

159

C₂×C₄

—

664

—

160

C₂×C₄

—

664

—

161

C₂×C₄

—

664

—

162

C₂×C₄

—

664

—

163

C₂×C₄

—

664

—

164

C₂×C₄

—

664

—

165

C₂×C₄

—

664

—

166

C₂×C₄

#14581772

yes

characteristic

167

C₂×C₄

#14581772

yes

—

168

C₂×C₄

#14581776

yes

characteristic

169

C₂×C₄

#14581776

yes

—

170

C₂×C₄

C₂×D₆₆₄

yes

—

171

C₂×C₄

#14581814

yes

—

172

C₂×C₄

#14581825

yes

—

173

C₂×C₄

#14581825

yes

—

174

C₈

—

175

C₈

—

176

C₂²

#19127638

yes

characteristic

177

C₂²

#19128078

yes

characteristic

178

C₂²

#19128085

yes

—

179

C₂²

#19128095

yes

characteristic

180

C₂²

#19128095

yes

—

181

C₄

—

1328

—

182

C₄

—

1328

—

183

C₄

—

664

—

184

C₄

#19127982

yes

—

185

C₄

#19128009

yes

—

186

C₄

#19128015

yes

—

187

C₄

#19128043

yes

—

188

C₂

#24907069

yes

characteristic

189

C₂

#24907109

yes

—

190

C₂

#24907134

yes

characteristic

191

C₂

#24908990

yes

characteristic

192

C₂

#24909010

yes

—

	subgroup	G/H
center	C₂³	#14581789
commutator, cosocle	C₂×C₄²×C₈₃	C₂³
Frattini	C₂×C₄²	C₂²×D₈₃
Fitting	C₈₃×#1273	C₂
socle, prime radical	C₂³×C₈₃	#127
hypercenter	#1273	D₈₃
hypocenter	C₈₃	#10676
LCS	G ▷ C₂×C₄²×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1273

subgroup

G/H

center

C₂³

#14581789

commutator, cosocle

C₂×C₄²×C₈₃

C₂³

Frattini

C₂×C₄²

C₂²×D₈₃

Fitting

C₈₃×#1273

C₂

C₂³×C₈₃

D₈₃

C₈₃

G ▷ C₂×C₄²×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1273

N	H ≅ G/N	type
102. C₈₃	#10676	semidirect

H ≅ G/N

type

102. C₈₃

#10676

semidirect

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

subgroup	index
#24908424	2
#24908265	2
#24907623	2
#24907591	2
C₈₃×#1273	2
#10676	83

subgroup

index

C₈₃×#1273

Subgroups — #35309469

Conjugacy classes of subgroups

Complements

Maximal subgroups