Group #35324278 — subgroups

Subgroups — #35324278

Order 21248.

284 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 87 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	#24909076	C₂	1	1	yes	maximal, characteristic
2	#24907528	C₂	1	2	yes	maximal
3	#24905571	C₂	1	2	yes	maximal, complemented
4	#24905557	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#366×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#14581798	C₂²	1	1	yes	characteristic
7	C₂×#14581732	C₂²	1	1	yes	characteristic
8	C₂×#14581728	C₂²	1	2	yes	—
9	#19127700	—	2	2	no	—
10	(C₂×D₈₃)⋊C₁₆	—	2	2	no	—
11	#19127470	—	2	2	no	—
12	C₂²×C₁₆×C₈₃	C₂²	1	1	yes	characteristic
13	C₂×#107×C₈₃	C₂²	1	2	yes	—
14	#366×C₈₃	—	2	2	no	—
15	C₂×#13050590	D₄	1	1	yes	characteristic
16	#14581798	—	2	2	no	—
17	#14581798	—	2	2	no	—
18	C₂×#13050574	—	2	2	no	—
19	C₂×#13050574	D₄	1	1	yes	characteristic
20	C₂×#13050573	—	2	1	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	#14581715	—	4	2	no	—
27	#14581712	—	4	2	no	—
28	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
29	C₂×C₄⋊C₄×C₈₃	—	2	2	no	—
30	C₂×C₁₆×C₈₃	—	2	2	no	—
31	C₂×C₁₆×C₈₃	D₄	1	2	yes	—
32	#107×C₈₃	—	2	2	no	—
33	#107×C₈₃	—	2	2	no	—
34	#13050590	—	2	2	no	—
35	#13050590	—	2	2	no	—
36	C₂×Dic₁₆₆	D₈	1	1	yes	characteristic
37	C₂×D₃₃₂	D₈	1	1	yes	characteristic
38	C₂×C₂²⋊D₈₃	—	2	1	no	—
39	C₂×C₄×D₈₃	—	2	1	no	—
40	#13050574	—	2	2	no	—
41	#13050574	—	4	2	no	—
42	#13050574	—	4	2	no	—
43	#13050573	—	4	2	no	—
44	C₂×C₈₃⋊C₈	—	2	1	no	—
45	C₂×C₈₃⋊C₈	—	4	2	no	—
46	C₈×D₈₃	—	4	1	no	—
47	C₈×D₈₃	—	4	1	no	—
48	C₈₃⋊C₁₆	—	4	1	no	—
49	C₂²×C₄×C₈₃	—	4	2	no	—
50	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
51	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
52	C₂×C₈×C₈₃	#41	1	1	yes	characteristic
53	C₂×C₈×C₈₃	#41	1	2	yes	—
54	C₂×C₈×C₈₃	#41	1	2	yes	—
55	C₄⋊C₄×C₈₃	—	4	2	no	—
56	C₄⋊C₄×C₈₃	—	4	2	no	—
57	C₁₆×C₈₃	—	2	2	no	—
58	C₂³×C₈₃	#135	1	1	yes	characteristic
59	C₂²×D₈₃	—	2	1	no	—
60	C₂×C₄×C₈₃	—	8	2	no	—
61	C₂×C₄×C₈₃	—	16	2	no	—
62	C₂×C₄×C₈₃	#121	1	1	yes	characteristic
63	C₂×C₄×C₈₃	#123	1	2	yes	—
64	C₂×C₄×C₈₃	C₂×D₈	1	1	yes	characteristic
65	C₂×C₄×C₈₃	#137	1	2	yes	—
66	C₂²⋊D₈₃	—	4	2	no	—
67	C₂×Dic₈₃	—	2	1	no	—
68	D₃₃₂	—	2	2	no	—
69	C₄×D₈₃	—	4	1	no	—
70	C₄×D₈₃	—	4	1	no	—
71	Dic₁₆₆	—	2	2	no	—
72	C₈×C₈₃	#123	1	1	yes	characteristic
73	C₈×C₈₃	#124	1	1	yes	characteristic
74	C₈×C₈₃	#126	1	2	yes	—
75	C₈₃⋊C₈	—	4	1	no	—
76	C₈₃⋊C₈	—	8	2	no	—
77	C₂²×C₈₃	—	2	1	no	—
78	C₂²×C₈₃	—	2	1	no	—
79	C₂²×C₈₃	#466	1	1	yes	characteristic
80	C₂²×C₈₃	#485	1	2	yes	—
81	C₂×D₈₃	—	4	1	no	—
82	C₂×D₈₃	—	4	1	no	—
83	C₂×D₈₃	—	4	1	no	—
84	C₄×C₈₃	—	16	2	no	—
85	C₄×C₈₃	#397	1	1	yes	characteristic
86	C₄×C₈₃	#470	1	1	yes	characteristic
87	C₄×C₈₃	#480	1	2	yes	—
88	Dic₈₃	—	4	1	no	—
89	#13495	—	83	1	no	maximal, Sylow
90	C₂×C₈₃	—	2	2	no	—
91	C₂×C₈₃	#2075	1	1	yes	characteristic
92	C₂×C₈₃	#2221	1	1	yes	characteristic
93	C₂×C₈₃	#2222	1	1	yes	characteristic
94	D₈₃	—	8	1	no	—
95	C₂×#366	D₈₃	1	1	yes	characteristic, complemented
96	#2113	—	83	1	no	—
97	#1816	—	83	2	no	—
98	#1387	—	83	2	no	—
99	#1381	—	83	1	no	—
100	C₈₃	#13495	1	1	yes	characteristic, complemented, Sylow
101	C₂×#131	—	83	1	no	—
102	C₂×M₃₂	—	83	1	no	—
103	C₂²×C₁₆	C₂×D₈₃	1	1	yes	characteristic
104	C₂×#108	—	83	2	no	—
105	C₂×#107	C₂×D₈₃	1	2	yes	—
106	#417	—	166	2	no	—
107	#366	—	2	2	no	—
108	C₂²⋊C₁₆	—	166	2	no	—
109	#343	—	166	2	no	—
110	C₂×#41	—	83	1	no	—
111	#131	—	166	2	no	—
112	#131	—	166	2	no	—
113	C₂×M₁₆	—	83	1	no	—
114	C₂×M₁₆	—	166	2	no	—
115	C₂×M₁₆	—	166	1	no	—
116	C₂²×C₈	—	166	1	no	—
117	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
118	C₂×C₄⋊C₄	—	2	2	no	—
119	M₃₂	—	166	2	no	—
120	C₂×C₁₆	—	2	2	no	—
121	C₂×C₁₆	—	166	1	no	—
122	C₂×C₁₆	C₂²⋊D₈₃	1	2	yes	—
123	#108	—	166	2	no	—
124	#108	—	166	2	no	—
125	#107	—	2	2	no	—
126	#107	—	2	2	no	—
127	#103	—	332	2	no	—
128	#102	—	332	2	no	—
129	#41	—	166	2	no	—
130	#41	—	166	2	no	—
131	C₂×Q₈	—	83	1	no	—
132	C₂×D₄	—	83	1	no	—
133	C₂×D₄	—	166	1	no	—
134	C₂²×C₄	—	4	2	no	—
135	C₂²×C₄	—	166	1	no	—
136	C₂²×C₄	D₄×D₈₃	1	1	yes	characteristic
137	M₁₆	—	332	2	no	—
138	M₁₆	—	332	2	no	—
139	M₁₆	—	332	2	no	—
140	M₁₆	—	166	2	no	—
141	C₂×C₈	—	332	1	no	—
142	C₂×C₈	—	332	1	no	—
143	C₂×C₈	—	332	2	no	—
144	C₂×C₈	—	166	1	no	—
145	C₂×C₈	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
146	C₂×C₈	#13050589	1	1	yes	characteristic
147	C₂×C₈	#13050590	1	2	yes	—
148	C₂×C₈	#13050591	1	2	yes	—
149	C₄⋊C₄	—	4	2	no	—
150	C₄⋊C₄	—	4	2	no	—
151	C₁₆	—	2	2	no	—
152	C₁₆	—	332	1	no	—
153	C₂³	—	166	1	no	—
154	C₂³	#14581811	1	1	yes	characteristic
155	Q₈	—	166	2	no	—
156	D₄	—	332	2	no	—
157	D₄	—	166	2	no	—
158	C₂×C₄	—	8	2	no	—
159	C₂×C₄	—	16	2	no	—
160	C₂×C₄	—	332	1	no	—
161	C₂×C₄	—	332	1	no	—
162	C₂×C₄	—	166	1	no	—
163	C₂×C₄	#14581763	1	1	yes	characteristic
164	C₂×C₄	#14581777	1	2	yes	—
165	C₂×C₄	D₈×D₈₃	1	1	yes	characteristic
166	C₂×C₄	#14581824	1	2	yes	—
167	C₈	—	332	1	no	—
168	C₈	—	664	2	no	—
169	C₈	#14581775	1	1	yes	characteristic
170	C₈	#14581781	1	1	yes	characteristic
171	C₈	#14581787	1	2	yes	—
172	C₂²	—	2	1	no	—
173	C₂²	—	2	1	no	—
174	C₂²	—	332	1	no	—
175	C₂²	—	332	1	no	—
176	C₂²	—	332	1	no	—
177	C₂²	#19127934	1	1	yes	characteristic
178	C₂²	#19128044	1	2	yes	—
179	C₄	—	16	2	no	—
180	C₄	—	332	1	no	—
181	C₄	#19127627	1	1	yes	characteristic
182	C₄	#19127956	1	1	yes	characteristic
183	C₄	#19128016	1	2	yes	—
184	C₂	—	2	2	no	—
185	C₂	—	664	1	no	—
186	C₂	#24908887	1	1	yes	characteristic
187	C₂	#24909580	1	1	yes	characteristic
188	C₂	#24909585	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#24909076

C₂

yes

maximal, characteristic

#24907528

C₂

yes

maximal

#24905571

C₂

yes

maximal, complemented

#24905557

C₂

yes

maximal, characteristic, complemented

C₂×#366×C₈₃

C₂

yes

maximal, characteristic, complemented

C₂×#14581798

C₂²

yes

characteristic

C₂×#14581732

C₂²

yes

characteristic

C₂×#14581728

C₂²

yes

—

#19127700

—

(C₂×D₈₃)⋊C₁₆

—

#19127470

—

C₂²×C₁₆×C₈₃

C₂²

yes

characteristic

C₂×#107×C₈₃

C₂²

yes

—

#366×C₈₃

—

C₂×#13050590

D₄

yes

—

—

—

D₄

yes

characteristic

C₂×#13050573

—

C₂×C₈×D₈₃

—

#14581732

—

C₂×C₈₃⋊C₁₆

—

—

—

—

—

C₂²×C₈×C₈₃

C₂³

yes

characteristic

C₂×C₄⋊C₄×C₈₃

—

C₂×C₁₆×C₈₃

—

C₂×C₁₆×C₈₃

D₄

yes

—

#107×C₈₃

—

#107×C₈₃

—

#13050590

—

#13050590

—

C₂×Dic₁₆₆

D₈

yes

characteristic

C₂×D₃₃₂

D₈

yes

characteristic

C₂×C₂²⋊D₈₃

—

C₂×C₄×D₈₃

—

—

—

—

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

—

C₈×D₈₃

—

C₈×D₈₃

—

C₈₃⋊C₁₆

—

C₂²×C₄×C₈₃

—

C₂²×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂×C₈×C₈₃

C₂×D₄

yes

C₂×C₈×C₈₃

yes

C₂×C₈×C₈₃

yes

—

C₂×C₈×C₈₃

#41

yes

—

C₄⋊C₄×C₈₃

—

C₄⋊C₄×C₈₃

—

C₁₆×C₈₃

—

C₂³×C₈₃

#135

yes

characteristic

C₂²×D₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

#121

yes

characteristic

C₂×C₄×C₈₃

#123

yes

—

C₂×C₄×C₈₃

C₂×D₈

yes

characteristic

C₂×C₄×C₈₃

#137

yes

—

C₂²⋊D₈₃

—

C₂×Dic₈₃

—

D₃₃₂

—

C₄×D₈₃

—

C₄×D₈₃

—

Dic₁₆₆

—

C₈×C₈₃

yes

C₈×C₈₃

yes

C₈×C₈₃

yes

—

C₈₃⋊C₈

—

C₈₃⋊C₈

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

#466

yes

characteristic

C₂²×C₈₃

#485

yes

—

C₂×D₈₃

—

C₂×D₈₃

—

C₂×D₈₃

—

C₄×C₈₃

—

C₄×C₈₃

yes

C₄×C₈₃

yes

C₄×C₈₃

yes

—

Dic₈₃

—

#13495

—

maximal, Sylow

C₂×C₈₃

—

C₂×C₈₃

yes

C₂×C₈₃

yes

C₂×C₈₃

yes

D₈₃

—

C₂×#366

D₈₃

yes

characteristic, complemented

—

—

—

—

100

C₈₃

#13495

yes

characteristic, complemented, Sylow

101

C₂×#131

—

102

C₂×M₃₂

—

103

C₂²×C₁₆

C₂×D₈₃

yes

characteristic

104

C₂×#108

—

105

C₂×#107

C₂×D₈₃

yes

—

106

#417

—

166

—

107

#366

—

108

C₂²⋊C₁₆

—

166

—

109

#343

—

166

—

110

C₂×#41

—

111

#131

—

166

—

112

#131

—

166

—

113

C₂×M₁₆

—

114

C₂×M₁₆

—

166

—

115

C₂×M₁₆

—

166

—

116

C₂²×C₈

—

166

—

117

C₂²×C₈

C₂²×D₈₃

yes

characteristic

118

C₂×C₄⋊C₄

—

119

M₃₂

—

166

—

120

C₂×C₁₆

—

121

C₂×C₁₆

—

166

—

122

C₂×C₁₆

C₂²⋊D₈₃

yes

—

123

#108

—

166

—

124

#108

—

166

—

125

#107

—

126

#107

—

127

#103

—

332

—

128

#102

—

332

—

129

#41

—

166

—

130

#41

—

166

—

131

C₂×Q₈

—

132

C₂×D₄

—

133

C₂×D₄

—

166

—

134

C₂²×C₄

—

135

C₂²×C₄

—

166

—

136

C₂²×C₄

D₄×D₈₃

yes

characteristic

137

M₁₆

—

332

—

138

M₁₆

—

332

—

139

M₁₆

—

332

—

140

M₁₆

—

166

—

141

C₂×C₈

—

332

—

142

C₂×C₈

—

332

—

143

C₂×C₈

—

332

—

144

C₂×C₈

—

166

—

145

C₂×C₈

C₂×C₂²⋊D₈₃

yes

characteristic

146

C₂×C₈

#13050589

yes

characteristic

147

C₂×C₈

#13050590

yes

—

148

C₂×C₈

#13050591

yes

—

149

C₄⋊C₄

—

150

C₄⋊C₄

—

151

C₁₆

—

152

C₁₆

—

332

—

153

C₂³

—

166

—

154

C₂³

#14581811

yes

characteristic

155

Q₈

—

166

—

156

D₄

—

332

—

157

D₄

—

166

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

332

—

161

C₂×C₄

—

332

—

162

C₂×C₄

—

166

—

163

C₂×C₄

#14581763

yes

characteristic

164

C₂×C₄

#14581777

yes

—

165

C₂×C₄

D₈×D₈₃

yes

characteristic

166

C₂×C₄

#14581824

yes

—

167

C₈

—

332

—

168

C₈

—

664

—

169

C₈

#14581775

yes

characteristic

170

C₈

#14581781

yes

characteristic

171

C₈

#14581787

yes

—

172

C₂²

—

173

C₂²

—

174

C₂²

—

332

—

175

C₂²

—

332

—

176

C₂²

—

332

—

177

C₂²

#19127934

yes

characteristic

178

C₂²

#19128044

yes

—

179

C₄

—

180

C₄

—

332

—

181

C₄

#19127627

yes

characteristic

182

C₄

#19127956

yes

characteristic

183

C₄

#19128016

yes

—

184

C₂

—

185

C₂

—

664

—

186

C₂

#24908887

yes

characteristic

187

C₂

#24909580

yes

characteristic

188

C₂

#24909585

yes

characteristic

	subgroup	G/H
center	C₂²	#19127934
commutator, cosocle	C₂²×C₈×C₈₃	C₂³
Frattini	C₂²×C₈	C₂²×D₈₃
Fitting	C₂×#366×C₈₃	C₂
socle	C₂²×C₈₃	#466
prime radical	C₂×#13050590	D₄
hypercenter	C₂×#366	D₈₃
hypocenter	C₈₃	#13495
LCS	G ▷ C₂²×C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#366

subgroup

G/H

center

C₂²

#19127934

commutator, cosocle

C₂²×C₈×C₈₃

C₂³

Frattini

C₂²×C₈

C₂²×D₈₃

Fitting

C₂×#366×C₈₃

C₂

C₂²×C₈₃

D₄

D₈₃

C₈₃

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

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#366

N	H ≅ G/N	type
3. #24905571	C₂	semidirect
4. #24905557	C₂	semidirect
5. C₂×#366×C₈₃	C₂	semidirect
95. C₂×#366	D₈₃	semidirect
100. C₈₃	#13495	semidirect

H ≅ G/N

type

3. #24905571

C₂

semidirect

4. #24905557

C₂

semidirect

5. C₂×#366×C₈₃

C₂

semidirect

95. C₂×#366

D₈₃

semidirect

100. C₈₃

#13495

semidirect

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

subgroup	index
#24909076	2
#24907528	2
#24905571	2
#24905557	2
C₂×#366×C₈₃	2
#13495	83

subgroup

index

C₂×#366×C₈₃

Subgroups — #35324278

Conjugacy classes of subgroups

Complements

Maximal subgroups