Group #28298542 — subgroups

Subgroups — #28298542

Order 13568.

214 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 64 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	#20291079	C₂	1	1	yes	maximal, characteristic
2	#20290636	C₂	1	1	yes	maximal, characteristic
3	#20273477	C₂	1	1	yes	maximal, characteristic
4	#16811569	—	2	1	no	—
5	C₄⋊#11719910	—	2	1	no	—
6	#16811518	—	2	1	no	—
7	#16811512	C₄	1	2	yes	complemented
8	#16811497	—	2	1	no	—
9	#16811497	C₄	1	2	yes	—
10	#16810679	—	2	1	no	—
11	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
12	#16810376	C₄	1	1	yes	characteristic, complemented
13	#16810373	—	2	1	no	—
14	C₅₃×#335	C₄	1	1	yes	characteristic, complemented
15	C₂×C₄⋊#6157	—	2	2	no	—
16	C₂×C₄⋊#6157	—	4	1	no	—
17	C₂×C₄⋊#6157	—	4	1	no	—
18	C₂×#11719910	—	2	2	no	—
19	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
20	C₂×C₈×D₅₃	D₄	1	1	yes	characteristic
21	C₄²×D₅₃	D₄	1	1	yes	characteristic
22	C₄⋊#88322	—	2	1	no	—
23	#13133129	—	4	1	no	—
24	#13133127	—	4	1	no	—
25	C₄×#88322	D₄	1	1	yes	characteristic
26	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
27	C₈×Dic₅₃	Q₈	1	1	yes	characteristic
28	C₄⋊C₈×C₅₃	—	2	1	no	—
29	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
30	C₂²×#6157	—	4	2	no	—
31	C₂×C₄×D₅₃	C₂²⋊C₄	1	1	yes	characteristic
32	C₂×C₄×D₅₃	D₈	1	1	yes	characteristic
33	C₂×C₄×D₅₃	SD₁₆	1	1	yes	characteristic
34	C₄⋊#6157	—	4	2	no	—
35	C₄⋊#6157	—	4	2	no	—
36	C₄⋊#6157	—	8	2	no	—
37	C₄⋊#6157	—	8	1	no	—
38	C₄⋊#6157	—	8	1	no	—
39	C₂×C₈×C₅₃	—	4	1	no	—
40	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
41	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
42	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
43	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
44	C₄×Dic₅₃	SD₁₆	1	1	yes	characteristic
45	C₄×Dic₅₃	Q₁₆	1	1	yes	characteristic
46	C₂×#88322	—	4	1	no	—
47	C₂×#88322	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×#88322	C₄⋊C₄	1	1	yes	characteristic
49	C₈×D₅₃	—	4	1	no	—
50	C₈×D₅₃	—	4	1	no	—
51	#11719910	—	4	2	no	—
52	C₂²×D₅₃	#102	1	1	yes	characteristic
53	C₂×#6157	—	8	2	no	—
54	C₂×#6157	—	8	2	no	—
55	C₂×#6157	—	8	2	no	—
56	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
57	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
58	C₂×C₄×C₅₃	#103	1	1	yes	characteristic
59	C₂×Dic₅₃	#103	1	1	yes	characteristic
60	C₂×Dic₅₃	#106	1	1	yes	characteristic
61	C₂×Dic₅₃	#107	1	1	yes	characteristic
62	C₄×D₅₃	—	2	1	no	—
63	C₄×D₅₃	—	2	1	no	—
64	C₄×D₅₃	—	2	1	no	—
65	C₄×D₅₃	—	2	1	no	—
66	C₄×D₅₃	—	2	1	no	—
67	C₄×D₅₃	—	2	1	no	—
68	C₈×C₅₃	—	2	1	no	—
69	C₈×C₅₃	—	2	1	no	—
70	C₈×C₅₃	—	8	1	no	—
71	#88322	—	2	1	no	—
72	#88322	—	2	1	no	—
73	#88322	—	8	1	no	—
74	#7222	—	53	1	no	maximal, Sylow
75	C₂²×C₅₃	#340	1	1	yes	characteristic
76	C₂×D₅₃	—	2	1	no	—
77	C₂×D₅₃	—	2	1	no	—
78	C₂×D₅₃	#361	1	1	yes	characteristic
79	C₂×D₅₃	#362	1	1	yes	characteristic
80	#6157	—	16	2	no	—
81	C₄×C₅₃	—	2	1	no	—
82	C₄×C₅₃	H(3,ℤ/4)	1	1	yes	characteristic
83	C₄×C₅₃	#344	1	1	yes	characteristic
84	C₄×C₅₃	#360	1	1	yes	characteristic
85	C₄×C₅₃	#360	1	1	yes	characteristic
86	Dic₅₃	—	2	1	no	—
87	Dic₅₃	—	2	1	no	—
88	Dic₅₃	—	2	1	no	—
89	#2163	—	53	1	no	—
90	#1834	—	53	1	no	—
91	#1568	—	53	1	no	—
92	C₂×C₅₃	#1286	1	1	yes	characteristic
93	C₂×C₅₃	#1386	1	1	yes	characteristic
94	C₂×C₅₃	#1387	1	1	yes	characteristic
95	D₅₃	—	2	1	no	—
96	D₅₃	—	2	1	no	—
97	#432	—	106	1	no	—
98	C₂×C₄×C₈	—	53	1	no	—
99	#384	—	106	1	no	—
100	C₄⋊C₁₆	—	106	1	no	—
101	C₂²⋊C₁₆	—	106	1	no	—
102	#346	—	53	2	no	—
103	#340	—	53	2	no	—
104	#340	—	106	1	no	—
105	#335	—	53	1	no	—
106	#335	#6157	1	1	yes	characteristic, complemented
107	#334	—	106	1	no	—
108	C₅₃	#7222	1	1	yes	characteristic, complemented, Sylow
109	C₂²×C₈	—	53	1	no	—
110	C₂²×C₈	—	53	1	no	—
111	C₂×C₄⋊C₄	—	106	2	no	—
112	C₂×C₄⋊C₄	—	212	1	no	—
113	C₂×C₄⋊C₄	—	212	1	no	—
114	C₂×C₄²	—	53	1	no	—
115	C₂×C₁₆	—	106	2	no	—
116	C₄⋊C₈	—	2	1	no	—
117	C₄⋊C₈	—	106	1	no	—
118	C₂²⋊C₈	—	212	1	no	—
119	#97	—	212	1	no	—
120	C₄×C₈	—	53	1	no	—
121	C₄×C₈	—	53	1	no	—
122	C₄×C₈	—	53	1	no	—
123	C₄×C₈	C₂×#6157	1	1	yes	characteristic
124	C₂²×C₄	—	53	1	no	—
125	C₂²×C₄	—	53	1	no	—
126	C₂²×C₄	—	53	1	no	—
127	C₂²×C₄	—	212	2	no	—
128	C₂×C₈	—	4	1	no	—
129	C₂×C₈	—	53	1	no	—
130	C₂×C₈	—	53	1	no	—
131	C₂×C₈	—	212	1	no	—
132	C₂×C₈	—	212	1	no	—
133	C₂×C₈	—	212	1	no	—
134	C₂×C₈	C₄×#6157	1	1	yes	characteristic
135	C₂×C₈	C₄⋊#6157	1	1	yes	characteristic
136	C₄⋊C₄	—	424	2	no	—
137	C₄⋊C₄	—	424	1	no	—
138	C₄⋊C₄	—	424	1	no	—
139	C₄⋊C₄	—	212	2	no	—
140	C₄⋊C₄	—	212	2	no	—
141	C₄²	—	53	1	no	—
142	C₄²	—	53	1	no	—
143	C₄²	—	53	1	no	—
144	C₄²	C₂²⋊#6157	1	1	yes	characteristic
145	C₁₆	—	212	2	no	—
146	C₂³	—	53	1	no	—
147	C₂×C₄	—	53	1	no	—
148	C₂×C₄	—	53	1	no	—
149	C₂×C₄	—	53	1	no	—
150	C₂×C₄	—	106	1	no	—
151	C₂×C₄	—	106	1	no	—
152	C₂×C₄	—	106	1	no	—
153	C₂×C₄	—	106	1	no	—
154	C₂×C₄	—	106	1	no	—
155	C₂×C₄	—	106	1	no	—
156	C₂×C₄	—	424	2	no	—
157	C₂×C₄	—	424	2	no	—
158	C₂×C₄	—	424	2	no	—
159	C₂×C₄	(C₂×#6157)⋊C₄	1	1	yes	characteristic
160	C₂×C₄	#13133279	1	1	yes	characteristic
161	C₂×C₄	#13133280	1	1	yes	characteristic
162	C₈	—	2	1	no	—
163	C₈	—	2	1	no	—
164	C₈	—	8	1	no	—
165	C₈	—	106	1	no	—
166	C₈	—	106	1	no	—
167	C₈	—	424	1	no	—
168	C₂²	—	53	1	no	—
169	C₂²	—	53	1	no	—
170	C₂²	—	106	1	no	—
171	C₂²	—	106	1	no	—
172	C₂²	#16811496	1	1	yes	characteristic
173	C₄	—	2	1	no	—
174	C₄	—	848	2	no	—
175	C₄	—	106	1	no	—
176	C₄	—	106	1	no	—
177	C₄	—	106	1	no	—
178	C₄	#16811491	1	1	yes	characteristic
179	C₄	#16811506	1	1	yes	characteristic
180	C₄	#16811539	1	1	yes	characteristic
181	C₄	#16811540	1	1	yes	characteristic
182	C₂	—	106	1	no	—
183	C₂	—	106	1	no	—
184	C₂	#20289570	1	1	yes	characteristic
185	C₂	#20289786	1	1	yes	characteristic
186	C₂	#20289791	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20291079

C₂

yes

maximal, characteristic

#20290636

C₂

yes

maximal, characteristic

#20273477

C₂

yes

maximal, characteristic

—

—

—

C₄

yes

complemented

#16811497

—

#16811497

C₄

yes

—

#16810679

—

C₄×C₈×D₅₃

C₂²

yes

characteristic

#16810376

C₄

yes

characteristic, complemented

#16810373

—

C₅₃×#335

C₄

yes

characteristic, complemented

C₂×C₄⋊#6157

—

C₂×C₄⋊#6157

—

C₂×C₄⋊#6157

—

C₂×#11719910

—

C₂×C₈×D₅₃

C₂×C₄

yes

characteristic

C₂×C₈×D₅₃

D₄

yes

characteristic

C₄²×D₅₃

D₄

yes

—

—

—

D₄

yes

characteristic

C₈×Dic₅₃

C₂×C₄

yes

characteristic

C₈×Dic₅₃

Q₈

yes

characteristic

C₄⋊C₈×C₅₃

—

C₄×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂²×#6157

—

C₂×C₄×D₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×D₅₃

D₈

yes

characteristic

C₂×C₄×D₅₃

SD₁₆

yes

—

—

—

—

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₄²

yes

characteristic

C₂×C₈×C₅₃

C₄⋊C₄

yes

characteristic

C₄²×C₅₃

C₂²⋊C₄

yes

characteristic

C₄×Dic₅₃

C₄⋊C₄

yes

characteristic

C₄×Dic₅₃

SD₁₆

yes

characteristic

C₄×Dic₅₃

Q₁₆

yes

characteristic

C₂×#88322

—

C₂×#88322

C₂²⋊C₄

yes

characteristic

C₂×#88322

C₄⋊C₄

yes

characteristic

C₈×D₅₃

—

C₈×D₅₃

—

#11719910

—

C₂²×D₅₃

yes

—

—

—

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×Dic₅₃

yes

C₂×Dic₅₃

yes

C₂×Dic₅₃

yes

C₄×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

—

—

—

—

C₂²×C₅₃

yes

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

yes

C₂×D₅₃

yes

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

#2163

—

#1834

—

#1568

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

D₅₃

—

D₅₃

—

#432

—

106

—

C₂×C₄×C₈

—

#384

—

106

—

100

C₄⋊C₁₆

—

106

—

101

C₂²⋊C₁₆

—

106

—

102

#346

—

103

#340

—

104

#340

—

106

—

105

#335

—

106

#335

#6157

yes

characteristic, complemented

107

#334

—

106

—

108

C₅₃

#7222

yes

characteristic, complemented, Sylow

109

C₂²×C₈

—

110

C₂²×C₈

—

111

C₂×C₄⋊C₄

—

106

—

112

C₂×C₄⋊C₄

—

212

—

113

C₂×C₄⋊C₄

—

212

—

114

C₂×C₄²

—

115

C₂×C₁₆

—

106

—

116

C₄⋊C₈

—

117

C₄⋊C₈

—

106

—

118

C₂²⋊C₈

—

212

—

119

#97

—

212

—

120

C₄×C₈

—

121

C₄×C₈

—

122

C₄×C₈

—

123

C₄×C₈

C₂×#6157

yes

characteristic

124

C₂²×C₄

—

125

C₂²×C₄

—

126

C₂²×C₄

—

127

C₂²×C₄

—

212

—

128

C₂×C₈

—

129

C₂×C₈

—

130

C₂×C₈

—

131

C₂×C₈

—

212

—

132

C₂×C₈

—

212

—

133

C₂×C₈

—

212

—

134

C₂×C₈

C₄×#6157

yes

characteristic

135

C₂×C₈

C₄⋊#6157

yes

characteristic

136

C₄⋊C₄

—

424

—

137

C₄⋊C₄

—

424

—

138

C₄⋊C₄

—

424

—

139

C₄⋊C₄

—

212

—

140

C₄⋊C₄

—

212

—

141

C₄²

—

142

C₄²

—

143

C₄²

—

144

C₄²

C₂²⋊#6157

yes

characteristic

145

C₁₆

—

212

—

146

C₂³

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

106

—

151

C₂×C₄

—

106

—

152

C₂×C₄

—

106

—

153

C₂×C₄

—

106

—

154

C₂×C₄

—

106

—

155

C₂×C₄

—

106

—

156

C₂×C₄

—

424

—

157

C₂×C₄

—

424

—

158

C₂×C₄

—

424

—

159

C₂×C₄

(C₂×#6157)⋊C₄

yes

characteristic

160

C₂×C₄

#13133279

yes

characteristic

161

C₂×C₄

#13133280

yes

characteristic

162

C₈

—

163

C₈

—

164

C₈

—

165

C₈

—

106

—

166

C₈

—

106

—

167

C₈

—

424

—

168

C₂²

—

169

C₂²

—

170

C₂²

—

106

—

171

C₂²

—

106

—

172

C₂²

#16811496

yes

characteristic

173

C₄

—

174

C₄

—

848

—

175

C₄

—

106

—

176

C₄

—

106

—

177

C₄

—

106

—

178

C₄

#16811491

yes

characteristic

179

C₄

#16811506

yes

characteristic

180

C₄

#16811539

yes

characteristic

181

C₄

#16811540

yes

characteristic

182

C₂

—

106

—

183

C₂

—

106

—

184

C₂

#20289570

yes

characteristic

185

C₂

#20289786

yes

characteristic

186

C₂

#20289791

yes

characteristic

	subgroup	G/H
center	C₂²	#16811496
commutator	C₂×C₈×C₅₃	C₄²
Frattini	C₄×C₈	C₂×#6157
Fitting	C₅₃×#335	C₄
socle	C₂²×C₅₃	#340
prime radical	C₂²×D₅₃	#102
cosocle	C₄×C₈×D₅₃	C₂²
hypercenter	#335	#6157
hypocenter	C₅₃	#7222
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #335

subgroup

G/H

center

C₂²

#16811496

commutator

C₂×C₈×C₅₃

C₄²

Frattini

C₄×C₈

C₂×#6157

Fitting

C₅₃×#335

C₄

C₂²×C₅₃

C₂²×D₅₃

C₄×C₈×D₅₃

C₂²

C₅₃

G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #335

N	H ≅ G/N	type
7. #16811512	C₄	semidirect
12. #16810376	C₄	semidirect
14. C₅₃×#335	C₄	semidirect
106. #335	#6157	semidirect
108. C₅₃	#7222	semidirect

H ≅ G/N

type

7. #16811512

C₄

semidirect

12. #16810376

C₄

semidirect

14. C₅₃×#335

C₄

semidirect

106. #335

#6157

semidirect

108. C₅₃

#7222

semidirect

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

subgroup	index
#20291079	2
#20290636	2
#20273477	2
#7222	53

subgroup

index

Subgroups — #28298542

Conjugacy classes of subgroups

Complements

Maximal subgroups