Group #28298313 — subgroups

Subgroups — #28298313

Order 13568.

234 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 88 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	#20290916	C₂	1	1	yes	maximal, characteristic
2	D₅₃⋊#334	C₂	1	1	yes	maximal, characteristic
3	#334×D₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂×D₅₃⋊C₁₆	C₄	1	2	yes	—
5	D₅₃⋊(C₄⋊C₈)	—	2	1	no	—
6	#16811512	—	2	1	no	—
7	#16811511	—	2	1	no	—
8	#16811485	—	2	1	no	—
9	#16811484	C₄	1	2	yes	—
10	C₄⋊C₈×D₅₃	—	2	1	no	—
11	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
12	#16810374	C₄	1	1	yes	characteristic
13	#16810373	—	2	1	no	—
14	C₅₃×#334	C₄	1	1	yes	characteristic
15	C₂×D₅₃⋊C₈	—	4	1	no	—
16	D₅₃⋊C₁₆	—	2	2	no	—
17	D₅₃⋊C₁₆	C₈	1	4	yes	complemented
18	C₂×#11719910	—	2	2	no	—
19	#13133284	—	4	1	no	—
20	C₄⋊#88324	—	2	2	no	—
21	C₂×C₈×D₅₃	—	4	1	no	—
22	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
23	C₂×C₈×D₅₃	Q₈	1	1	yes	characteristic
24	C₄²×D₅₃	D₄	1	1	yes	characteristic
25	C₄⋊#88322	—	2	1	no	—
26	#13133146	—	4	1	no	—
27	C₄×#88322	D₄	1	1	yes	characteristic
28	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
29	C₈×Dic₅₃	D₄	1	1	yes	characteristic
30	C₄⋊C₈×C₅₃	—	2	1	no	—
31	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
33	C₂×C₄×D₅₃	SD₁₆	1	1	yes	characteristic
34	C₂×C₄×D₅₃	SD₁₆	1	1	yes	characteristic
35	C₂×#88324	—	4	2	no	—
36	D₅₃⋊C₈	—	8	2	no	—
37	C₂×C₈×C₅₃	—	4	1	no	—
38	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
39	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
40	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
41	C₄×Dic₅₃	C₂²⋊C₄	1	1	yes	characteristic
42	C₄×Dic₅₃	D₈	1	1	yes	characteristic
43	C₄×Dic₅₃	Q₁₆	1	1	yes	characteristic
44	C₂×#88322	—	4	1	no	—
45	C₂×#88322	C₂²⋊C₄	1	1	yes	characteristic
46	C₂×#88322	C₄⋊C₄	1	1	yes	characteristic
47	C₈×D₅₃	—	2	2	no	—
48	C₈×D₅₃	—	8	2	no	—
49	C₈×D₅₃	C₂×C₈	1	2	yes	—
50	C₈×D₅₃	M₁₆	1	2	yes	—
51	#11719910	—	2	4	no	—
52	C₂²×D₅₃	#106	1	1	yes	characteristic
53	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
54	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
55	C₂×C₄×C₅₃	#103	1	1	yes	characteristic
56	C₂×Dic₅₃	#102	1	1	yes	characteristic
57	C₂×Dic₅₃	#103	1	1	yes	characteristic
58	C₂×Dic₅₃	#107	1	1	yes	characteristic
59	C₄×D₅₃	—	2	2	no	—
60	C₄×D₅₃	—	2	2	no	—
61	C₄×D₅₃	C₄⋊C₈	1	2	yes	—
62	C₄×D₅₃	#108	1	2	yes	—
63	#88324	—	8	2	no	—
64	C₈×C₅₃	—	2	1	no	—
65	C₈×C₅₃	—	8	1	no	—
66	C₈×C₅₃	C₄×C₈	1	1	yes	characteristic
67	C₈×C₅₃	#97	1	1	yes	characteristic
68	#88322	—	2	1	no	—
69	#88322	—	8	1	no	—
70	#88322	C₂²⋊C₈	1	2	yes	—
71	#7156	—	53	1	no	maximal, Sylow
72	C₂²×C₅₃	#340	1	1	yes	characteristic
73	C₂×D₅₃	#334	1	2	yes	—
74	C₂×D₅₃	#365	1	2	yes	—
75	C₂×D₅₃	#365	1	2	yes	—
76	C₄×C₅₃	—	2	1	no	—
77	C₄×C₅₃	—	2	1	no	—
78	C₄×C₅₃	#336	1	1	yes	characteristic
79	C₄×C₅₃	#341	1	1	yes	characteristic
80	Dic₅₃	—	2	1	no	—
81	Dic₅₃	—	2	1	no	—
82	Dic₅₃	#335	1	2	yes	—
83	C₂×#346	—	53	1	no	—
84	C₂×#334	—	53	1	no	—
85	C₂×#334	—	53	1	no	—
86	C₂×C₅₃	#1277	1	1	yes	characteristic
87	C₂×C₅₃	#1383	1	1	yes	characteristic
88	C₂×C₅₃	#1383	1	1	yes	characteristic
89	D₅₃	#1368	1	4	yes	—
90	C₂²×C₁₆	—	53	2	no	—
91	C₂×C₄⋊C₈	—	106	1	no	—
92	C₂×C₄⋊C₈	—	106	1	no	—
93	C₂×C₄×C₈	—	53	1	no	—
94	#346	—	106	1	no	—
95	#346	—	106	1	no	—
96	#334	—	53	2	no	—
97	#334	—	53	1	no	—
98	#334	—	106	1	no	—
99	#334	—	106	1	no	—
100	#334	#6157	1	1	yes	characteristic
101	C₅₃	#7156	1	1	yes	characteristic, complemented, Sylow
102	C₂²×C₈	—	53	1	no	—
103	C₂²×C₈	—	53	1	no	—
104	C₂²×C₈	—	212	1	no	—
105	C₂²×C₈	—	212	1	no	—
106	C₂×C₄²	—	53	1	no	—
107	C₂×C₁₆	—	53	4	no	—
108	C₂×C₁₆	—	106	2	no	—
109	C₂×C₁₆	—	106	2	no	—
110	C₄⋊C₈	—	2	1	no	—
111	C₄⋊C₈	—	106	2	no	—
112	C₄⋊C₈	—	106	1	no	—
113	C₄⋊C₈	—	212	1	no	—
114	C₄⋊C₈	—	212	1	no	—
115	C₄×C₈	—	53	1	no	—
116	C₄×C₈	—	53	1	no	—
117	C₄×C₈	—	53	1	no	—
118	C₄×C₈	C₂×#6157	1	1	yes	characteristic
119	C₂²×C₄	—	53	1	no	—
120	C₂²×C₄	—	53	1	no	—
121	C₂²×C₄	—	53	1	no	—
122	C₂×C₈	—	4	1	no	—
123	C₂×C₈	—	53	1	no	—
124	C₂×C₈	—	53	2	no	—
125	C₂×C₈	—	53	2	no	—
126	C₂×C₈	—	53	1	no	—
127	C₂×C₈	—	106	2	no	—
128	C₂×C₈	—	424	2	no	—
129	C₂×C₈	—	424	2	no	—
130	C₂×C₈	—	212	2	no	—
131	C₂×C₈	—	212	1	no	—
132	C₂×C₈	C₄×#6157	1	1	yes	characteristic
133	C₂×C₈	C₄⋊#6157	1	1	yes	characteristic
134	C₄²	—	53	1	no	—
135	C₄²	—	53	1	no	—
136	C₄²	—	53	1	no	—
137	C₄²	C₂²⋊#6157	1	1	yes	characteristic
138	C₁₆	—	106	4	no	—
139	C₂³	—	53	1	no	—
140	C₂×C₄	—	53	1	no	—
141	C₂×C₄	—	53	2	no	—
142	C₂×C₄	—	53	2	no	—
143	C₂×C₄	—	53	1	no	—
144	C₂×C₄	—	53	1	no	—
145	C₂×C₄	—	106	2	no	—
146	C₂×C₄	—	106	2	no	—
147	C₂×C₄	(C₂×#6157)⋊C₄	1	1	yes	characteristic
148	C₂×C₄	#13133279	1	1	yes	characteristic
149	C₂×C₄	#13133280	1	1	yes	characteristic
150	C₈	—	2	1	no	—
151	C₈	—	8	1	no	—
152	C₈	—	53	2	no	—
153	C₈	—	106	1	no	—
154	C₈	—	424	1	no	—
155	C₈	—	424	2	no	—
156	C₈	C₈×#6157	1	1	yes	characteristic
157	C₈	#13133271	1	1	yes	characteristic
158	C₂²	—	53	2	no	—
159	C₂²	—	53	2	no	—
160	C₂²	—	53	2	no	—
161	C₂²	#16811496	1	1	yes	characteristic
162	C₄	—	2	1	no	—
163	C₄	—	2	1	no	—
164	C₄	—	53	2	no	—
165	C₄	—	106	1	no	—
166	C₄	—	106	1	no	—
167	C₄	(C₂×#6157)⋊C₈	1	1	yes	characteristic
168	C₄	#16811498	1	1	yes	characteristic
169	C₂	—	53	4	no	—
170	C₂	#20289544	1	1	yes	characteristic
171	C₂	#20289779	1	1	yes	characteristic
172	C₂	#20289781	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20290916

C₂

yes

maximal, characteristic

D₅₃⋊#334

C₂

yes

maximal, characteristic

#334×D₅₃

C₂

yes

maximal, characteristic

C₂×D₅₃⋊C₁₆

C₄

yes

—

D₅₃⋊(C₄⋊C₈)

—

—

—

—

C₄

yes

—

C₄⋊C₈×D₅₃

—

C₄×C₈×D₅₃

C₂²

yes

characteristic

#16810374

C₄

yes

characteristic

#16810373

—

C₅₃×#334

C₄

yes

characteristic

C₂×D₅₃⋊C₈

—

D₅₃⋊C₁₆

—

D₅₃⋊C₁₆

C₈

yes

—

—

—

C₂×C₈×D₅₃

—

C₂×C₈×D₅₃

C₂×C₄

yes

characteristic

C₂×C₈×D₅₃

Q₈

yes

characteristic

C₄²×D₅₃

D₄

yes

—

—

D₄

yes

characteristic

C₈×Dic₅₃

C₂×C₄

yes

characteristic

C₈×Dic₅₃

D₄

yes

characteristic

C₄⋊C₈×C₅₃

—

C₄×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂×C₄×D₅₃

C₄⋊C₄

yes

characteristic

C₂×C₄×D₅₃

SD₁₆

yes

characteristic

C₂×C₄×D₅₃

SD₁₆

yes

characteristic

C₂×#88324

—

D₅₃⋊C₈

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₄²

yes

characteristic

C₂×C₈×C₅₃

C₄⋊C₄

yes

characteristic

C₄²×C₅₃

C₂²⋊C₄

yes

characteristic

C₄×Dic₅₃

C₂²⋊C₄

yes

characteristic

C₄×Dic₅₃

D₈

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

—

C₈×D₅₃

C₂×C₈

yes

—

C₈×D₅₃

M₁₆

yes

—

#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⋊C₈

yes

—

C₄×D₅₃

#108

yes

—

#88324

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

C₄×C₈

yes

C₈×C₅₃

yes

—

—

C₂²⋊C₈

yes

—

—

C₂²×C₅₃

yes

C₂×D₅₃

yes

—

C₂×D₅₃

#365

yes

—

C₂×D₅₃

#365

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

Dic₅₃

—

Dic₅₃

#335

yes

—

C₂×#346

—

C₂×#334

—

C₂×#334

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

D₅₃

yes

—

C₂²×C₁₆

—

C₂×C₄⋊C₈

—

106

—

C₂×C₄⋊C₈

—

106

—

C₂×C₄×C₈

—

#346

—

106

—

#346

—

106

—

#334

—

#334

—

#334

—

106

—

#334

—

106

—

100

#334

#6157

yes

characteristic

101

C₅₃

#7156

yes

characteristic, complemented, Sylow

102

C₂²×C₈

—

103

C₂²×C₈

—

104

C₂²×C₈

—

212

—

105

C₂²×C₈

—

212

—

106

C₂×C₄²

—

107

C₂×C₁₆

—

108

C₂×C₁₆

—

106

—

109

C₂×C₁₆

—

106

—

110

C₄⋊C₈

—

111

C₄⋊C₈

—

106

—

112

C₄⋊C₈

—

106

—

113

C₄⋊C₈

—

212

—

114

C₄⋊C₈

—

212

—

115

C₄×C₈

—

116

C₄×C₈

—

117

C₄×C₈

—

118

C₄×C₈

C₂×#6157

yes

characteristic

119

C₂²×C₄

—

120

C₂²×C₄

—

121

C₂²×C₄

—

122

C₂×C₈

—

123

C₂×C₈

—

124

C₂×C₈

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

—

106

—

128

C₂×C₈

—

424

—

129

C₂×C₈

—

424

—

130

C₂×C₈

—

212

—

131

C₂×C₈

—

212

—

132

C₂×C₈

C₄×#6157

yes

characteristic

133

C₂×C₈

C₄⋊#6157

yes

characteristic

134

C₄²

—

135

C₄²

—

136

C₄²

—

137

C₄²

C₂²⋊#6157

yes

characteristic

138

C₁₆

—

106

—

139

C₂³

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

106

—

146

C₂×C₄

—

106

—

147

C₂×C₄

(C₂×#6157)⋊C₄

yes

characteristic

148

C₂×C₄

#13133279

yes

characteristic

149

C₂×C₄

#13133280

yes

characteristic

150

C₈

—

151

C₈

—

152

C₈

—

153

C₈

—

106

—

154

C₈

—

424

—

155

C₈

—

424

—

156

C₈

C₈×#6157

yes

characteristic

157

C₈

#13133271

yes

characteristic

158

C₂²

—

159

C₂²

—

160

C₂²

—

161

C₂²

#16811496

yes

characteristic

162

C₄

—

163

C₄

—

164

C₄

—

165

C₄

—

106

—

166

C₄

—

106

—

167

C₄

(C₂×#6157)⋊C₈

yes

characteristic

168

C₄

#16811498

yes

characteristic

169

C₂

—

170

C₂

#20289544

yes

characteristic

171

C₂

#20289779

yes

characteristic

172

C₂

#20289781

yes

characteristic

	subgroup	G/H
center	C₂²	#16811496
commutator	C₈×C₅₃	C₄×C₈
Frattini	C₄×C₈	C₂×#6157
Fitting	C₅₃×#334	C₄
socle	C₂²×C₅₃	#340
prime radical	C₂²×D₅₃	#106
cosocle	C₄×C₈×D₅₃	C₂²
hypercenter	#334	#6157
hypocenter	C₅₃	#7156
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #334

subgroup

G/H

center

C₂²

#16811496

commutator

C₈×C₅₃

C₄×C₈

Frattini

C₄×C₈

C₂×#6157

Fitting

C₅₃×#334

C₄

C₂²×C₅₃

C₂²×D₅₃

C₄×C₈×D₅₃

C₂²

C₅₃

G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #334

N	H ≅ G/N	type
17. D₅₃⋊C₁₆	C₈	semidirect
101. C₅₃	#7156	semidirect

H ≅ G/N

type

17. D₅₃⋊C₁₆

C₈

semidirect

101. C₅₃

#7156

semidirect

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

subgroup	index
#20290916	2
D₅₃⋊#334	2
#334×D₅₃	2
#7156	53

subgroup

index

#20290916

D₅₃⋊#334

#334×D₅₃

#7156

Subgroups — #28298313

Conjugacy classes of subgroups

Complements

Maximal subgroups