Group #28298539 — subgroups

Subgroups — #28298539

Order 13568.

206 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 60 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	#20290637	C₂	1	1	yes	maximal, characteristic
3	#20273469	C₂	1	1	yes	maximal, characteristic
4	#16811567	—	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	#16810645	—	2	1	no	—
11	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
12	#16810374	C₄	1	1	yes	characteristic, complemented
13	#16810373	—	2	1	no	—
14	C₅₃×#334	C₄	1	1	yes	characteristic, complemented
15	C₂×C₄⋊#6157	—	2	2	no	—
16	C₂×#11719910	—	2	2	no	—
17	(C₂×#6157)⋊C₄	—	4	1	no	—
18	(C₂×#6157)⋊C₄	—	4	1	no	—
19	C₂×#11719912	—	4	1	no	—
20	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
21	C₂×C₈×D₅₃	D₄	1	1	yes	characteristic
22	C₄²×D₅₃	D₄	1	1	yes	characteristic
23	C₄⋊#88322	—	2	1	no	—
24	#13133146	—	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	C₂×C₈×C₅₃	—	4	1	no	—
37	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
38	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
39	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
40	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
41	C₄×Dic₅₃	SD₁₆	1	1	yes	characteristic
42	C₄×Dic₅₃	Q₁₆	1	1	yes	characteristic
43	C₂×#88322	—	4	1	no	—
44	C₂×#88322	C₂²⋊C₄	1	1	yes	characteristic
45	C₂×#88322	C₄⋊C₄	1	1	yes	characteristic
46	#11719912	—	8	2	no	—
47	C₈×D₅₃	—	4	1	no	—
48	C₈×D₅₃	—	4	1	no	—
49	#11719910	—	4	2	no	—
50	C₂²×D₅₃	#102	1	1	yes	characteristic
51	C₂×#6157	—	8	2	no	—
52	C₂×#6157	—	8	2	no	—
53	C₂×#6157	—	8	2	no	—
54	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
55	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
56	C₂×C₄×C₅₃	#103	1	1	yes	characteristic
57	C₂×Dic₅₃	#103	1	1	yes	characteristic
58	C₂×Dic₅₃	#106	1	1	yes	characteristic
59	C₂×Dic₅₃	#107	1	1	yes	characteristic
60	C₄×D₅₃	—	2	2	no	—
61	C₄×D₅₃	—	2	1	no	—
62	C₄×D₅₃	—	2	1	no	—
63	C₄×D₅₃	—	4	1	no	—
64	C₈×C₅₃	—	2	1	no	—
65	C₈×C₅₃	—	2	1	no	—
66	C₈×C₅₃	—	8	1	no	—
67	#88322	—	2	1	no	—
68	#88322	—	2	1	no	—
69	#88322	—	8	1	no	—
70	#7221	—	53	1	no	maximal, Sylow
71	C₂²×C₅₃	#340	1	1	yes	characteristic
72	C₂×D₅₃	—	2	1	no	—
73	C₂×D₅₃	—	2	1	no	—
74	C₂×D₅₃	#360	1	2	yes	—
75	#6157	—	16	2	no	—
76	C₄×C₅₃	—	2	1	no	—
77	C₄×C₅₃	—	2	1	no	—
78	C₄×C₅₃	H(3,ℤ/4)	1	1	yes	characteristic
79	C₄×C₅₃	#344	1	1	yes	characteristic
80	Dic₅₃	—	2	1	no	—
81	Dic₅₃	—	2	1	no	—
82	Dic₅₃	—	2	1	no	—
83	#2163	—	53	1	no	—
84	#1835	—	53	1	no	—
85	#1566	—	53	1	no	—
86	C₂×C₅₃	#1286	1	1	yes	characteristic
87	C₂×C₅₃	#1386	1	1	yes	characteristic
88	C₂×C₅₃	#1389	1	1	yes	characteristic
89	D₅₃	—	2	2	no	—
90	#423	—	106	1	no	—
91	C₂×C₄×C₈	—	53	1	no	—
92	#382	—	106	1	no	—
93	C₄⋊C₁₆	—	106	1	no	—
94	C₂²⋊C₁₆	—	106	1	no	—
95	#346	—	53	2	no	—
96	#340	—	53	2	no	—
97	#340	—	106	1	no	—
98	#334	—	53	1	no	—
99	#334	—	106	1	no	—
100	#334	#6157	1	1	yes	characteristic, complemented
101	C₅₃	#7221	1	1	yes	characteristic, complemented, Sylow
102	C₂×M₁₆	—	212	1	no	—
103	C₂²×C₈	—	53	1	no	—
104	C₂²×C₈	—	53	1	no	—
105	C₂×C₄⋊C₄	—	106	2	no	—
106	C₂×C₄²	—	53	1	no	—
107	C₂×C₁₆	—	106	2	no	—
108	C₄⋊C₈	—	2	1	no	—
109	C₄⋊C₈	—	106	1	no	—
110	C₄⋊C₈	—	212	1	no	—
111	C₄×C₈	—	53	1	no	—
112	C₄×C₈	—	53	1	no	—
113	C₄×C₈	—	53	1	no	—
114	C₄×C₈	C₂×#6157	1	1	yes	characteristic
115	#95	—	212	1	no	—
116	#95	—	212	1	no	—
117	C₂²×C₄	—	53	1	no	—
118	C₂²×C₄	—	53	1	no	—
119	C₂²×C₄	—	53	1	no	—
120	C₂²×C₄	—	212	2	no	—
121	M₁₆	—	424	2	no	—
122	C₂×C₈	—	4	1	no	—
123	C₂×C₈	—	53	1	no	—
124	C₂×C₈	—	53	1	no	—
125	C₂×C₈	—	212	1	no	—
126	C₂×C₈	—	212	1	no	—
127	C₂×C₈	—	212	1	no	—
128	C₂×C₈	C₄×#6157	1	1	yes	characteristic
129	C₂×C₈	C₄⋊#6157	1	1	yes	characteristic
130	C₄⋊C₄	—	212	2	no	—
131	C₄⋊C₄	—	212	2	no	—
132	C₄²	—	53	1	no	—
133	C₄²	—	53	1	no	—
134	C₄²	—	53	1	no	—
135	C₄²	C₂²⋊#6157	1	1	yes	characteristic
136	C₁₆	—	212	2	no	—
137	C₂³	—	53	1	no	—
138	C₂×C₄	—	53	1	no	—
139	C₂×C₄	—	53	1	no	—
140	C₂×C₄	—	53	1	no	—
141	C₂×C₄	—	106	2	no	—
142	C₂×C₄	—	106	1	no	—
143	C₂×C₄	—	106	1	no	—
144	C₂×C₄	—	424	2	no	—
145	C₂×C₄	—	424	2	no	—
146	C₂×C₄	—	424	2	no	—
147	C₂×C₄	—	212	1	no	—
148	C₂×C₄	(C₂×#6157)⋊C₄	1	1	yes	characteristic
149	C₂×C₄	#13133279	1	1	yes	characteristic
150	C₂×C₄	#13133280	1	1	yes	characteristic
151	C₈	—	2	1	no	—
152	C₈	—	2	1	no	—
153	C₈	—	8	1	no	—
154	C₈	—	106	1	no	—
155	C₈	—	106	1	no	—
156	C₈	—	424	1	no	—
157	C₂²	—	53	2	no	—
158	C₂²	—	106	1	no	—
159	C₂²	—	106	1	no	—
160	C₂²	#16811496	1	1	yes	characteristic
161	C₄	—	2	1	no	—
162	C₄	—	2	1	no	—
163	C₄	—	848	2	no	—
164	C₄	—	106	1	no	—
165	C₄	—	106	1	no	—
166	C₄	—	106	1	no	—
167	C₄	#16811491	1	1	yes	characteristic
168	C₄	#16811506	1	1	yes	characteristic
169	C₂	—	106	2	no	—
170	C₂	#20289570	1	1	yes	characteristic
171	C₂	#20289788	1	1	yes	characteristic
172	C₂	#20289795	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20291079

C₂

yes

maximal, characteristic

#20290637

C₂

yes

maximal, characteristic

#20273469

C₂

yes

maximal, characteristic

—

—

—

C₄

yes

complemented

#16811497

—

#16811497

C₄

yes

—

#16810645

—

C₄×C₈×D₅₃

C₂²

yes

characteristic

#16810374

C₄

yes

characteristic, complemented

#16810373

—

C₅₃×#334

C₄

yes

characteristic, complemented

C₂×C₄⋊#6157

—

—

—

—

—

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

characteristic

C₄⋊#6157

—

C₄⋊#6157

—

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

#11719912

—

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

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

—

—

—

—

C₂²×C₅₃

yes

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

#360

yes

—

#6157

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

#2163

—

#1835

—

#1566

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

D₅₃

—

#423

—

106

—

C₂×C₄×C₈

—

#382

—

106

—

C₄⋊C₁₆

—

106

—

C₂²⋊C₁₆

—

106

—

#346

—

#340

—

#340

—

106

—

#334

—

#334

—

106

—

100

#334

#6157

yes

characteristic, complemented

101

C₅₃

#7221

yes

characteristic, complemented, Sylow

102

C₂×M₁₆

—

212

—

103

C₂²×C₈

—

104

C₂²×C₈

—

105

C₂×C₄⋊C₄

—

106

—

106

C₂×C₄²

—

107

C₂×C₁₆

—

106

—

108

C₄⋊C₈

—

109

C₄⋊C₈

—

106

—

110

C₄⋊C₈

—

212

—

111

C₄×C₈

—

112

C₄×C₈

—

113

C₄×C₈

—

114

C₄×C₈

C₂×#6157

yes

characteristic

115

#95

—

212

—

116

#95

—

212

—

117

C₂²×C₄

—

118

C₂²×C₄

—

119

C₂²×C₄

—

120

C₂²×C₄

—

212

—

121

M₁₆

—

424

—

122

C₂×C₈

—

123

C₂×C₈

—

124

C₂×C₈

—

125

C₂×C₈

—

212

—

126

C₂×C₈

—

212

—

127

C₂×C₈

—

212

—

128

C₂×C₈

C₄×#6157

yes

characteristic

129

C₂×C₈

C₄⋊#6157

yes

characteristic

130

C₄⋊C₄

—

212

—

131

C₄⋊C₄

—

212

—

132

C₄²

—

133

C₄²

—

134

C₄²

—

135

C₄²

C₂²⋊#6157

yes

characteristic

136

C₁₆

—

212

—

137

C₂³

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

106

—

142

C₂×C₄

—

106

—

143

C₂×C₄

—

106

—

144

C₂×C₄

—

424

—

145

C₂×C₄

—

424

—

146

C₂×C₄

—

424

—

147

C₂×C₄

—

212

—

148

C₂×C₄

(C₂×#6157)⋊C₄

yes

characteristic

149

C₂×C₄

#13133279

yes

characteristic

150

C₂×C₄

#13133280

yes

characteristic

151

C₈

—

152

C₈

—

153

C₈

—

154

C₈

—

106

—

155

C₈

—

106

—

156

C₈

—

424

—

157

C₂²

—

158

C₂²

—

106

—

159

C₂²

—

106

—

160

C₂²

#16811496

yes

characteristic

161

C₄

—

162

C₄

—

163

C₄

—

848

—

164

C₄

—

106

—

165

C₄

—

106

—

166

C₄

—

106

—

167

C₄

#16811491

yes

characteristic

168

C₄

#16811506

yes

characteristic

169

C₂

—

106

—

170

C₂

#20289570

yes

characteristic

171

C₂

#20289788

yes

characteristic

172

C₂

#20289795

yes

characteristic

	subgroup	G/H
center	C₂²	#16811496
commutator	C₂×C₈×C₅₃	C₄²
Frattini	C₄×C₈	C₂×#6157
Fitting	C₅₃×#334	C₄
socle	C₂²×C₅₃	#340
prime radical	C₂²×D₅₃	#102
cosocle	C₄×C₈×D₅₃	C₂²
hypercenter	#334	#6157
hypocenter	C₅₃	#7221
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #334

subgroup

G/H

center

C₂²

#16811496

commutator

C₂×C₈×C₅₃

C₄²

Frattini

C₄×C₈

C₂×#6157

Fitting

C₅₃×#334

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 ◁ #334

N	H ≅ G/N	type
7. #16811512	C₄	semidirect
12. #16810374	C₄	semidirect
14. C₅₃×#334	C₄	semidirect
100. #334	#6157	semidirect
101. C₅₃	#7221	semidirect

H ≅ G/N

type

7. #16811512

C₄

semidirect

12. #16810374

C₄

semidirect

14. C₅₃×#334

C₄

semidirect

100. #334

#6157

semidirect

101. C₅₃

#7221

semidirect

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

subgroup	index
#20291079	2
#20290637	2
#20273469	2
#7221	53

subgroup

index

Subgroups — #28298539

Conjugacy classes of subgroups

Complements

Maximal subgroups