Group #28298757 — subgroups

Subgroups — #28298757

Order 13568.

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

normal subgroups only

This group has 198 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	#20291078	C₂	1	1	yes	maximal, characteristic
2	D₅₃⋊#335	C₂	1	1	yes	maximal, characteristic
3	#20273469	C₂	1	1	yes	maximal, characteristic
4	D₅₃⋊(C₄⋊C₈)	—	2	1	no	—
5	#16811544	—	2	1	no	—
6	#16811518	C₄	1	2	yes	—
7	#16811511	—	2	1	no	—
8	#16811487	C₄	1	2	yes	—
9	#16811486	—	2	1	no	—
10	#16810645	—	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	C₂×#11719910	—	2	2	no	—
17	#13133284	—	4	1	no	—
18	C₄⋊#88324	—	2	2	no	—
19	C₂×#11719912	—	4	1	no	—
20	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
21	C₂×C₈×D₅₃	Q₈	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	D₄	1	1	yes	characteristic
28	C₄⋊C₈×C₅₃	—	2	1	no	—
29	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
31	C₂×C₄×D₅₃	SD₁₆	1	1	yes	characteristic
32	C₂×C₄×D₅₃	SD₁₆	1	1	yes	characteristic
33	C₂×#88324	—	4	2	no	—
34	D₅₃⋊C₈	—	8	2	no	—
35	C₂×C₈×C₅₃	—	4	1	no	—
36	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
37	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
38	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
39	C₄×Dic₅₃	C₂²⋊C₄	1	1	yes	characteristic
40	C₄×Dic₅₃	D₈	1	1	yes	characteristic
41	C₄×Dic₅₃	Q₁₆	1	1	yes	characteristic
42	C₂×#88322	—	4	1	no	—
43	C₂×#88322	C₂²⋊C₄	1	1	yes	characteristic
44	C₂×#88322	C₄⋊C₄	1	1	yes	characteristic
45	#11719912	—	8	2	no	—
46	C₈×D₅₃	—	2	2	no	—
47	C₈×D₅₃	—	4	1	no	—
48	#11719910	—	4	2	no	—
49	C₂²×D₅₃	#106	1	1	yes	characteristic
50	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
51	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
52	C₂×C₄×C₅₃	#103	1	1	yes	characteristic
53	C₂×Dic₅₃	#102	1	1	yes	characteristic
54	C₂×Dic₅₃	#103	1	1	yes	characteristic
55	C₂×Dic₅₃	#107	1	1	yes	characteristic
56	C₄×D₅₃	—	2	2	no	—
57	C₄×D₅₃	—	2	1	no	—
58	C₄×D₅₃	—	2	1	no	—
59	C₄×D₅₃	—	2	1	no	—
60	C₄×D₅₃	—	2	1	no	—
61	#88324	—	8	2	no	—
62	C₈×C₅₃	—	2	1	no	—
63	C₈×C₅₃	—	2	1	no	—
64	C₈×C₅₃	—	8	1	no	—
65	#88322	—	2	1	no	—
66	#88322	—	8	1	no	—
67	#88322	#108	1	2	yes	—
68	#7292	—	53	1	no	maximal, Sylow
69	C₂²×C₅₃	#340	1	1	yes	characteristic
70	C₂×D₅₃	—	2	1	no	—
71	C₂×D₅₃	—	2	1	no	—
72	C₂×D₅₃	#365	1	2	yes	—
73	C₄×C₅₃	—	2	1	no	—
74	C₄×C₅₃	—	2	1	no	—
75	C₄×C₅₃	#341	1	1	yes	characteristic
76	C₄×C₅₃	#344	1	1	yes	characteristic
77	Dic₅₃	—	2	1	no	—
78	Dic₅₃	—	2	1	no	—
79	Dic₅₃	#361	1	1	yes	characteristic
80	Dic₅₃	#362	1	1	yes	characteristic
81	#2163	—	53	1	no	—
82	#1566	—	53	1	no	—
83	C₂×#335	—	53	1	no	—
84	C₂×C₅₃	#1301	1	1	yes	characteristic
85	C₂×C₅₃	#1383	1	1	yes	characteristic
86	C₂×C₅₃	#1387	1	1	yes	characteristic
87	D₅₃	—	2	2	no	—
88	#423	—	106	1	no	—
89	C₂×C₄⋊C₈	—	106	1	no	—
90	C₂×C₄×C₈	—	53	1	no	—
91	C₄⋊C₁₆	—	106	1	no	—
92	C₂²⋊C₁₆	—	53	2	no	—
93	#346	—	106	1	no	—
94	#335	—	53	2	no	—
95	#335	—	106	1	no	—
96	#334	—	53	1	no	—
97	#334	—	106	1	no	—
98	#334	#6157	1	1	yes	characteristic
99	C₅₃	#7292	1	1	yes	characteristic, complemented, Sylow
100	C₂×M₁₆	—	212	1	no	—
101	C₂²×C₈	—	53	1	no	—
102	C₂²×C₈	—	53	1	no	—
103	C₂²×C₈	—	212	1	no	—
104	C₂×C₄²	—	53	1	no	—
105	C₂×C₁₆	—	106	2	no	—
106	C₄⋊C₈	—	2	1	no	—
107	C₄⋊C₈	—	106	2	no	—
108	C₄⋊C₈	—	106	1	no	—
109	C₄⋊C₈	—	212	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	C₂²×C₄	—	53	1	no	—
116	C₂²×C₄	—	53	1	no	—
117	C₂²×C₄	—	53	1	no	—
118	M₁₆	—	424	2	no	—
119	C₂×C₈	—	4	1	no	—
120	C₂×C₈	—	53	1	no	—
121	C₂×C₈	—	53	1	no	—
122	C₂×C₈	—	106	2	no	—
123	C₂×C₈	—	424	2	no	—
124	C₂×C₈	—	212	1	no	—
125	C₂×C₈	—	212	2	no	—
126	C₂×C₈	—	212	1	no	—
127	C₂×C₈	C₄×#6157	1	1	yes	characteristic
128	C₂×C₈	C₄⋊#6157	1	1	yes	characteristic
129	C₄²	—	53	1	no	—
130	C₄²	—	53	1	no	—
131	C₄²	—	53	1	no	—
132	C₄²	C₂²⋊#6157	1	1	yes	characteristic
133	C₁₆	—	212	2	no	—
134	C₂³	—	53	1	no	—
135	C₂×C₄	—	53	1	no	—
136	C₂×C₄	—	53	1	no	—
137	C₂×C₄	—	53	1	no	—
138	C₂×C₄	—	106	1	no	—
139	C₂×C₄	—	106	1	no	—
140	C₂×C₄	—	106	2	no	—
141	C₂×C₄	—	106	1	no	—
142	C₂×C₄	—	106	1	no	—
143	C₂×C₄	(C₂×#6157)⋊C₄	1	1	yes	characteristic
144	C₂×C₄	#13133279	1	1	yes	characteristic
145	C₂×C₄	#13133280	1	1	yes	characteristic
146	C₈	—	2	1	no	—
147	C₈	—	2	1	no	—
148	C₈	—	8	1	no	—
149	C₈	—	53	2	no	—
150	C₈	—	106	1	no	—
151	C₈	—	424	1	no	—
152	C₈	—	424	2	no	—
153	C₂²	—	53	2	no	—
154	C₂²	—	106	1	no	—
155	C₂²	—	106	1	no	—
156	C₂²	#16811496	1	1	yes	characteristic
157	C₄	—	2	1	no	—
158	C₄	—	2	1	no	—
159	C₄	—	53	1	no	—
160	C₄	—	53	1	no	—
161	C₄	—	106	1	no	—
162	C₄	—	106	1	no	—
163	C₄	#16811498	1	1	yes	characteristic
164	C₄	#16811505	1	1	yes	characteristic
165	C₂	—	106	2	no	—
166	C₂	#20289605	1	1	yes	characteristic
167	C₂	#20289779	1	1	yes	characteristic
168	C₂	#20289791	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20291078

C₂

yes

maximal, characteristic

D₅₃⋊#335

C₂

yes

maximal, characteristic

#20273469

C₂

yes

maximal, characteristic

D₅₃⋊(C₄⋊C₈)

—

#16811544

—

#16811518

C₄

yes

—

#16811511

—

#16811487

C₄

yes

—

#16811486

—

#16810645

—

C₄×C₈×D₅₃

C₂²

yes

characteristic

#16810374

C₄

yes

characteristic

#16810373

—

C₅₃×#334

C₄

yes

characteristic

C₂×D₅₃⋊C₈

—

—

—

—

—

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

#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

—

C₄×D₅₃

—

#88324

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

—

—

yes

—

—

C₂²×C₅₃

yes

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

#365

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

Dic₅₃

—

Dic₅₃

yes

Dic₅₃

yes

—

—

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

D₅₃

—

#423

—

106

—

C₂×C₄⋊C₈

—

106

—

C₂×C₄×C₈

—

C₄⋊C₁₆

—

106

—

C₂²⋊C₁₆

—

#346

—

106

—

#335

—

#335

—

106

—

#334

—

#334

—

106

—

yes

C₅₃

yes

characteristic, complemented, Sylow

100

C₂×M₁₆

—

212

—

101

C₂²×C₈

—

102

C₂²×C₈

—

103

C₂²×C₈

—

212

—

104

C₂×C₄²

—

105

C₂×C₁₆

—

106

—

106

C₄⋊C₈

—

107

C₄⋊C₈

—

106

—

108

C₄⋊C₈

—

106

—

109

C₄⋊C₈

—

212

—

110

C₄⋊C₈

—

212

—

111

C₄×C₈

—

112

C₄×C₈

—

113

C₄×C₈

—

114

C₄×C₈

C₂×#6157

yes

characteristic

115

C₂²×C₄

—

116

C₂²×C₄

—

117

C₂²×C₄

—

118

M₁₆

—

424

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

—

122

C₂×C₈

—

106

—

123

C₂×C₈

—

424

—

124

C₂×C₈

—

212

—

125

C₂×C₈

—

212

—

126

C₂×C₈

—

212

—

127

C₂×C₈

C₄×#6157

yes

characteristic

128

C₂×C₈

C₄⋊#6157

yes

characteristic

129

C₄²

—

130

C₄²

—

131

C₄²

—

132

C₄²

C₂²⋊#6157

yes

characteristic

133

C₁₆

—

212

—

134

C₂³

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

106

—

139

C₂×C₄

—

106

—

140

C₂×C₄

—

106

—

141

C₂×C₄

—

106

—

142

C₂×C₄

—

106

—

143

C₂×C₄

(C₂×#6157)⋊C₄

yes

characteristic

144

C₂×C₄

#13133279

yes

characteristic

145

C₂×C₄

#13133280

yes

characteristic

146

C₈

—

147

C₈

—

148

C₈

—

149

C₈

—

150

C₈

—

106

—

151

C₈

—

424

—

152

C₈

—

424

—

153

C₂²

—

154

C₂²

—

106

—

155

C₂²

—

106

—

156

C₂²

#16811496

yes

characteristic

157

C₄

—

158

C₄

—

159

C₄

—

160

C₄

—

161

C₄

—

106

—

162

C₄

—

106

—

163

C₄

#16811498

yes

characteristic

164

C₄

#16811505

yes

characteristic

165

C₂

—

106

—

166

C₂

#20289605

yes

characteristic

167

C₂

#20289779

yes

characteristic

168

C₂

#20289791

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	#106
cosocle	C₄×C₈×D₅₃	C₂²
hypercenter	#334	#6157
hypocenter	C₅₃	#7292
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₂×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 ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #334

N	H ≅ G/N	type
99. C₅₃	#7292	semidirect

H ≅ G/N

type

99. C₅₃

#7292

semidirect

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

subgroup	index
#20291078	2
D₅₃⋊#335	2
#20273469	2
#7292	53

subgroup

index

#20291078

D₅₃⋊#335

#20273469

#7292

Subgroups — #28298757

Conjugacy classes of subgroups

Complements

Maximal subgroups