Group #28298323 — subgroups

Subgroups — #28298323

Order 13568.

210 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 78 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	D₅₃⋊#364	C₂	1	1	yes	maximal, characteristic
2	C₄×D₅₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	#364×D₅₃	C₂	1	1	yes	maximal, characteristic
4	D₅₃⋊M₃₂	—	2	1	no	—
5	C₂×D₅₃⋊C₁₆	C₄	1	1	yes	characteristic
6	C₂×D₅₃⋊C₁₆	C₄	1	1	yes	characteristic
7	#16811546	—	2	1	no	—
8	#16811545	C₄	1	2	yes	—
9	#16811509	—	2	1	no	—
10	C₄×#11719910	—	2	1	no	—
11	M₃₂×D₅₃	—	2	1	no	—
12	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
13	#16810444	C₄	1	1	yes	characteristic
14	#16810443	—	2	1	no	—
15	C₅₃×#364	C₄	1	1	yes	characteristic
16	#13133292	—	4	1	no	—
17	#13133291	—	2	2	no	—
18	D₅₃⋊C₁₆	—	2	1	no	—
19	D₅₃⋊C₁₆	—	2	1	no	—
20	D₅₃⋊C₁₆	—	4	1	no	—
21	D₅₃⋊C₁₆	C₈	1	2	yes	—
22	D₅₃⋊C₁₆	C₈	1	2	yes	—
23	C₂×#11719910	—	2	1	no	—
24	C₂×#11719910	—	2	1	no	—
25	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
26	C₂×C₈×D₅₃	D₄	1	1	yes	characteristic
27	C₄²×D₅₃	Q₈	1	1	yes	characteristic
28	#13133157	—	2	1	no	—
29	#13133156	—	4	1	no	—
30	C₁₆×D₅₃	—	4	1	no	—
31	C₄×#88322	D₄	1	1	yes	characteristic
32	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
33	C₈×Dic₅₃	D₄	1	1	yes	characteristic
34	M₃₂×C₅₃	—	2	1	no	—
35	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄×D₅₃	—	2	1	no	—
37	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
39	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
40	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
41	C₄×Dic₅₃	—	2	1	no	—
42	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
43	C₂×#88322	C₂²⋊C₄	1	1	yes	characteristic
44	C₂×#88322	C₂²⋊C₄	1	1	yes	characteristic
45	C₈×D₅₃	C₂×C₈	1	1	yes	characteristic
46	C₈×D₅₃	C₂×C₈	1	1	yes	characteristic
47	C₈×D₅₃	M₁₆	1	1	yes	characteristic
48	C₈×D₅₃	M₁₆	1	1	yes	characteristic
49	C₈×D₅₃	D₈	1	1	yes	characteristic
50	C₈×D₅₃	SD₁₆	1	1	yes	characteristic
51	C₈×D₅₃	SD₁₆	1	1	yes	characteristic
52	C₈×D₅₃	Q₁₆	1	1	yes	characteristic
53	#11719910	—	2	2	no	—
54	#11719910	—	2	2	no	—
55	#11719910	—	4	2	no	—
56	C₁₆×C₅₃	—	4	1	no	—
57	#11719908	—	4	1	no	—
58	C₂²×D₅₃	#108	1	1	yes	characteristic
59	C₂×C₄×C₅₃	—	2	1	no	—
60	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
61	C₂×Dic₅₃	—	2	1	no	—
62	C₂×Dic₅₃	#108	1	1	yes	characteristic
63	C₄×D₅₃	—	2	4	no	—
64	C₄×D₅₃	C₄⋊C₈	1	1	yes	characteristic
65	C₄×D₅₃	C₄⋊C₈	1	1	yes	characteristic
66	C₄×D₅₃	#106	1	1	yes	characteristic
67	C₄×D₅₃	#107	1	1	yes	characteristic
68	C₈×C₅₃	C₄×C₈	1	1	yes	characteristic
69	C₈×C₅₃	#97	1	1	yes	characteristic
70	C₈×C₅₃	#102	1	1	yes	characteristic
71	C₈×C₅₃	#103	1	1	yes	characteristic
72	#88322	C₂²⋊C₈	1	1	yes	characteristic
73	#88322	C₂²⋊C₈	1	1	yes	characteristic
74	#88322	#102	1	1	yes	characteristic
75	#88322	#103	1	1	yes	characteristic
76	#7160	—	53	1	no	maximal, Sylow
77	C₂²×C₅₃	#341	1	1	yes	characteristic
78	C₂×D₅₃	—	2	2	no	—
79	C₂×D₅₃	#334	1	1	yes	characteristic
80	C₂×D₅₃	#335	1	1	yes	characteristic
81	C₄×C₅₃	—	2	2	no	—
82	C₄×C₅₃	#336	1	1	yes	characteristic
83	C₄×C₅₃	#340	1	1	yes	characteristic
84	Dic₅₃	—	2	2	no	—
85	Dic₅₃	#334	1	1	yes	characteristic
86	Dic₅₃	#335	1	1	yes	characteristic
87	C₂×#364	—	53	1	no	—
88	C₂×#364	—	53	1	no	—
89	C₂×C₄×C₁₆	—	53	1	no	—
90	C₂×C₅₃	—	2	1	no	—
91	C₂×C₅₃	#1277	1	1	yes	characteristic
92	D₅₃	—	2	1	no	—
93	D₅₃	#1373	1	2	yes	—
94	C₂×M₃₂	—	106	1	no	—
95	C₂×M₃₂	—	106	1	no	—
96	C₂²×C₁₆	—	53	1	no	—
97	C₂²×C₁₆	—	53	1	no	—
98	C₂×C₄×C₈	—	53	1	no	—
99	#364	—	53	2	no	—
100	#364	—	53	1	no	—
101	#364	—	106	1	no	—
102	#364	—	106	1	no	—
103	#364	#6157	1	1	yes	characteristic
104	C₄×C₁₆	—	106	1	no	—
105	C₄×C₁₆	—	106	1	no	—
106	C₅₃	#7160	1	1	yes	characteristic, complemented, Sylow
107	C₂²×C₈	—	53	1	no	—
108	C₂²×C₈	—	53	1	no	—
109	C₂×C₄²	—	53	1	no	—
110	M₃₂	—	2	1	no	—
111	M₃₂	—	106	2	no	—
112	M₃₂	—	106	1	no	—
113	M₃₂	—	212	1	no	—
114	M₃₂	—	212	1	no	—
115	C₂×C₁₆	—	53	2	no	—
116	C₂×C₁₆	—	53	2	no	—
117	C₂×C₁₆	—	106	1	no	—
118	C₂×C₁₆	—	106	1	no	—
119	C₂×C₁₆	—	106	1	no	—
120	C₂×C₁₆	—	106	1	no	—
121	C₂×C₁₆	—	212	1	no	—
122	C₂×C₁₆	—	212	1	no	—
123	C₄×C₈	—	53	1	no	—
124	C₄×C₈	—	53	1	no	—
125	C₄×C₈	—	53	1	no	—
126	C₄×C₈	C₂×#6157	1	1	yes	characteristic
127	C₂²×C₄	—	53	1	no	—
128	C₂²×C₄	—	106	1	no	—
129	C₂×C₈	—	53	1	no	—
130	C₂×C₈	—	53	1	no	—
131	C₂×C₈	—	53	1	no	—
132	C₂×C₈	—	53	1	no	—
133	C₂×C₈	—	53	1	no	—
134	C₂×C₈	—	53	1	no	—
135	C₂×C₈	—	53	1	no	—
136	C₂×C₈	—	53	1	no	—
137	C₂×C₈	—	53	1	no	—
138	C₂×C₈	—	53	1	no	—
139	C₂×C₈	C₄×#6157	1	1	yes	characteristic
140	C₂×C₈	C₂²⋊#6157	1	1	yes	characteristic
141	C₄²	—	53	1	no	—
142	C₄²	—	106	1	no	—
143	C₄²	C₄⋊#6157	1	1	yes	characteristic
144	C₁₆	—	4	1	no	—
145	C₁₆	—	106	2	no	—
146	C₁₆	—	106	2	no	—
147	C₁₆	—	212	2	no	—
148	C₁₆	—	212	1	no	—
149	C₂³	—	53	1	no	—
150	C₂×C₄	—	2	1	no	—
151	C₂×C₄	—	53	1	no	—
152	C₂×C₄	—	53	1	no	—
153	C₂×C₄	—	53	1	no	—
154	C₂×C₄	—	53	1	no	—
155	C₂×C₄	—	53	1	no	—
156	C₂×C₄	—	106	4	no	—
157	C₂×C₄	—	106	1	no	—
158	C₂×C₄	(C₂×#6157)⋊C₄	1	1	yes	characteristic
159	C₈	—	53	1	no	—
160	C₈	—	53	1	no	—
161	C₈	—	53	1	no	—
162	C₈	—	53	1	no	—
163	C₈	C₈×#6157	1	1	yes	characteristic
164	C₈	#13133271	1	1	yes	characteristic
165	C₈	#13133279	1	1	yes	characteristic
166	C₈	#13133280	1	1	yes	characteristic
167	C₂²	—	53	1	no	—
168	C₂²	—	53	1	no	—
169	C₂²	—	106	2	no	—
170	C₂²	#16811498	1	1	yes	characteristic
171	C₄	—	2	2	no	—
172	C₄	—	53	1	no	—
173	C₄	—	53	1	no	—
174	C₄	—	106	2	no	—
175	C₄	(C₂×#6157)⋊C₈	1	1	yes	characteristic
176	C₄	#16811496	1	1	yes	characteristic
177	C₂	—	2	1	no	—
178	C₂	—	53	2	no	—
179	C₂	—	106	1	no	—
180	C₂	#20289544	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

D₅₃⋊#364

C₂

yes

maximal, characteristic

C₄×D₅₃⋊C₁₆

C₂

yes

maximal, characteristic

#364×D₅₃

C₂

yes

maximal, characteristic

D₅₃⋊M₃₂

—

C₂×D₅₃⋊C₁₆

C₄

yes

characteristic

C₂×D₅₃⋊C₁₆

C₄

yes

characteristic

#16811546

—

#16811545

C₄

yes

—

#16811509

—

C₄×#11719910

—

M₃₂×D₅₃

—

C₄×C₈×D₅₃

C₂²

yes

characteristic

#16810444

C₄

yes

characteristic

#16810443

—

C₅₃×#364

C₄

yes

characteristic

#13133292

—

#13133291

—

D₅₃⋊C₁₆

—

D₅₃⋊C₁₆

—

D₅₃⋊C₁₆

—

D₅₃⋊C₁₆

C₈

yes

—

D₅₃⋊C₁₆

C₈

yes

—

C₂×#11719910

—

C₂×#11719910

—

C₂×C₈×D₅₃

C₂×C₄

yes

characteristic

C₂×C₈×D₅₃

D₄

yes

characteristic

C₄²×D₅₃

Q₈

yes

characteristic

#13133157

—

#13133156

—

C₁₆×D₅₃

—

C₄×#88322

D₄

yes

characteristic

C₈×Dic₅₃

C₂×C₄

yes

characteristic

C₈×Dic₅₃

D₄

yes

characteristic

M₃₂×C₅₃

—

C₄×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂×C₄×D₅₃

—

C₂×C₄×D₅₃

C₄⋊C₄

yes

characteristic

C₂×C₈×C₅₃

C₄²

yes

characteristic

C₂×C₈×C₅₃

C₂²⋊C₄

yes

characteristic

C₄²×C₅₃

C₄⋊C₄

yes

characteristic

C₄×Dic₅₃

—

C₄×Dic₅₃

C₄⋊C₄

yes

characteristic

C₂×#88322

C₂²⋊C₄

yes

characteristic

C₂×#88322

C₂²⋊C₄

yes

characteristic

C₈×D₅₃

C₂×C₈

yes

characteristic

C₈×D₅₃

C₂×C₈

yes

characteristic

C₈×D₅₃

M₁₆

yes

characteristic

C₈×D₅₃

M₁₆

yes

characteristic

C₈×D₅₃

D₈

yes

characteristic

C₈×D₅₃

SD₁₆

yes

characteristic

C₈×D₅₃

SD₁₆

yes

characteristic

C₈×D₅₃

Q₁₆

yes

—

—

—

C₁₆×C₅₃

—

#11719908

—

C₂²×D₅₃

#108

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

#95

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

#108

yes

characteristic

C₄×D₅₃

—

C₄×D₅₃

C₄⋊C₈

yes

characteristic

C₄×D₅₃

C₄⋊C₈

yes

C₄×D₅₃

yes

C₄×D₅₃

yes

C₈×C₅₃

C₄×C₈

yes

C₈×C₅₃

yes

C₈×C₅₃

yes

C₈×C₅₃

yes

C₂²⋊C₈

yes

characteristic

#88322

C₂²⋊C₈

yes

yes

yes

—

C₂²×C₅₃

yes

C₂×D₅₃

—

C₂×D₅₃

yes

C₂×D₅₃

yes

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

Dic₅₃

yes

Dic₅₃

yes

—

—

C₂×C₄×C₁₆

—

C₂×C₅₃

—

C₂×C₅₃

#1277

yes

characteristic

D₅₃

—

D₅₃

#1373

yes

—

C₂×M₃₂

—

106

—

C₂×M₃₂

—

106

—

C₂²×C₁₆

—

C₂²×C₁₆

—

C₂×C₄×C₈

—

#364

—

100

#364

—

101

#364

—

106

—

102

#364

—

106

—

103

#364

#6157

yes

characteristic

104

C₄×C₁₆

—

106

—

105

C₄×C₁₆

—

106

—

106

C₅₃

#7160

yes

characteristic, complemented, Sylow

107

C₂²×C₈

—

108

C₂²×C₈

—

109

C₂×C₄²

—

110

M₃₂

—

111

M₃₂

—

106

—

112

M₃₂

—

106

—

113

M₃₂

—

212

—

114

M₃₂

—

212

—

115

C₂×C₁₆

—

116

C₂×C₁₆

—

117

C₂×C₁₆

—

106

—

118

C₂×C₁₆

—

106

—

119

C₂×C₁₆

—

106

—

120

C₂×C₁₆

—

106

—

121

C₂×C₁₆

—

212

—

122

C₂×C₁₆

—

212

—

123

C₄×C₈

—

124

C₄×C₈

—

125

C₄×C₈

—

126

C₄×C₈

C₂×#6157

yes

characteristic

127

C₂²×C₄

—

128

C₂²×C₄

—

106

—

129

C₂×C₈

—

130

C₂×C₈

—

131

C₂×C₈

—

132

C₂×C₈

—

133

C₂×C₈

—

134

C₂×C₈

—

135

C₂×C₈

—

136

C₂×C₈

—

137

C₂×C₈

—

138

C₂×C₈

—

139

C₂×C₈

C₄×#6157

yes

characteristic

140

C₂×C₈

C₂²⋊#6157

yes

characteristic

141

C₄²

—

142

C₄²

—

106

—

143

C₄²

C₄⋊#6157

yes

characteristic

144

C₁₆

—

145

C₁₆

—

106

—

146

C₁₆

—

106

—

147

C₁₆

—

212

—

148

C₁₆

—

212

—

149

C₂³

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

106

—

157

C₂×C₄

—

106

—

158

C₂×C₄

(C₂×#6157)⋊C₄

yes

characteristic

159

C₈

—

160

C₈

—

161

C₈

—

162

C₈

—

163

C₈

C₈×#6157

yes

characteristic

164

C₈

#13133271

yes

characteristic

165

C₈

#13133279

yes

characteristic

166

C₈

#13133280

yes

characteristic

167

C₂²

—

168

C₂²

—

169

C₂²

—

106

—

170

C₂²

#16811498

yes

characteristic

171

C₄

—

172

C₄

—

173

C₄

—

174

C₄

—

106

—

175

C₄

(C₂×#6157)⋊C₈

yes

characteristic

176

C₄

#16811496

yes

characteristic

177

C₂

—

178

C₂

—

179

C₂

—

106

—

180

C₂

#20289544

yes

characteristic

	subgroup	G/H
center	C₈	#13133279
commutator	C₈×C₅₃	C₄×C₈
Frattini	C₄×C₈	C₂×#6157
Fitting	C₅₃×#364	C₄
socle	C₂×C₅₃	#1277
prime radical	C₂²×D₅₃	#108
cosocle	C₄×C₈×D₅₃	C₂²
hypercenter	#364	#6157
hypocenter	C₅₃	#7160
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #364

subgroup

G/H

center

C₈

#13133279

commutator

C₈×C₅₃

C₄×C₈

Frattini

C₄×C₈

C₂×#6157

Fitting

C₅₃×#364

C₄

C₂×C₅₃

C₂²×D₅₃

C₄×C₈×D₅₃

C₂²

C₅₃

G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #364

N	H ≅ G/N	type
106. C₅₃	#7160	semidirect

H ≅ G/N

type

106. C₅₃

#7160

semidirect

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

subgroup	index
D₅₃⋊#364	2
C₄×D₅₃⋊C₁₆	2
#364×D₅₃	2
#7160	53

subgroup

index

D₅₃⋊#364

C₄×D₅₃⋊C₁₆

#364×D₅₃

#7160

Subgroups — #28298323

Conjugacy classes of subgroups

Complements

Maximal subgroups