Group #28298311 — subgroups

Subgroups — #28298311

Order 13568.

222 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 222 conjugacy classes of subgroups — too many to draw. Only its 86 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	C₄×D₅₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic
2	#20290265	C₂	1	1	yes	maximal, characteristic
3	#20273462	C₂	1	1	yes	maximal, characteristic
4	C₂×D₅₃⋊C₁₆	C₄	1	2	yes	—
5	#16811642	—	2	1	no	—
6	#16811509	—	2	1	no	—
7	C₄×#11719910	—	2	1	no	—
8	#16811486	C₄	1	2	yes	—
9	#16811485	—	2	1	no	—
10	#16810675	—	2	1	no	—
11	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
12	#16810376	C₄	1	1	yes	characteristic
13	#16810373	—	2	1	no	—
14	C₅₃×#335	C₄	1	1	yes	characteristic
15	D₅₃⋊C₁₆	—	2	2	no	—
16	D₅₃⋊C₁₆	C₈	1	4	yes	complemented
17	C₂×#11719910	—	2	2	no	—
18	C₄⋊#88324	—	2	2	no	—
19	#13133274	—	4	1	no	—
20	C₄×#88324	—	4	1	no	—
21	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
22	C₂×C₈×D₅₃	Q₈	1	1	yes	characteristic
23	C₄²×D₅₃	D₄	1	1	yes	characteristic
24	C₄⋊#88322	—	2	1	no	—
25	#13133129	—	4	1	no	—
26	C₄×#88322	D₄	1	1	yes	characteristic
27	C₈×Dic₅₃	—	4	1	no	—
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	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	C₂²⋊C₄	1	1	yes	characteristic
41	C₄×Dic₅₃	D₈	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	C₈×D₅₃	—	2	2	no	—
47	C₈×D₅₃	C₂×C₈	1	2	yes	—
48	C₈×D₅₃	M₁₆	1	2	yes	—
49	#11719910	—	2	4	no	—
50	C₂²×D₅₃	#106	1	1	yes	characteristic
51	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
52	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
53	C₂×C₄×C₅₃	#103	1	1	yes	characteristic
54	C₂×Dic₅₃	#102	1	1	yes	characteristic
55	C₂×Dic₅₃	#103	1	1	yes	characteristic
56	C₂×Dic₅₃	#107	1	1	yes	characteristic
57	C₄×D₅₃	—	2	1	no	—
58	C₄×D₅₃	—	2	1	no	—
59	C₄×D₅₃	—	2	2	no	—
60	C₄×D₅₃	C₄⋊C₈	1	2	yes	—
61	C₄×D₅₃	#108	1	2	yes	—
62	#88324	—	8	2	no	—
63	C₈×C₅₃	—	2	1	no	—
64	C₈×C₅₃	—	8	1	no	—
65	C₈×C₅₃	C₄×C₈	1	1	yes	characteristic
66	C₈×C₅₃	#97	1	1	yes	characteristic
67	#88322	—	2	1	no	—
68	#88322	—	8	1	no	—
69	#88322	C₂²⋊C₈	1	2	yes	—
70	#7155	—	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	#334	1	2	yes	—
75	C₄×C₅₃	—	2	1	no	—
76	C₄×C₅₃	#336	1	1	yes	characteristic
77	C₄×C₅₃	#341	1	1	yes	characteristic
78	C₄×C₅₃	#359	1	1	yes	characteristic
79	C₄×C₅₃	#359	1	1	yes	characteristic
80	Dic₅₃	—	2	1	no	—
81	Dic₅₃	#335	1	2	yes	—
82	Dic₅₃	#359	1	2	yes	—
83	C₂×C₄×C₁₆	—	53	1	no	—
84	#1564	—	53	1	no	—
85	#1564	—	53	1	no	—
86	C₂×C₅₃	#1277	1	1	yes	characteristic
87	C₂×C₅₃	#1382	1	1	yes	characteristic
88	C₂×C₅₃	#1382	1	1	yes	characteristic
89	D₅₃	—	2	2	no	—
90	C₂²×C₁₆	—	53	2	no	—
91	#431	—	106	1	no	—
92	#431	—	106	1	no	—
93	C₂×C₄×C₈	—	53	1	no	—
94	C₄×C₁₆	—	106	1	no	—
95	C₄×C₁₆	—	106	1	no	—
96	#335	—	53	2	no	—
97	#335	—	53	1	no	—
98	#335	#6157	1	1	yes	characteristic
99	#334	—	106	1	no	—
100	#334	—	106	1	no	—
101	C₅₃	#7155	1	1	yes	characteristic, complemented, Sylow
102	C₂²×C₈	—	53	1	no	—
103	C₂²×C₈	—	53	1	no	—
104	C₂×C₄²	—	53	1	no	—
105	C₂×C₁₆	—	53	4	no	—
106	C₂×C₁₆	—	106	2	no	—
107	C₂×C₁₆	—	106	2	no	—
108	C₄⋊C₈	—	2	1	no	—
109	C₄⋊C₈	—	106	2	no	—
110	C₄⋊C₈	—	106	1	no	—
111	C₂²⋊C₈	—	212	1	no	—
112	C₂²⋊C₈	—	212	1	no	—
113	C₄×C₈	—	53	1	no	—
114	C₄×C₈	—	53	1	no	—
115	C₄×C₈	—	53	1	no	—
116	C₄×C₈	—	212	1	no	—
117	C₄×C₈	—	212	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	—	212	2	no	—
129	C₂×C₈	—	212	1	no	—
130	C₂×C₈	C₄×#6157	1	1	yes	characteristic
131	C₂×C₈	C₄⋊#6157	1	1	yes	characteristic
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	—	106	4	no	—
137	C₂³	—	53	1	no	—
138	C₂×C₄	—	53	1	no	—
139	C₂×C₄	—	53	1	no	—
140	C₂×C₄	—	53	2	no	—
141	C₂×C₄	—	53	2	no	—
142	C₂×C₄	—	53	1	no	—
143	C₂×C₄	—	106	2	no	—
144	C₂×C₄	—	106	1	no	—
145	C₂×C₄	—	106	1	no	—
146	C₂×C₄	(C₂×#6157)⋊C₄	1	1	yes	characteristic
147	C₂×C₄	#13133279	1	1	yes	characteristic
148	C₂×C₄	#13133280	1	1	yes	characteristic
149	C₈	—	2	1	no	—
150	C₈	—	8	1	no	—
151	C₈	—	53	2	no	—
152	C₈	—	106	1	no	—
153	C₈	—	424	2	no	—
154	C₈	—	424	1	no	—
155	C₈	C₈×#6157	1	1	yes	characteristic
156	C₈	#13133271	1	1	yes	characteristic
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	—	53	2	no	—
163	C₄	—	53	2	no	—
164	C₄	—	106	1	no	—
165	C₄	(C₂×#6157)⋊C₈	1	1	yes	characteristic
166	C₄	#16811498	1	1	yes	characteristic
167	C₄	#16811537	1	1	yes	characteristic
168	C₄	#16811538	1	1	yes	characteristic
169	C₂	—	106	2	no	—
170	C₂	#20289544	1	1	yes	characteristic
171	C₂	#20289776	1	1	yes	characteristic
172	C₂	#20289778	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×D₅₃⋊C₁₆

C₂

yes

maximal, characteristic

#20290265

C₂

yes

maximal, characteristic

#20273462

C₂

yes

maximal, characteristic

C₂×D₅₃⋊C₁₆

C₄

yes

—

—

—

—

C₄

yes

—

#16811485

—

#16810675

—

C₄×C₈×D₅₃

C₂²

yes

characteristic

#16810376

C₄

yes

characteristic

#16810373

—

C₅₃×#335

C₄

yes

characteristic

D₅₃⋊C₁₆

—

D₅₃⋊C₁₆

C₈

yes

—

—

—

—

C₂×C₈×D₅₃

C₂×C₄

yes

characteristic

C₂×C₈×D₅₃

Q₈

yes

characteristic

C₄²×D₅₃

D₄

yes

—

—

D₄

yes

characteristic

C₈×Dic₅₃

—

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

—

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

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×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

—

C₂×D₅₃

—

C₂×D₅₃

#334

yes

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

Dic₅₃

#335

yes

—

Dic₅₃

#359

yes

—

C₂×C₄×C₁₆

—

#1564

—

#1564

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

D₅₃

—

C₂²×C₁₆

—

#431

—

106

—

#431

—

106

—

C₂×C₄×C₈

—

C₄×C₁₆

—

106

—

C₄×C₁₆

—

106

—

—

—

yes

—

106

—

100

#334

—

106

—

101

C₅₃

#7155

yes

characteristic, complemented, Sylow

102

C₂²×C₈

—

103

C₂²×C₈

—

104

C₂×C₄²

—

105

C₂×C₁₆

—

106

C₂×C₁₆

—

106

—

107

C₂×C₁₆

—

106

—

108

C₄⋊C₈

—

109

C₄⋊C₈

—

106

—

110

C₄⋊C₈

—

106

—

111

C₂²⋊C₈

—

212

—

112

C₂²⋊C₈

—

212

—

113

C₄×C₈

—

114

C₄×C₈

—

115

C₄×C₈

—

116

C₄×C₈

—

212

—

117

C₄×C₈

—

212

—

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

—

212

—

129

C₂×C₈

—

212

—

130

C₂×C₈

C₄×#6157

yes

characteristic

131

C₂×C₈

C₄⋊#6157

yes

characteristic

132

C₄²

—

133

C₄²

—

134

C₄²

—

135

C₄²

C₂²⋊#6157

yes

characteristic

136

C₁₆

—

106

—

137

C₂³

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

106

—

144

C₂×C₄

—

106

—

145

C₂×C₄

—

106

—

146

C₂×C₄

(C₂×#6157)⋊C₄

yes

characteristic

147

C₂×C₄

#13133279

yes

characteristic

148

C₂×C₄

#13133280

yes

characteristic

149

C₈

—

150

C₈

—

151

C₈

—

152

C₈

—

106

—

153

C₈

—

424

—

154

C₈

—

424

—

155

C₈

C₈×#6157

yes

characteristic

156

C₈

#13133271

yes

characteristic

157

C₂²

—

158

C₂²

—

106

—

159

C₂²

—

106

—

160

C₂²

#16811496

yes

characteristic

161

C₄

—

162

C₄

—

163

C₄

—

164

C₄

—

106

—

165

C₄

(C₂×#6157)⋊C₈

yes

characteristic

166

C₄

#16811498

yes

characteristic

167

C₄

#16811537

yes

characteristic

168

C₄

#16811538

yes

characteristic

169

C₂

—

106

—

170

C₂

#20289544

yes

characteristic

171

C₂

#20289776

yes

characteristic

172

C₂

#20289778

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#13133279
commutator	C₈×C₅₃	C₄×C₈
Frattini	C₄×C₈	C₂×#6157
Fitting	C₅₃×#335	C₄
socle	C₂²×C₅₃	#340
prime radical	C₂²×D₅₃	#106
cosocle	C₄×C₈×D₅₃	C₂²
hypercenter	#335	#6157
hypocenter	C₅₃	#7155
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ #335

subgroup

G/H

center

C₂×C₄

#13133279

commutator

C₈×C₅₃

C₄×C₈

Frattini

C₄×C₈

C₂×#6157

Fitting

C₅₃×#335

C₄

C₂²×C₅₃

C₂²×D₅₃

C₄×C₈×D₅₃

C₂²

C₅₃

G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ #335

N	H ≅ G/N	type
16. D₅₃⋊C₁₆	C₈	semidirect
101. C₅₃	#7155	semidirect

H ≅ G/N

type

16. D₅₃⋊C₁₆

C₈

semidirect

101. C₅₃

#7155

semidirect

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

subgroup	index
C₄×D₅₃⋊C₁₆	2
#20290265	2
#20273462	2
#7155	53

subgroup

index

C₄×D₅₃⋊C₁₆

#20290265

#20273462

#7155

Subgroups — #28298311

Conjugacy classes of subgroups

Complements

Maximal subgroups