Group #28298601 — subgroups

Subgroups — #28298601

Order 13568.

190 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 190 conjugacy classes of subgroups — too many to draw. Only its 62 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⋊(C₄⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
2	#20290278	C₂	1	1	yes	maximal, characteristic
3	#20273475	C₂	1	1	yes	maximal, characteristic
4	C₂×D₅₃⋊C₁₆	C₄	1	2	yes	—
5	#16811645	—	2	1	no	—
6	#16811544	—	2	1	no	—
7	C₄⋊#11719910	—	2	1	no	—
8	#16811487	—	2	1	no	—
9	#16811485	C₄	1	2	yes	—
10	#16810679	—	2	1	no	—
11	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
12	#16810375	—	2	1	no	—
13	#16810374	C₄	1	1	yes	characteristic
14	C₅₃×#334	C₄	1	1	yes	characteristic
15	D₅₃⋊C₁₆	—	4	2	no	—
16	C₂×#11719910	—	2	2	no	—
17	C₄⋊#88324	—	2	2	no	—
18	#13133274	—	4	1	no	—
19	#13133272	—	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	#13133129	—	4	1	no	—
25	#13133127	—	4	1	no	—
26	C₄×#88322	D₄	1	1	yes	characteristic
27	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
28	C₈×Dic₅₃	D₄	1	1	yes	characteristic
29	C₄⋊C₈×C₅₃	—	2	1	no	—
30	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
32	C₂×C₄×D₅₃	D₈	1	1	yes	characteristic
33	C₂×C₄×D₅₃	Q₁₆	1	1	yes	characteristic
34	C₂×#88324	—	4	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	SD₁₆	1	1	yes	characteristic
41	C₄×Dic₅₃	SD₁₆	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	C₈×D₅₃	—	2	1	no	—
46	C₈×D₅₃	—	2	1	no	—
47	C₈×D₅₃	—	4	1	no	—
48	#11719910	—	4	2	no	—
49	C₂²×D₅₃	#107	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	#106	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	—	4	1	no	—
60	#88324	—	8	2	no	—
61	C₈×C₅₃	—	2	1	no	—
62	C₈×C₅₃	—	2	1	no	—
63	C₈×C₅₃	—	8	1	no	—
64	#88322	—	2	1	no	—
65	#88322	—	8	1	no	—
66	#88322	#100	1	1	yes	characteristic
67	#88322	#101	1	1	yes	characteristic
68	#7242	—	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	#368	1	2	yes	—
73	C₄×C₅₃	—	2	1	no	—
74	C₄×C₅₃	—	2	1	no	—
75	C₄×C₅₃	#338	1	1	yes	characteristic
76	C₄×C₅₃	#343	1	1	yes	characteristic
77	Dic₅₃	—	2	1	no	—
78	Dic₅₃	—	2	1	no	—
79	Dic₅₃	—	2	1	no	—
80	C₂×C₄⋊C₁₆	—	53	1	no	—
81	#1568	—	53	1	no	—
82	#1568	—	53	1	no	—
83	C₂×C₅₃	#1292	1	1	yes	characteristic
84	C₂×C₅₃	#1381	1	1	yes	characteristic
85	C₂×C₅₃	#1384	1	1	yes	characteristic
86	D₅₃	—	2	2	no	—
87	C₂²×C₁₆	—	53	2	no	—
88	#432	—	106	1	no	—
89	#432	—	106	1	no	—
90	C₂×C₄×C₈	—	53	1	no	—
91	C₄⋊C₁₆	—	106	1	no	—
92	C₄⋊C₁₆	—	106	1	no	—
93	#335	—	106	1	no	—
94	#335	—	106	1	no	—
95	#334	—	53	2	no	—
96	#334	—	53	1	no	—
97	#334	#6157	1	1	yes	characteristic
98	C₅₃	#7242	1	1	yes	characteristic, complemented, Sylow
99	C₂²×C₈	—	53	1	no	—
100	C₂²×C₈	—	53	1	no	—
101	C₂×C₄²	—	53	1	no	—
102	C₂×C₁₆	—	106	2	no	—
103	C₂×C₁₆	—	212	2	no	—
104	C₄⋊C₈	—	2	1	no	—
105	C₄⋊C₈	—	106	2	no	—
106	C₄⋊C₈	—	106	1	no	—
107	C₂²⋊C₈	—	212	1	no	—
108	C₂²⋊C₈	—	212	1	no	—
109	#97	—	212	1	no	—
110	#97	—	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	C₂×C₈	—	4	1	no	—
119	C₂×C₈	—	53	1	no	—
120	C₂×C₈	—	53	1	no	—
121	C₂×C₈	—	106	1	no	—
122	C₂×C₈	—	106	1	no	—
123	C₂×C₈	—	212	1	no	—
124	C₂×C₈	—	212	2	no	—
125	C₂×C₈	—	212	1	no	—
126	C₂×C₈	C₄×#6157	1	1	yes	characteristic
127	C₂×C₈	C₄⋊#6157	1	1	yes	characteristic
128	C₄²	—	53	1	no	—
129	C₄²	—	53	1	no	—
130	C₄²	—	53	1	no	—
131	C₄²	C₂²⋊#6157	1	1	yes	characteristic
132	C₁₆	—	212	2	no	—
133	C₂³	—	53	1	no	—
134	C₂×C₄	—	53	1	no	—
135	C₂×C₄	—	53	1	no	—
136	C₂×C₄	—	53	1	no	—
137	C₂×C₄	—	106	2	no	—
138	C₂×C₄	—	106	1	no	—
139	C₂×C₄	—	106	1	no	—
140	C₂×C₄	—	212	1	no	—
141	C₂×C₄	(C₂×#6157)⋊C₄	1	1	yes	characteristic
142	C₂×C₄	#13133279	1	1	yes	characteristic
143	C₂×C₄	#13133280	1	1	yes	characteristic
144	C₈	—	2	1	no	—
145	C₈	—	2	1	no	—
146	C₈	—	8	1	no	—
147	C₈	—	53	1	no	—
148	C₈	—	53	1	no	—
149	C₈	—	106	1	no	—
150	C₈	—	424	1	no	—
151	C₈	—	424	2	no	—
152	C₂²	—	53	2	no	—
153	C₂²	—	106	1	no	—
154	C₂²	—	106	1	no	—
155	C₂²	#16811496	1	1	yes	characteristic
156	C₄	—	2	1	no	—
157	C₄	—	2	1	no	—
158	C₄	—	106	1	no	—
159	C₄	—	106	1	no	—
160	C₄	—	106	1	no	—
161	C₄	#16811492	1	1	yes	characteristic
162	C₄	#16811502	1	1	yes	characteristic
163	C₂	—	106	2	no	—
164	C₂	#20289584	1	1	yes	characteristic
165	C₂	#20289774	1	1	yes	characteristic
166	C₂	#20289782	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

D₅₃⋊(C₄⋊C₁₆)

C₂

yes

maximal, characteristic

#20290278

C₂

yes

maximal, characteristic

#20273475

C₂

yes

maximal, characteristic

C₂×D₅₃⋊C₁₆

C₄

yes

—

—

—

—

—

C₄

yes

—

#16810679

—

C₄×C₈×D₅₃

C₂²

yes

characteristic

#16810375

—

#16810374

C₄

yes

characteristic

C₅₃×#334

C₄

yes

characteristic

D₅₃⋊C₁₆

—

—

—

—

—

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

D₈

yes

characteristic

C₂×C₄×D₅₃

Q₁₆

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

SD₁₆

yes

characteristic

C₄×Dic₅₃

SD₁₆

yes

characteristic

C₂×#88322

—

C₂×#88322

C₂²⋊C₄

yes

characteristic

C₂×#88322

C₄⋊C₄

yes

characteristic

C₈×D₅₃

—

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

—

#88324

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

—

—

yes

yes

—

C₂²×C₅₃

yes

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

#368

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

C₂×C₄⋊C₁₆

—

#1568

—

#1568

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

D₅₃

—

C₂²×C₁₆

—

#432

—

106

—

#432

—

106

—

C₂×C₄×C₈

—

C₄⋊C₁₆

—

106

—

C₄⋊C₁₆

—

106

—

#335

—

106

—

#335

—

106

—

—

—

yes

C₅₃

yes

characteristic, complemented, Sylow

C₂²×C₈

—

100

C₂²×C₈

—

101

C₂×C₄²

—

102

C₂×C₁₆

—

106

—

103

C₂×C₁₆

—

212

—

104

C₄⋊C₈

—

105

C₄⋊C₈

—

106

—

106

C₄⋊C₈

—

106

—

107

C₂²⋊C₈

—

212

—

108

C₂²⋊C₈

—

212

—

109

#97

—

212

—

110

#97

—

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

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

—

106

—

122

C₂×C₈

—

106

—

123

C₂×C₈

—

212

—

124

C₂×C₈

—

212

—

125

C₂×C₈

—

212

—

126

C₂×C₈

C₄×#6157

yes

characteristic

127

C₂×C₈

C₄⋊#6157

yes

characteristic

128

C₄²

—

129

C₄²

—

130

C₄²

—

131

C₄²

C₂²⋊#6157

yes

characteristic

132

C₁₆

—

212

—

133

C₂³

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

106

—

138

C₂×C₄

—

106

—

139

C₂×C₄

—

106

—

140

C₂×C₄

—

212

—

141

C₂×C₄

(C₂×#6157)⋊C₄

yes

characteristic

142

C₂×C₄

#13133279

yes

characteristic

143

C₂×C₄

#13133280

yes

characteristic

144

C₈

—

145

C₈

—

146

C₈

—

147

C₈

—

148

C₈

—

149

C₈

—

106

—

150

C₈

—

424

—

151

C₈

—

424

—

152

C₂²

—

153

C₂²

—

106

—

154

C₂²

—

106

—

155

C₂²

#16811496

yes

characteristic

156

C₄

—

157

C₄

—

158

C₄

—

106

—

159

C₄

—

106

—

160

C₄

—

106

—

161

C₄

#16811492

yes

characteristic

162

C₄

#16811502

yes

characteristic

163

C₂

—

106

—

164

C₂

#20289584

yes

characteristic

165

C₂

#20289774

yes

characteristic

166

C₂

#20289782

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	#107
cosocle	C₄×C₈×D₅₃	C₂²
hypercenter	#334	#6157
hypocenter	C₅₃	#7242
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
98. C₅₃	#7242	semidirect

H ≅ G/N

type

98. C₅₃

#7242

semidirect

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

subgroup	index
D₅₃⋊(C₄⋊C₁₆)	2
#20290278	2
#20273475	2
#7242	53

subgroup

index

D₅₃⋊(C₄⋊C₁₆)

#20290278

#20273475

#7242

Subgroups — #28298601

Conjugacy classes of subgroups

Complements

Maximal subgroups