Group #27275263 — subgroups

Subgroups — #27275263

Order 13568.

366 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 155 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×#13133127	C₂	1	1	yes	maximal, characteristic
2	#20273461	C₂	1	2	yes	maximal
3	#20273007	C₂	1	1	yes	maximal, characteristic
4	#20272547	C₂	1	2	yes	maximal
5	C₅₃×#1452	C₂	1	1	yes	maximal, characteristic
6	#16810676	C₂²	1	2	yes	—
7	C₂×#13133127	C₄	1	2	yes	—
8	C₂×#13133127	C₄	1	2	yes	—
9	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
10	C₄²×Dic₅₃	C₂²	1	1	yes	characteristic
11	#16810375	C₄	1	4	yes	complemented
12	#16810373	C₄	1	4	yes	complemented
13	#16810345	—	2	2	no	—
14	C₅₃×#431	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₅₃×#322	—	2	2	no	—
17	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
18	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×Dic₅₃	—	2	2	no	—
20	C₂×C₄×Dic₅₃	C₂×C₄	1	2	yes	—
21	C₄⋊#88322	C₂×C₄	1	4	yes	—
22	C₂²⋊#88322	—	2	2	no	—
23	#13133127	—	2	4	no	—
24	#13133127	—	2	4	no	—
25	C₄×#88322	—	2	2	no	—
26	C₄×#88322	C₂×C₄	1	2	yes	—
27	C₄×#88322	C₂×C₄	1	2	yes	—
28	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
29	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
30	C₄⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
31	C₂²⋊C₈×C₅₃	—	2	2	no	—
32	C₄×C₈×C₅₃	—	2	2	no	—
33	C₄×C₈×C₅₃	D₄	1	2	yes	—
34	C₄×C₈×C₅₃	Q₈	1	2	yes	—
35	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
36	C₂²×C₄×C₅₃	#41	1	2	yes	—
37	C₂²×Dic₅₃	—	2	2	no	—
38	C₂×C₈×C₅₃	—	2	2	no	—
39	C₂×C₈×C₅₃	—	2	2	no	—
40	C₂×C₈×C₅₃	—	2	4	no	—
41	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	—
42	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
43	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
44	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
45	C₄×Dic₅₃	—	2	2	no	—
46	C₄×Dic₅₃	—	4	4	no	—
47	C₄×Dic₅₃	M₁₆	1	4	yes	—
48	C₂×#88322	—	2	2	no	—
49	C₂×#88322	—	2	4	no	—
50	C₂×#88322	C₄²	1	1	yes	characteristic
51	C₂×#88322	C₄²	1	1	yes	characteristic
52	C₂×#88322	C₄²	1	2	yes	—
53	C₂×#88322	M₁₆	1	4	yes	—
54	C₂³×C₅₃	#117	1	1	yes	characteristic
55	C₂×C₄×C₅₃	—	2	4	no	—
56	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
57	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
58	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
59	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
60	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
61	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
62	C₂×Dic₅₃	—	2	4	no	—
63	C₂×Dic₅₃	—	4	4	no	—
64	C₈×C₅₃	—	2	4	no	—
65	C₈×C₅₃	—	4	4	no	—
66	#88322	—	2	1	no	—
67	#88322	—	2	1	no	—
68	#88322	—	4	4	no	—
69	#88322	#97	1	4	yes	—
70	#8269	—	53	1	no	maximal, Sylow
71	C₂²×C₅₃	—	2	2	no	—
72	C₂²×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
73	C₂²×C₅₃	#432	1	2	yes	—
74	C₄×C₅₃	—	2	2	no	—
75	C₄×C₅₃	C₂×#97	1	1	yes	characteristic
76	C₄×C₅₃	C₄×M₁₆	1	1	yes	characteristic
77	C₄×C₅₃	#423	1	2	yes	—
78	C₄×C₅₃	#444	1	4	yes	—
79	Dic₅₃	—	4	4	no	—
80	C₄×#97	—	53	1	no	—
81	#1564	—	53	2	no	—
82	#1452	—	53	1	no	—
83	#1452	D₅₃	1	1	yes	characteristic
84	#1276	—	53	2	no	—
85	C₂×C₅₃	—	2	2	no	—
86	C₂×C₅₃	#1767	1	1	yes	characteristic
87	C₂×C₅₃	#1773	1	2	yes	—
88	#431	—	53	2	no	—
89	#431	C₂×D₅₃	1	2	yes	—
90	C₂×#97	—	53	2	no	—
91	C₂×#97	—	53	2	no	—
92	C₂×C₄×C₈	—	53	1	no	—
93	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
94	C₄³	—	53	1	no	—
95	#335	—	53	4	no	—
96	#334	—	53	4	no	—
97	#322	—	2	2	no	—
98	#322	—	106	2	no	—
99	C₅₃	#8269	1	1	yes	characteristic, complemented, Sylow
100	C₂²×C₈	—	53	1	no	—
101	C₂²×C₈	—	53	1	no	—
102	C₂²×C₈	C₄×D₅₃	1	2	yes	—
103	C₂×C₄²	—	53	2	no	—
104	C₂×C₄²	—	106	2	no	—
105	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
106	C₄⋊C₈	—	53	4	no	—
107	C₄⋊C₈	C₄×D₅₃	1	4	yes	—
108	C₂²⋊C₈	—	2	2	no	—
109	C₂²⋊C₈	—	106	2	no	—
110	#97	—	106	4	no	—
111	#97	—	106	4	no	—
112	C₄×C₈	—	2	2	no	—
113	C₄×C₈	—	53	2	no	—
114	C₄×C₈	—	53	2	no	—
115	C₄×C₈	—	106	2	no	—
116	C₄×C₈	Dic₁₀₆	1	2	yes	—
117	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
118	C₂²×C₄	—	106	2	no	—
119	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
120	C₂²×C₄	#11719945	1	2	yes	—
121	C₂×C₈	—	2	2	no	—
122	C₂×C₈	—	2	2	no	—
123	C₂×C₈	—	2	4	no	—
124	C₂×C₈	—	53	2	no	—
125	C₂×C₈	—	53	4	no	—
126	C₂×C₈	—	53	1	no	—
127	C₂×C₈	—	53	1	no	—
128	C₂×C₈	—	106	2	no	—
129	C₂×C₈	—	106	4	no	—
130	C₂×C₈	#11719919	1	4	yes	—
131	C₄²	—	53	4	no	—
132	C₄²	—	106	2	no	—
133	C₄²	—	212	4	no	—
134	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
135	C₄²	C₂×C₄×D₅₃	1	2	yes	—
136	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
137	C₂³	#13133174	1	1	yes	characteristic
138	C₂×C₄	—	2	4	no	—
139	C₂×C₄	—	106	4	no	—
140	C₂×C₄	—	212	4	no	—
141	C₂×C₄	C₄²×D₅₃	1	1	yes	characteristic
142	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
143	C₂×C₄	C₄×C₂²⋊D₅₃	1	2	yes	—
144	C₂×C₄	C₄×Dic₁₀₆	1	2	yes	—
145	C₂×C₄	M₁₆×D₅₃	1	2	yes	—
146	C₂×C₄	M₁₆×D₅₃	1	2	yes	—
147	C₈	—	2	4	no	—
148	C₈	—	4	4	no	—
149	C₈	—	53	4	no	—
150	C₈	—	106	1	no	—
151	C₈	—	106	1	no	—
152	C₈	—	212	4	no	—
153	C₂²	—	2	2	no	—
154	C₂²	C₄×#11719919	1	1	yes	characteristic
155	C₂²	#16810678	1	2	yes	—
156	C₄	—	2	2	no	—
157	C₄	—	212	4	no	—
158	C₄	#97×D₅₃	1	1	yes	characteristic
159	C₄	#16810575	1	1	yes	characteristic
160	C₄	#16810646	1	2	yes	—
161	C₄	#16810743	1	4	yes	—
162	C₂	—	2	2	no	—
163	C₂	#20274569	1	1	yes	characteristic
164	C₂	#20274600	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#13133127

C₂

yes

maximal, characteristic

#20273461

C₂

yes

maximal

#20273007

C₂

yes

maximal, characteristic

#20272547

C₂

yes

maximal

C₅₃×#1452

C₂

yes

maximal, characteristic

#16810676

C₂²

yes

—

C₂×#13133127

C₄

yes

—

C₂×#13133127

C₄

yes

—

C₂×C₄×#88322

C₂²

yes

characteristic

C₄²×Dic₅₃

C₂²

yes

characteristic

#16810375

C₄

yes

complemented

#16810373

C₄

yes

complemented

#16810345

—

C₅₃×#431

C₂²

yes

—

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₅₃×#322

—

C₂²×#88322

C₂×C₄

yes

characteristic

C₂²×#88322

C₂×C₄

yes

characteristic

C₂×C₄×Dic₅₃

—

C₂×C₄×Dic₅₃

C₂×C₄

yes

—

C₄⋊#88322

C₂×C₄

yes

—

—

—

—

—

C₂×C₄

yes

—

C₄×#88322

C₂×C₄

yes

—

C₂²×C₈×C₅₃

C₂×C₄

yes

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

C₄⋊C₈×C₅₃

C₂×C₄

yes

—

C₂²⋊C₈×C₅₃

—

C₄×C₈×C₅₃

—

C₄×C₈×C₅₃

D₄

yes

—

C₄×C₈×C₅₃

Q₈

yes

—

C₂²×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₅₃

#41

yes

—

C₂²×Dic₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₄⋊C₄

yes

—

C₄²×C₅₃

C₂²×C₄

yes

—

C₄²×C₅₃

C₂×D₄

yes

characteristic

C₄²×C₅₃

C₂×Q₈

yes

characteristic

C₄×Dic₅₃

—

C₄×Dic₅₃

—

C₄×Dic₅₃

M₁₆

yes

—

C₂×#88322

—

C₂×#88322

—

C₂×#88322

C₄²

yes

characteristic

C₂×#88322

C₄²

yes

characteristic

C₂×#88322

C₄²

yes

—

C₂×#88322

M₁₆

yes

—

C₂³×C₅₃

#117

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×C₄²

yes

characteristic

C₂×C₄×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₄×D₄

yes

—

C₂×C₄×C₅₃

C₄×Q₈

yes

—

C₂×C₄×C₅₃

C₂×M₁₆

yes

—

C₂×C₄×C₅₃

C₂×M₁₆

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

—

—

—

yes

—

#8269

—

maximal, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

C₄×C₄⋊C₄

yes

characteristic

C₂²×C₅₃

#432

yes

—

C₄×C₅₃

—

C₄×C₅₃

C₂×#97

yes

characteristic

C₄×C₅₃

C₄×M₁₆

yes

characteristic

C₄×C₅₃

#423

yes

—

C₄×C₅₃

#444

yes

—

Dic₅₃

—

C₄×#97

—

#1564

—

#1452

—

#1452

D₅₃

yes

characteristic

#1276

—

C₂×C₅₃

—

C₂×C₅₃

#1767

yes

characteristic

C₂×C₅₃

#1773

yes

—

#431

—

#431

C₂×D₅₃

yes

—

C₂×#97

—

C₂×#97

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₅₃

yes

characteristic

C₄³

—

—

—

—

—

106

—

C₅₃

#8269

yes

characteristic, complemented, Sylow

100

C₂²×C₈

—

101

C₂²×C₈

—

102

C₂²×C₈

C₄×D₅₃

yes

—

103

C₂×C₄²

—

104

C₂×C₄²

—

106

—

105

C₂×C₄²

C₂²×D₅₃

yes

characteristic

106

C₄⋊C₈

—

107

C₄⋊C₈

C₄×D₅₃

yes

—

108

C₂²⋊C₈

—

109

C₂²⋊C₈

—

106

—

110

#97

—

106

—

111

#97

—

106

—

112

C₄×C₈

—

113

C₄×C₈

—

114

C₄×C₈

—

115

C₄×C₈

—

106

—

116

C₄×C₈

Dic₁₀₆

yes

—

117

C₄×C₈

C₂²⋊D₅₃

yes

—

118

C₂²×C₄

—

106

—

119

C₂²×C₄

C₂×C₄×D₅₃

yes

characteristic

120

C₂²×C₄

#11719945

yes

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

—

124

C₂×C₈

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

—

128

C₂×C₈

—

106

—

129

C₂×C₈

—

106

—

130

C₂×C₈

#11719919

yes

—

131

C₄²

—

132

C₄²

—

106

—

133

C₄²

—

212

—

134

C₄²

C₂×Dic₁₀₆

yes

characteristic

135

C₄²

C₂×C₄×D₅₃

yes

—

136

C₄²

C₂×C₂²⋊D₅₃

yes

characteristic

137

C₂³

#13133174

yes

characteristic

138

C₂×C₄

—

139

C₂×C₄

—

106

—

140

C₂×C₄

—

212

—

141

C₂×C₄

C₄²×D₅₃

yes

characteristic

142

C₂×C₄

C₂×#11719919

yes

characteristic

143

C₂×C₄

C₄×C₂²⋊D₅₃

yes

—

144

C₂×C₄

C₄×Dic₁₀₆

yes

—

145

C₂×C₄

M₁₆×D₅₃

yes

—

146

C₂×C₄

M₁₆×D₅₃

yes

—

147

C₈

—

148

C₈

—

149

C₈

—

150

C₈

—

106

—

151

C₈

—

106

—

152

C₈

—

212

—

153

C₂²

—

154

C₂²

C₄×#11719919

yes

characteristic

155

C₂²

#16810678

yes

—

156

C₄

—

157

C₄

—

212

—

158

C₄

#97×D₅₃

yes

characteristic

159

C₄

#16810575

yes

characteristic

160

C₄

#16810646

yes

—

161

C₄

#16810743

yes

—

162

C₂

—

163

C₂

#20274569

yes

characteristic

164

C₂

#20274600

yes

—

	subgroup	G/H
center	C₄²	C₂×C₂²⋊D₅₃
commutator	C₂×C₄×C₅₃	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1452	C₂
socle	C₂²×C₅₃	C₄×C₄⋊C₄
prime radical	C₂³×C₅₃	#117
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1452	D₅₃
hypocenter	C₅₃	#8269
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄² ◁ C₂×C₄×C₈ ◁ #1452

subgroup

G/H

center

C₄²

C₂×C₂²⋊D₅₃

commutator

C₂×C₄×C₅₃

C₂×C₄²

Frattini

C₂×C₄²

C₂²×D₅₃

Fitting

C₅₃×#1452

C₂

socle

C₂²×C₅₃

C₄×C₄⋊C₄

prime radical

C₂³×C₅₃

#117

cosocle

C₂×C₄²×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₄² ◁ C₂×C₄×C₈ ◁ #1452

N	H ≅ G/N	type
11. #16810375	C₄	semidirect
12. #16810373	C₄	semidirect
99. C₅₃	#8269	semidirect

H ≅ G/N

type

11. #16810375

C₄

semidirect

12. #16810373

C₄

semidirect

99. C₅₃

#8269

semidirect

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

subgroup	index
C₄×#13133127	2
#20273461	2
#20273007	2
#20272547	2
C₅₃×#1452	2
#8269	53

subgroup

index

C₅₃×#1452

Subgroups — #27275263

Conjugacy classes of subgroups

Complements

Maximal subgroups