Group #27270684 — subgroups

Subgroups — #27270684

Order 13568.

292 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 131 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	#20274832	C₂	1	2	yes	maximal
2	C₅₃×#1842	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄⋊#88322	—	2	2	no	—
4	#16810378	—	2	2	no	—
5	#16810378	C₄	1	4	yes	—
6	C₂×C₄⋊C₈×C₅₃	—	2	2	no	—
7	C₂×C₄⋊C₈×C₅₃	C₄	1	2	yes	—
8	C₄³×C₅₃	C₂²	1	1	yes	characteristic
9	C₂²×#88322	—	2	4	no	—
10	C₄⋊#88322	—	4	2	no	—
11	C₄⋊#88322	—	4	2	no	—
12	C₂²×C₈×C₅₃	—	2	2	no	—
13	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
14	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
15	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
16	C₂×C₄²×C₅₃	D₄	1	2	yes	—
17	C₂×C₄²×C₅₃	Q₈	1	1	yes	characteristic
18	C₄⋊C₈×C₅₃	—	4	2	no	—
19	C₄⋊C₈×C₅₃	—	4	2	no	—
20	C₄⋊C₈×C₅₃	—	4	2	no	—
21	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
22	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
23	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
24	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
25	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
26	C₂×C₈×C₅₃	—	4	2	no	—
27	C₂×C₈×C₅₃	—	4	4	no	—
28	C₄²×C₅₃	—	2	1	no	—
29	C₄²×C₅₃	—	2	1	no	—
30	C₄²×C₅₃	—	2	2	no	—
31	C₄²×C₅₃	—	4	2	no	—
32	C₄²×C₅₃	—	4	1	no	—
33	C₄²×C₅₃	D₈	1	2	yes	—
34	C₄²×C₅₃	SD₁₆	1	2	yes	—
35	C₄²×C₅₃	SD₁₆	1	2	yes	—
36	C₄²×C₅₃	Q₁₆	1	2	yes	—
37	C₂×#88322	—	4	4	no	—
38	C₂×#88322	—	4	4	no	—
39	C₂×#88322	—	4	4	no	—
40	C₂³×C₅₃	#95	1	1	yes	characteristic
41	C₂×C₄×C₅₃	—	2	2	no	—
42	C₂×C₄×C₅₃	—	2	1	no	—
43	C₂×C₄×C₅₃	—	2	2	no	—
44	C₂×C₄×C₅₃	—	2	1	no	—
45	C₂×C₄×C₅₃	—	2	2	no	—
46	C₂×C₄×C₅₃	—	2	2	no	—
47	C₂×C₄×C₅₃	—	2	1	no	—
48	C₂×C₄×C₅₃	—	2	2	no	—
49	C₂×C₄×C₅₃	#100	1	1	yes	characteristic
50	C₂×C₄×C₅₃	#101	1	1	yes	characteristic
51	C₂×C₄×C₅₃	#102	1	2	yes	—
52	C₂×C₄×C₅₃	#102	1	2	yes	—
53	C₂×C₄×C₅₃	#103	1	2	yes	—
54	C₂×C₄×C₅₃	#103	1	2	yes	—
55	C₂×C₄×C₅₃	#106	1	2	yes	—
56	C₂×C₄×C₅₃	#107	1	2	yes	—
57	C₂×C₄×C₅₃	#108	1	2	yes	—
58	C₈×C₅₃	—	8	2	no	—
59	#88322	—	8	4	no	—
60	#7040	—	53	1	no	maximal, Sylow
61	C₂²×C₅₃	H(3,ℤ/4)	1	1	yes	characteristic
62	C₂²×C₅₃	#340	1	2	yes	—
63	C₂²×C₅₃	#341	1	1	yes	characteristic
64	C₂²×C₅₃	#343	1	1	yes	characteristic
65	C₂²×C₅₃	#344	1	2	yes	—
66	C₄×C₅₃	—	2	2	no	—
67	C₄×C₅₃	—	2	2	no	—
68	C₄×C₅₃	—	2	2	no	—
69	C₄×C₅₃	—	2	2	no	—
70	C₄×C₅₃	—	2	2	no	—
71	C₄×C₅₃	—	4	1	no	—
72	C₄×C₅₃	#331	1	1	yes	characteristic
73	C₄×C₅₃	#332	1	2	yes	—
74	C₄×C₅₃	#333	1	1	yes	characteristic
75	#1842	D₅₃	1	1	yes	characteristic
76	#1831	—	53	2	no	—
77	C₂×C₅₃	#1286	1	2	yes	—
78	C₂×C₅₃	#1290	1	1	yes	characteristic
79	C₂×C₅₃	#1296	1	1	yes	characteristic
80	C₂×C₅₃	#1301	1	2	yes	—
81	C₂×C₅₃	#1307	1	1	yes	characteristic
82	C₂×C₄⋊C₈	—	2	2	no	—
83	C₂×C₄⋊C₈	—	106	2	no	—
84	C₂×C₄⋊C₈	Dic₅₃	1	2	yes	—
85	C₄³	C₂×D₅₃	1	1	yes	characteristic
86	#336	—	53	4	no	—
87	#336	—	106	2	no	—
88	C₅₃	#7040	1	1	yes	characteristic, complemented, Sylow
89	C₂²×C₈	—	2	2	no	—
90	C₂²×C₈	—	106	4	no	—
91	C₂×C₄²	Dic₁₀₆	1	1	yes	characteristic
92	C₂×C₄²	C₄×D₅₃	1	2	yes	—
93	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
94	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
95	C₂×C₄²	C₂²⋊D₅₃	1	2	yes	—
96	C₄⋊C₈	—	4	2	no	—
97	C₄⋊C₈	—	4	2	no	—
98	C₄⋊C₈	—	4	2	no	—
99	C₄⋊C₈	—	212	2	no	—
100	C₄⋊C₈	—	212	2	no	—
101	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
102	C₂²×C₄	#11719919	1	2	yes	—
103	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
104	C₂²×C₄	#11719921	1	2	yes	—
105	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
106	C₂×C₈	—	4	2	no	—
107	C₂×C₈	—	4	4	no	—
108	C₂×C₈	—	212	4	no	—
109	C₂×C₈	—	212	4	no	—
110	C₂×C₈	—	212	4	no	—
111	C₄²	—	2	1	no	—
112	C₄²	—	2	1	no	—
113	C₄²	—	2	2	no	—
114	C₄²	—	4	2	no	—
115	C₄²	—	4	1	no	—
116	C₄²	#11719922	1	2	yes	—
117	C₄²	#11719923	1	2	yes	—
118	C₄²	#11719924	1	2	yes	—
119	C₄²	#11719925	1	2	yes	—
120	C₂³	#13133124	1	1	yes	characteristic
121	C₂×C₄	—	2	2	no	—
122	C₂×C₄	—	2	1	no	—
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	—	2	1	no	—
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	—	2	1	no	—
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	#13133134	1	1	yes	characteristic
130	C₂×C₄	#13133136	1	1	yes	characteristic
131	C₂×C₄	#13133137	1	2	yes	—
132	C₂×C₄	#13133138	1	2	yes	—
133	C₂×C₄	#13133140	1	2	yes	—
134	C₂×C₄	#13133141	1	2	yes	—
135	C₂×C₄	#13133148	1	2	yes	—
136	C₂×C₄	#13133150	1	2	yes	—
137	C₂×C₄	#13133153	1	2	yes	—
138	C₈	—	8	2	no	—
139	C₈	—	424	4	no	—
140	C₂²	#16810380	1	1	yes	characteristic
141	C₂²	#16810385	1	2	yes	—
142	C₂²	#16810388	1	1	yes	characteristic
143	C₂²	#16810392	1	1	yes	characteristic
144	C₂²	#16810395	1	2	yes	—
145	C₄	—	2	2	no	—
146	C₄	—	2	2	no	—
147	C₄	—	2	2	no	—
148	C₄	—	2	2	no	—
149	C₄	—	2	2	no	—
150	C₄	—	4	1	no	—
151	C₄	#16810367	1	1	yes	characteristic
152	C₄	#16810369	1	2	yes	—
153	C₄	#16810372	1	1	yes	characteristic
154	C₂	#20272575	1	2	yes	—
155	C₂	#20272585	1	1	yes	characteristic
156	C₂	#20272598	1	1	yes	characteristic
157	C₂	#20272612	1	2	yes	—
158	C₂	#20272626	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20274832

C₂

yes

maximal

C₅₃×#1842

C₂

yes

maximal, characteristic

C₂×C₄⋊#88322

—

#16810378

—

#16810378

C₄

yes

—

C₂×C₄⋊C₈×C₅₃

—

C₂×C₄⋊C₈×C₅₃

C₄

yes

—

C₄³×C₅₃

C₂²

yes

characteristic

C₂²×#88322

—

C₄⋊#88322

—

C₄⋊#88322

—

C₂²×C₈×C₅₃

—

C₂×C₄²×C₅₃

C₂×C₄

yes

—

C₂×C₄²×C₅₃

C₂×C₄

yes

characteristic

C₂×C₄²×C₅₃

D₄

yes

characteristic

C₂×C₄²×C₅₃

D₄

yes

—

C₂×C₄²×C₅₃

Q₈

yes

characteristic

C₄⋊C₈×C₅₃

—

C₄⋊C₈×C₅₃

—

C₄⋊C₈×C₅₃

—

C₂²×C₄×C₅₃

C₄²

yes

characteristic

C₂²×C₄×C₅₃

C₂²⋊C₄

yes

—

C₂²×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₅₃

C₄⋊C₄

yes

—

C₂²×C₄×C₅₃

C₄⋊C₄

yes

characteristic

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

D₈

yes

—

C₄²×C₅₃

SD₁₆

yes

—

C₄²×C₅₃

SD₁₆

yes

—

C₄²×C₅₃

Q₁₆

yes

—

C₂×#88322

—

C₂×#88322

—

C₂×#88322

—

C₂³×C₅₃

#95

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

—

C₂×C₄×C₅₃

#102

yes

—

C₂×C₄×C₅₃

#103

yes

—

C₂×C₄×C₅₃

#103

yes

—

C₂×C₄×C₅₃

#106

yes

—

C₂×C₄×C₅₃

#107

yes

—

C₂×C₄×C₅₃

#108

yes

—

C₈×C₅₃

—

—

—

C₂²×C₅₃

yes

C₂²×C₅₃

yes

—

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#331

yes

characteristic

C₄×C₅₃

#332

yes

—

C₄×C₅₃

#333

yes

characteristic

#1842

D₅₃

yes

characteristic

#1831

—

C₂×C₅₃

#1286

yes

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

C₂×C₅₃

#1307

yes

characteristic

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

106

—

C₂×C₄⋊C₈

Dic₅₃

yes

—

C₄³

C₂×D₅₃

yes

characteristic

#336

—

#336

—

106

—

C₅₃

#7040

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

—

106

—

C₂×C₄²

Dic₁₀₆

yes

characteristic

C₂×C₄²

C₄×D₅₃

yes

—

C₂×C₄²

D₂₁₂

yes

characteristic

C₂×C₄²

C₂×Dic₅₃

yes

characteristic

C₂×C₄²

C₂²⋊D₅₃

yes

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

212

—

100

C₄⋊C₈

—

212

—

101

C₂²×C₄

C₄×Dic₅₃

yes

characteristic

102

C₂²×C₄

#11719919

yes

—

103

C₂²×C₄

C₄⋊Dic₅₃

yes

characteristic

104

C₂²×C₄

#11719921

yes

—

105

C₂²×C₄

C₂²⋊Dic₅₃

yes

characteristic

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

212

—

109

C₂×C₈

—

212

—

110

C₂×C₈

—

212

—

111

C₄²

—

112

C₄²

—

113

C₄²

—

114

C₄²

—

115

C₄²

—

116

C₄²

#11719922

yes

—

117

C₄²

#11719923

yes

—

118

C₄²

#11719924

yes

—

119

C₄²

#11719925

yes

—

120

C₂³

#13133124

yes

characteristic

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

#13133134

yes

characteristic

130

C₂×C₄

#13133136

yes

characteristic

131

C₂×C₄

#13133137

yes

—

132

C₂×C₄

#13133138

yes

—

133

C₂×C₄

#13133140

yes

—

134

C₂×C₄

#13133141

yes

—

135

C₂×C₄

#13133148

yes

—

136

C₂×C₄

#13133150

yes

—

137

C₂×C₄

#13133153

yes

—

138

C₈

—

139

C₈

—

424

—

140

C₂²

#16810380

yes

characteristic

141

C₂²

#16810385

yes

—

142

C₂²

#16810388

yes

characteristic

143

C₂²

#16810392

yes

characteristic

144

C₂²

#16810395

yes

—

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

—

151

C₄

#16810367

yes

characteristic

152

C₄

#16810369

yes

—

153

C₄

#16810372

yes

characteristic

154

C₂

#20272575

yes

—

155

C₂

#20272585

yes

characteristic

156

C₂

#20272598

yes

characteristic

157

C₂

#20272612

yes

—

158

C₂

#20272626

yes

characteristic

	subgroup	G/H
center	C₂³	#13133124
commutator	C₂²×C₄×C₅₃	C₄²
Frattini	C₄³	C₂×D₅₃
Fitting	C₅₃×#1842	C₂
socle, prime radical	C₂³×C₅₃	#95
cosocle	C₄³×C₅₃	C₂²
hypercenter	#1842	D₅₃
hypocenter	C₅₃	#7040
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #1842

subgroup

G/H

center

C₂³

#13133124

commutator

C₂²×C₄×C₅₃

C₄²

Frattini

C₄³

C₂×D₅₃

Fitting

C₅₃×#1842

C₂

socle, prime radical

C₂³×C₅₃

#95

cosocle

C₄³×C₅₃

C₂²

D₅₃

C₅₃

G ▷ C₂²×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂³ ◁ C₄³ ◁ #1842

N	H ≅ G/N	type
88. C₅₃	#7040	semidirect

H ≅ G/N

type

88. C₅₃

#7040

semidirect

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

subgroup	index
#20274832	2
C₅₃×#1842	2
#7040	53

subgroup

index

#20274832

C₅₃×#1842

#7040

Subgroups — #27270684

Conjugacy classes of subgroups

Complements

Maximal subgroups