Group #27274522 — subgroups

Subgroups — #27274522

Order 13568.

446 conjugacy classes of subgroups; only the 227 normal subgroups are shown.

normal subgroups only

This group has 446 conjugacy classes of subgroups — too many to draw. Only its 227 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	#20274461	C₂	1	1	yes	maximal, characteristic
2	C₄²×#88322	C₂	1	1	yes	maximal, characteristic
3	#20272542	C₂	1	4	yes	maximal
4	C₅₃×#1744	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄×#88322	C₄	1	2	yes	—
6	C₂×C₄×#88322	C₄	1	2	yes	—
7	C₂×C₄×#88322	C₂²	1	2	yes	—
8	C₂×C₄×#88322	C₂²	1	2	yes	—
9	#16810378	C₄	1	4	yes	—
10	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
11	C₄³×C₅₃	C₂²	1	1	yes	characteristic
12	C₅₃×#336	C₄	1	4	yes	—
13	C₂²×#88322	C₂×C₄	1	4	yes	—
14	C₂²×#88322	C₂×C₄	1	4	yes	—
15	C₄×#88322	—	2	4	no	—
16	C₄×#88322	—	2	4	no	—
17	C₄×#88322	—	2	4	no	—
18	C₄×#88322	C₈	1	8	yes	complemented
19	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
20	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
21	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
24	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
25	C₄×C₈×C₅₃	—	2	4	no	—
26	C₂²×C₄×C₅₃	C₄²	1	2	yes	—
27	C₂²×C₄×C₅₃	C₄²	1	2	yes	—
28	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
29	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
30	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
31	C₂×C₈×C₅₃	—	2	4	no	—
32	C₂×C₈×C₅₃	—	2	8	no	—
33	C₄²×C₅₃	—	2	4	no	—
34	C₄²×C₅₃	—	2	4	no	—
35	C₄²×C₅₃	C₂×C₈	1	4	yes	—
36	C₄²×C₅₃	M₁₆	1	4	yes	—
37	C₄²×C₅₃	#41	1	4	yes	—
38	C₂×#88322	—	2	4	no	—
39	C₂×#88322	—	2	4	no	—
40	C₂×#88322	—	2	4	no	—
41	C₂×#88322	—	2	8	no	—
42	C₂×#88322	C₂×C₈	1	8	yes	—
43	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
44	C₂×C₄×C₅₃	—	2	2	no	—
45	C₂×C₄×C₅₃	—	2	2	no	—
46	C₂×C₄×C₅₃	—	2	2	no	—
47	C₂×C₄×C₅₃	—	2	2	no	—
48	C₂×C₄×C₅₃	C₄×C₈	1	4	yes	—
49	C₂×C₄×C₅₃	#97	1	4	yes	—
50	C₂×C₄×C₅₃	#117	1	2	yes	—
51	C₂×C₄×C₅₃	#117	1	2	yes	—
52	C₂×C₄×C₅₃	#117	1	2	yes	—
53	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
54	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
55	C₂×C₄×C₅₃	#131	1	4	yes	—
56	C₂×C₄×C₅₃	#131	1	4	yes	—
57	C₈×C₅₃	—	4	4	no	—
58	#88322	—	2	8	no	—
59	#88322	—	4	4	no	—
60	#8133	—	53	1	no	maximal, Sylow
61	C₂²×C₅₃	#376	1	1	yes	characteristic
62	C₂²×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
63	C₂²×C₅₃	C₂×#97	1	1	yes	characteristic
64	C₂²×C₅₃	#405	1	2	yes	—
65	C₂²×C₅₃	#405	1	2	yes	—
66	C₄×C₅₃	—	2	4	no	—
67	C₄×C₅₃	—	2	4	no	—
68	C₄×C₅₃	#431	1	4	yes	—
69	C₄×C₅₃	#433	1	4	yes	—
70	C₄×C₅₃	#433	1	4	yes	—
71	#1744	—	53	1	no	—
72	#1744	D₅₃	1	1	yes	characteristic
73	C₄²×C₈	—	53	1	no	—
74	#1273	—	53	4	no	—
75	C₂×C₅₃	#1450	1	2	yes	—
76	C₂×C₅₃	#1452	1	2	yes	—
77	C₂×C₅₃	#1744	1	1	yes	characteristic
78	C₂×C₅₃	#1746	1	1	yes	characteristic
79	C₂×C₅₃	#1746	1	1	yes	characteristic
80	C₂×C₄×C₈	—	53	2	no	—
81	C₂×C₄×C₈	—	53	2	no	—
82	C₂×C₄×C₈	—	53	2	no	—
83	C₂×C₄×C₈	—	53	2	no	—
84	C₂×C₄×C₈	C₂×D₅₃	1	2	yes	—
85	C₄³	C₂×D₅₃	1	1	yes	characteristic
86	#336	—	53	4	no	—
87	#336	Dic₅₃	1	4	yes	—
88	C₅₃	#8133	1	1	yes	characteristic, complemented, Sylow
89	C₂²×C₈	—	53	4	no	—
90	C₂²×C₈	—	53	4	no	—
91	C₂²×C₈	C₂×Dic₅₃	1	4	yes	—
92	C₂×C₄²	C₄×D₅₃	1	2	yes	—
93	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
94	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
95	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
96	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
97	C₄×C₈	—	2	4	no	—
98	C₄×C₈	—	53	8	no	—
99	C₄×C₈	—	106	4	no	—
100	C₄×C₈	—	106	4	no	—
101	C₄×C₈	—	106	4	no	—
102	C₂²×C₄	C₄×Dic₅₃	1	2	yes	—
103	C₂²×C₄	C₄×Dic₅₃	1	2	yes	—
104	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
105	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
106	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
107	C₂×C₈	—	2	4	no	—
108	C₂×C₈	—	2	8	no	—
109	C₂×C₈	—	53	8	no	—
110	C₂×C₈	—	106	4	no	—
111	C₂×C₈	—	106	4	no	—
112	C₂×C₈	—	106	4	no	—
113	C₂×C₈	—	106	8	no	—
114	C₄²	—	2	4	no	—
115	C₄²	—	2	4	no	—
116	C₄²	C₈×D₅₃	1	4	yes	—
117	C₄²	#11719912	1	4	yes	—
118	C₄²	#11719945	1	4	yes	—
119	C₂³	C₂×C₄×Dic₅₃	1	1	yes	characteristic
120	C₂×C₄	—	2	2	no	—
121	C₂×C₄	—	2	2	no	—
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	C₈×Dic₅₃	1	4	yes	—
125	C₂×C₄	#13133127	1	4	yes	—
126	C₂×C₄	#13133174	1	2	yes	—
127	C₂×C₄	#13133174	1	2	yes	—
128	C₂×C₄	#13133176	1	2	yes	—
129	C₂×C₄	C₂×C₈×D₅₃	1	2	yes	—
130	C₂×C₄	C₂×#11719912	1	2	yes	—
131	C₂×C₄	#13133224	1	4	yes	—
132	C₂×C₄	#13133225	1	4	yes	—
133	C₈	—	4	4	no	—
134	C₈	—	106	8	no	—
135	C₈	—	212	4	no	—
136	C₂²	#16810469	1	1	yes	characteristic
137	C₂²	C₂×C₈×Dic₅₃	1	1	yes	characteristic
138	C₂²	C₂×#13133127	1	1	yes	characteristic
139	C₂²	#16810579	1	2	yes	—
140	C₂²	#16810581	1	2	yes	—
141	C₄	—	2	4	no	—
142	C₄	—	2	4	no	—
143	C₄	#16810674	1	4	yes	—
144	C₄	#16810684	1	4	yes	—
145	C₄	#16810685	1	4	yes	—
146	C₂	#20272999	1	2	yes	—
147	C₂	#20273008	1	2	yes	—
148	C₂	#20274460	1	1	yes	characteristic
149	C₂	#20274467	1	1	yes	characteristic
150	C₂	#20274468	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20274461

C₂

yes

maximal, characteristic

C₄²×#88322

C₂

yes

maximal, characteristic

#20272542

C₂

yes

maximal

C₅₃×#1744

C₂

yes

maximal, characteristic

C₂×C₄×#88322

C₄

yes

—

C₂×C₄×#88322

C₄

yes

—

C₂×C₄×#88322

C₂²

yes

—

C₂×C₄×#88322

C₂²

yes

—

#16810378

C₄

yes

—

C₂×C₄×C₈×C₅₃

C₂²

yes

—

C₄³×C₅₃

C₂²

yes

characteristic

C₅₃×#336

C₄

yes

—

C₂²×#88322

C₂×C₄

yes

—

C₂²×#88322

C₂×C₄

yes

—

—

—

—

C₈

yes

complemented

C₂²×C₈×C₅₃

C₂×C₄

yes

—

C₂×C₄²×C₅₃

C₂×C₄

yes

—

C₂×C₄²×C₅₃

C₂×C₄

yes

characteristic

C₂×C₄²×C₅₃

C₂×C₄

yes

characteristic

C₂×C₄²×C₅₃

C₂×C₄

yes

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

C₄×C₈×C₅₃

—

C₂²×C₄×C₅₃

C₄²

yes

—

C₂²×C₄×C₅₃

C₄²

yes

—

C₂²×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

C₂×C₈

yes

—

C₄²×C₅₃

M₁₆

yes

—

C₄²×C₅₃

#41

yes

—

—

—

—

—

C₂×C₈

yes

—

C₂³×C₅₃

C₂×C₄²

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₄×C₈

yes

—

C₂×C₄×C₅₃

#97

yes

—

C₂×C₄×C₅₃

#117

yes

—

C₂×C₄×C₅₃

#117

yes

—

C₂×C₄×C₅₃

#117

yes

—

C₂×C₄×C₅₃

C₂²×C₈

yes

—

C₂×C₄×C₅₃

C₂×M₁₆

yes

—

C₂×C₄×C₅₃

#131

yes

—

C₂×C₄×C₅₃

#131

yes

—

C₈×C₅₃

—

—

—

—

C₂²×C₅₃

yes

C₂²×C₅₃

C₂×C₄×C₈

yes

characteristic

C₂²×C₅₃

C₂×#97

yes

characteristic

C₂²×C₅₃

#405

yes

—

C₂²×C₅₃

#405

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#431

yes

—

C₄×C₅₃

#433

yes

—

C₄×C₅₃

#433

yes

—

#1744

—

#1744

D₅₃

yes

characteristic

C₄²×C₈

—

#1273

—

C₂×C₅₃

#1450

yes

—

C₂×C₅₃

#1452

yes

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₅₃

yes

—

C₄³

C₂×D₅₃

yes

characteristic

#336

—

#336

Dic₅₃

yes

—

C₅₃

#8133

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₂×Dic₅₃

yes

—

C₂×C₄²

C₄×D₅₃

yes

—

C₂×C₄²

C₄×D₅₃

yes

characteristic

C₂×C₄²

C₄×D₅₃

yes

characteristic

C₂×C₄²

C₂×Dic₅₃

yes

—

C₂×C₄²

C₂²×D₅₃

yes

characteristic

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

106

—

100

C₄×C₈

—

106

—

101

C₄×C₈

—

106

—

102

C₂²×C₄

C₄×Dic₅₃

yes

—

103

C₂²×C₄

C₄×Dic₅₃

yes

—

104

C₂²×C₄

C₂×C₄×D₅₃

yes

characteristic

105

C₂²×C₄

C₂×C₄×D₅₃

yes

characteristic

106

C₂²×C₄

C₂²×Dic₅₃

yes

characteristic

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

—

106

—

111

C₂×C₈

—

106

—

112

C₂×C₈

—

106

—

113

C₂×C₈

—

106

—

114

C₄²

—

115

C₄²

—

116

C₄²

C₈×D₅₃

yes

—

117

C₄²

#11719912

yes

—

118

C₄²

#11719945

yes

—

119

C₂³

C₂×C₄×Dic₅₃

yes

characteristic

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

C₈×Dic₅₃

yes

—

125

C₂×C₄

#13133127

yes

—

126

C₂×C₄

#13133174

yes

—

127

C₂×C₄

#13133174

yes

—

128

C₂×C₄

#13133176

yes

—

129

C₂×C₄

C₂×C₈×D₅₃

yes

—

130

C₂×C₄

C₂×#11719912

yes

—

131

C₂×C₄

#13133224

yes

—

132

C₂×C₄

#13133225

yes

—

133

C₈

—

134

C₈

—

106

—

135

C₈

—

212

—

136

C₂²

#16810469

yes

characteristic

137

C₂²

C₂×C₈×Dic₅₃

yes

characteristic

138

C₂²

C₂×#13133127

yes

characteristic

139

C₂²

#16810579

yes

—

140

C₂²

#16810581

yes

—

141

C₄

—

142

C₄

—

143

C₄

#16810674

yes

—

144

C₄

#16810684

yes

—

145

C₄

#16810685

yes

—

146

C₂

#20272999

yes

—

147

C₂

#20273008

yes

—

148

C₂

#20274460

yes

characteristic

149

C₂

#20274467

yes

characteristic

150

C₂

#20274468

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂²×D₅₃
commutator	C₂²×C₅₃	C₂×C₄×C₈
Fitting	C₅₃×#1744	C₂
socle, prime radical	C₂³×C₅₃	C₂×C₄²
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1744	D₅₃
hypocenter	C₅₃	#8133
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄² ◁ #1744

subgroup

G/H

center, Frattini

C₂×C₄²

C₂²×D₅₃

commutator

C₂²×C₅₃

C₂×C₄×C₈

Fitting

C₅₃×#1744

C₂

socle, prime radical

C₂³×C₅₃

C₂×C₄²

cosocle

C₂×C₄²×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₂²×C₅₃ ▷ C₅₃

C₁ ◁ C₂×C₄² ◁ #1744

N	H ≅ G/N	type
18. C₄×#88322	C₈	semidirect
88. C₅₃	#8133	semidirect

H ≅ G/N

type

18. C₄×#88322

C₈

semidirect

88. C₅₃

#8133

semidirect

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

subgroup	index
#20274461	2
C₄²×#88322	2
#20272542	2
C₅₃×#1744	2
#8133	53

subgroup

index

#20274461

C₄²×#88322

#20272542

C₅₃×#1744

#8133

Subgroups — #27274522

Conjugacy classes of subgroups

Complements

Maximal subgroups