Group #27273482 — subgroups

Subgroups — #27273482

Order 13568.

610 conjugacy classes of subgroups; only the 207 normal subgroups are shown.

normal subgroups only

This group has 610 conjugacy classes of subgroups — too many to draw. Only its 207 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	C₂×#16810674	C₂	1	3	yes	maximal
2	#20272542	C₂	1	3	yes	maximal, complemented
3	C₅₃×#1273	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₄²×D₅₃	C₂²	1	1	yes	characteristic
5	#16810674	—	2	12	no	—
6	C₂×#13133146	C₄	1	6	yes	—
7	C₂×#13133129	C₄	1	6	yes	—
8	C₂×C₄×#88322	C₂²	1	3	yes	—
9	C₂×C₄×C₈×C₅₃	C₂²	1	3	yes	—
10	C₂²×C₄×D₅₃	C₂×C₄	1	3	yes	—
11	C₂²×#88322	C₂×C₄	1	6	yes	—
12	C₂×C₄×Dic₅₃	C₂×C₄	1	3	yes	—
13	C₄²×D₅₃	D₄	1	6	yes	—
14	C₄²×D₅₃	Q₈	1	2	yes	—
15	#13133146	—	2	12	no	—
16	#13133129	—	2	12	no	—
17	C₄×#88322	—	2	6	no	—
18	C₂²×C₈×C₅₃	C₂×C₄	1	6	yes	—
19	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
20	C₄×C₈×C₅₃	—	2	6	no	—
21	C₂³×D₅₃	C₄²	1	1	yes	characteristic
22	C₂²×C₄×C₅₃	C₂²×C₄	1	3	yes	—
23	C₂²×Dic₅₃	C₄²	1	3	yes	—
24	C₂×C₄×D₅₃	—	2	6	no	—
25	C₂×C₄×D₅₃	—	2	6	no	—
26	C₂×C₄×D₅₃	C₂²⋊C₄	1	6	yes	—
27	C₂×C₄×D₅₃	C₄⋊C₄	1	6	yes	—
28	C₂×C₈×C₅₃	—	2	6	no	—
29	C₂×C₈×C₅₃	—	2	6	no	—
30	C₂×C₈×C₅₃	—	2	6	no	—
31	C₄²×C₅₃	C₂×D₄	1	3	yes	—
32	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
33	C₄×Dic₅₃	C₂²⋊C₄	1	6	yes	—
34	C₄×Dic₅₃	C₄⋊C₄	1	6	yes	—
35	C₂×#88322	—	2	6	no	—
36	C₂×#88322	—	2	6	no	—
37	C₂×#88322	—	2	6	no	—
38	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
39	C₂²×D₅₃	—	2	3	no	—
40	C₂²×D₅₃	—	2	3	no	—
41	C₂²×D₅₃	#95	1	2	yes	—
42	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	3	yes	—
43	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	3	yes	—
44	C₂×C₄×C₅₃	#131	1	6	yes	—
45	C₂×C₄×C₅₃	#131	1	6	yes	—
46	C₂×Dic₅₃	—	2	3	no	—
47	C₂×Dic₅₃	—	2	3	no	—
48	C₂×Dic₅₃	#95	1	6	yes	—
49	C₄×D₅₃	—	2	12	no	—
50	C₄×D₅₃	—	2	12	no	—
51	C₈×C₅₃	—	4	6	no	—
52	#88322	—	4	6	no	—
53	#7851	—	53	1	no	maximal, Sylow
54	C₂²×C₅₃	C₂×#95	1	1	yes	characteristic
55	C₂²×C₅₃	#405	1	3	yes	—
56	C₂²×C₅₃	#405	1	3	yes	—
57	C₂×D₅₃	—	2	6	no	—
58	C₂×D₅₃	—	4	1	no	—
59	C₂×D₅₃	—	4	3	no	—
60	C₄×C₅₃	#408	1	6	yes	—
61	C₄×C₅₃	#424	1	6	yes	—
62	Dic₅₃	—	2	6	no	—
63	C₂×#431	—	53	3	no	—
64	#1273	—	53	3	no	—
65	#1273	D₅₃	1	1	yes	characteristic, complemented
66	C₂×C₅₃	#1454	1	1	yes	characteristic
67	C₂×C₅₃	#1454	1	3	yes	—
68	C₂×C₅₃	#1729	1	3	yes	—
69	D₅₃	—	4	2	no	—
70	C₂²×C₄²	—	53	1	no	—
71	#431	—	106	12	no	—
72	C₂×C₄⋊C₈	—	53	6	no	—
73	C₂×C₂²⋊C₈	—	53	6	no	—
74	C₂×C₄×C₈	—	53	3	no	—
75	C₂×C₄×C₈	C₂×D₅₃	1	3	yes	—
76	C₅₃	#7851	1	1	yes	characteristic, complemented, Sylow
77	C₂³×C₄	—	53	3	no	—
78	C₂²×C₈	—	53	6	no	—
79	C₂²×C₈	C₄×D₅₃	1	6	yes	—
80	C₂×C₄²	—	53	3	no	—
81	C₂×C₄²	—	53	6	no	—
82	C₂×C₄²	—	53	2	no	—
83	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
84	C₄⋊C₈	—	106	12	no	—
85	C₂²⋊C₈	—	106	12	no	—
86	C₄×C₈	—	2	6	no	—
87	C₄×C₈	—	106	6	no	—
88	C₂⁴	—	53	1	no	—
89	C₂²×C₄	—	53	3	no	—
90	C₂²×C₄	—	53	6	no	—
91	C₂²×C₄	—	53	6	no	—
92	C₂²×C₄	—	106	6	no	—
93	C₂²×C₄	—	106	6	no	—
94	C₂²×C₄	C₂×C₄×D₅₃	1	3	yes	—
95	C₂×C₈	—	2	6	no	—
96	C₂×C₈	—	2	6	no	—
97	C₂×C₈	—	2	6	no	—
98	C₂×C₈	—	106	6	no	—
99	C₂×C₈	—	106	6	no	—
100	C₂×C₈	—	106	6	no	—
101	C₄²	—	53	6	no	—
102	C₄²	—	53	6	no	—
103	C₄²	D₄×D₅₃	1	3	yes	—
104	C₄²	Q₈×D₅₃	1	1	yes	characteristic
105	C₂³	—	53	2	no	—
106	C₂³	—	106	3	no	—
107	C₂³	—	106	3	no	—
108	C₂³	C₄²×D₅₃	1	1	yes	characteristic
109	C₂×C₄	—	53	6	no	—
110	C₂×C₄	—	106	3	no	—
111	C₂×C₄	—	106	3	no	—
112	C₂×C₄	—	106	12	no	—
113	C₂×C₄	—	106	12	no	—
114	C₂×C₄	C₂²⋊C₄×D₅₃	1	3	yes	—
115	C₂×C₄	C₄⋊C₄×D₅₃	1	3	yes	—
116	C₂×C₄	#13133223	1	6	yes	—
117	C₂×C₄	#13133225	1	6	yes	—
118	C₈	—	4	6	no	—
119	C₈	—	212	6	no	—
120	C₂²	—	106	6	no	—
121	C₂²	—	212	1	no	—
122	C₂²	—	212	3	no	—
123	C₂²	#95×D₅₃	1	1	yes	characteristic
124	C₂²	#16810577	1	3	yes	—
125	C₂²	#16810580	1	3	yes	—
126	C₄	—	106	6	no	—
127	C₄	#16810588	1	6	yes	—
128	C₄	#16810648	1	6	yes	—
129	C₂	—	212	2	no	—
130	C₂	#20273014	1	1	yes	characteristic
131	C₂	#20273016	1	3	yes	—
132	C₂	#20274410	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16810674

C₂

yes

maximal

#20272542

C₂

yes

maximal, complemented

C₅₃×#1273

C₂

yes

maximal, characteristic, complemented

C₂×C₄²×D₅₃

C₂²

yes

characteristic

#16810674

—

C₂×#13133146

C₄

yes

—

C₂×#13133129

C₄

yes

—

C₂×C₄×#88322

C₂²

yes

—

C₂×C₄×C₈×C₅₃

C₂²

yes

—

C₂²×C₄×D₅₃

C₂×C₄

yes

—

C₂²×#88322

C₂×C₄

yes

—

C₂×C₄×Dic₅₃

C₂×C₄

yes

—

C₄²×D₅₃

D₄

yes

—

C₄²×D₅₃

Q₈

yes

—

#13133146

—

#13133129

—

C₄×#88322

—

C₂²×C₈×C₅₃

C₂×C₄

yes

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

C₄×C₈×C₅₃

—

C₂³×D₅₃

C₄²

yes

characteristic

C₂²×C₄×C₅₃

C₂²×C₄

yes

—

C₂²×Dic₅₃

C₄²

yes

—

C₂×C₄×D₅₃

—

C₂×C₄×D₅₃

—

C₂×C₄×D₅₃

C₂²⋊C₄

yes

—

C₂×C₄×D₅₃

C₄⋊C₄

yes

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₄²×C₅₃

C₂×D₄

yes

—

C₄²×C₅₃

C₂×Q₈

yes

characteristic

C₄×Dic₅₃

C₂²⋊C₄

yes

—

C₄×Dic₅₃

C₄⋊C₄

yes

—

C₂×#88322

—

C₂×#88322

—

C₂×#88322

—

C₂³×C₅₃

C₂×C₄²

yes

characteristic

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂²×D₅₃

#95

yes

—

C₂×C₄×C₅₃

C₂×C₂²⋊C₄

yes

—

C₂×C₄×C₅₃

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₅₃

#131

yes

—

C₂×C₄×C₅₃

#131

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

#95

yes

—

C₄×D₅₃

—

C₄×D₅₃

—

C₈×C₅₃

—

#88322

—

#7851

—

maximal, Sylow

C₂²×C₅₃

C₂×#95

yes

characteristic

C₂²×C₅₃

#405

yes

—

C₂²×C₅₃

#405

yes

—

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

#408

yes

—

C₄×C₅₃

#424

yes

—

Dic₅₃

—

C₂×#431

—

#1273

—

#1273

D₅₃

yes

characteristic, complemented

C₂×C₅₃

#1454

yes

characteristic

C₂×C₅₃

#1454

yes

—

C₂×C₅₃

#1729

yes

—

D₅₃

—

C₂²×C₄²

—

#431

—

106

—

C₂×C₄⋊C₈

—

C₂×C₂²⋊C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₅₃

yes

—

C₅₃

#7851

yes

characteristic, complemented, Sylow

C₂³×C₄

—

C₂²×C₈

—

C₂²×C₈

C₄×D₅₃

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂²×D₅₃

yes

characteristic

C₄⋊C₈

—

106

—

C₂²⋊C₈

—

106

—

C₄×C₈

—

C₄×C₈

—

106

—

C₂⁴

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

106

—

C₂²×C₄

—

106

—

C₂²×C₄

C₂×C₄×D₅₃

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

106

—

C₂×C₈

—

106

—

100

C₂×C₈

—

106

—

101

C₄²

—

102

C₄²

—

103

C₄²

D₄×D₅₃

yes

—

104

C₄²

Q₈×D₅₃

yes

characteristic

105

C₂³

—

106

C₂³

—

106

—

107

C₂³

—

106

—

108

C₂³

C₄²×D₅₃

yes

characteristic

109

C₂×C₄

—

110

C₂×C₄

—

106

—

111

C₂×C₄

—

106

—

112

C₂×C₄

—

106

—

113

C₂×C₄

—

106

—

114

C₂×C₄

C₂²⋊C₄×D₅₃

yes

—

115

C₂×C₄

C₄⋊C₄×D₅₃

yes

—

116

C₂×C₄

#13133223

yes

—

117

C₂×C₄

#13133225

yes

—

118

C₈

—

119

C₈

—

212

—

120

C₂²

—

106

—

121

C₂²

—

212

—

122

C₂²

—

212

—

123

C₂²

#95×D₅₃

yes

characteristic

124

C₂²

#16810577

yes

—

125

C₂²

#16810580

yes

—

126

C₄

—

106

—

127

C₄

#16810588

yes

—

128

C₄

#16810648

yes

—

129

C₂

—

212

—

130

C₂

#20273014

yes

characteristic

131

C₂

#20273016

yes

—

132

C₂

#20274410

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂²×D₅₃
commutator, socle	C₂³×C₅₃	C₂×C₄²
Fitting	C₅₃×#1273	C₂
prime radical	C₂³×D₅₃	C₄²
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1273	D₅₃
hypocenter	C₅₃	#7851
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄² ◁ #1273

subgroup

G/H

center, Frattini

C₂×C₄²

C₂²×D₅₃

commutator, socle

C₂³×C₅₃

C₂×C₄²

Fitting

C₅₃×#1273

C₂

prime radical

C₂³×D₅₃

C₄²

cosocle

C₂×C₄²×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₂³×C₅₃ ▷ C₅₃

C₁ ◁ C₂×C₄² ◁ #1273

N	H ≅ G/N	type
2. #20272542	C₂	semidirect
3. C₅₃×#1273	C₂	semidirect
65. #1273	D₅₃	semidirect
76. C₅₃	#7851	semidirect

H ≅ G/N

type

2. #20272542

C₂

semidirect

3. C₅₃×#1273

C₂

semidirect

65. #1273

D₅₃

semidirect

76. C₅₃

#7851

semidirect

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

subgroup	index
C₂×#16810674	2
#20272542	2
C₅₃×#1273	2
#7851	53

subgroup

index

C₂×#16810674

#20272542

C₅₃×#1273

#7851

Subgroups — #27273482

Conjugacy classes of subgroups

Complements

Maximal subgroups