Group #27276548 — subgroups

Subgroups — #27276548

Order 13568.

656 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 656 conjugacy classes of subgroups — too many to draw. Only its 147 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	#20275685	C₂	1	3	yes	maximal
2	C₂×#16810365	C₂	1	3	yes	maximal
3	C₂×C₅₃×#564	C₂	1	1	yes	maximal, characteristic
4	C₂×#13133128	C₂²	1	3	yes	—
5	#16810493	C₂²	1	1	yes	characteristic
6	#16810385	—	2	6	no	—
7	#16810365	—	2	6	no	—
8	C₅₃×#564	—	2	6	no	—
9	C₅₃×#564	C₄	1	4	yes	complemented
10	C₂×#125×C₅₃	—	2	6	no	—
11	C₂×#125×C₅₃	C₂²	1	3	yes	—
12	C₂²×#88322	—	2	3	no	—
13	C₂×C₄⋊Dic₅₃	—	2	3	no	—
14	#13133128	—	2	6	no	—
15	#125×C₅₃	—	2	24	no	—
16	#125×C₅₃	—	2	24	no	—
17	#125×C₅₃	—	2	6	no	—
18	#125×C₅₃	C₂×C₄	1	6	yes	—
19	#125×C₅₃	D₄	1	6	yes	—
20	C₂×C₄⋊C₄×C₅₃	—	2	6	no	—
21	C₂×C₄⋊C₄×C₅₃	—	2	6	no	—
22	C₂×C₄⋊C₄×C₅₃	D₄	1	6	yes	—
23	C₂×C₄²×C₅₃	—	2	2	no	—
24	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
25	C₂²×C₄×C₅₃	—	2	6	no	—
26	C₂²×C₄×C₅₃	C₂×D₄	1	3	yes	—
27	C₂²×Dic₅₃	—	4	1	no	—
28	C₄⋊C₄×C₅₃	—	2	24	no	—
29	C₄⋊C₄×C₅₃	—	2	12	no	—
30	C₄⋊C₄×C₅₃	—	2	12	no	—
31	C₄⋊C₄×C₅₃	—	2	6	no	—
32	C₄⋊C₄×C₅₃	C₂²⋊C₄	1	12	yes	—
33	C₄²×C₅₃	—	2	8	no	—
34	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
35	C₄²×C₅₃	C₂×D₄	1	3	yes	—
36	C₄⋊Dic₅₃	—	4	6	no	—
37	C₂×#88322	—	2	6	no	—
38	C₂×#88322	—	4	3	no	—
39	C₂×#88322	—	4	3	no	—
40	C₂³×C₅₃	#120	1	1	yes	characteristic
41	C₂×C₄×C₅₃	—	2	3	no	—
42	C₂×C₄×C₅₃	—	2	3	no	—
43	C₂×C₄×C₅₃	—	2	12	no	—
44	C₂×C₄×C₅₃	—	4	6	no	—
45	C₂×C₄×C₅₃	—	4	6	no	—
46	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	3	yes	—
47	C₂×C₄×C₅₃	#120	1	3	yes	—
48	C₂×Dic₅₃	—	8	3	no	—
49	#88322	—	4	6	no	—
50	#8509	—	53	1	no	maximal, Sylow
51	C₂²×C₅₃	#379	1	1	yes	characteristic
52	C₂²×C₅₃	#454	1	3	yes	—
53	C₂²×C₅₃	#456	1	3	yes	—
54	C₄×C₅₃	—	2	3	no	—
55	C₄×C₅₃	—	2	3	no	—
56	C₄×C₅₃	—	4	12	no	—
57	Dic₅₃	—	16	1	no	—
58	C₂×#564	D₅₃	1	1	yes	characteristic
59	#1992	—	53	3	no	—
60	C₂×#330	—	53	3	no	—
61	C₂×C₅₃	#1658	1	1	yes	characteristic
62	C₂×C₅₃	#1663	1	3	yes	—
63	C₂×C₅₃	#1793	1	3	yes	—
64	#564	—	2	6	no	—
65	#564	Dic₅₃	1	4	yes	complemented
66	C₂×#125	—	2	6	no	—
67	C₂×#125	C₂×D₅₃	1	3	yes	—
68	C₂×#97	—	53	3	no	—
69	#384	—	53	1	no	—
70	#340	—	106	6	no	—
71	#330	—	106	6	no	—
72	C₅₃	#8509	1	1	yes	characteristic, complemented, Sylow
73	C₂²×C₈	—	106	3	no	—
74	#125	—	2	24	no	—
75	#125	—	2	24	no	—
76	#125	—	2	6	no	—
77	#125	C₂×Dic₅₃	1	6	yes	—
78	#125	C₂²⋊D₅₃	1	6	yes	—
79	C₂×C₄⋊C₄	—	2	6	no	—
80	C₂×C₄⋊C₄	—	2	6	no	—
81	C₂×C₄⋊C₄	—	106	3	no	—
82	C₂×C₄⋊C₄	C₂²⋊D₅₃	1	6	yes	—
83	C₂×C₄²	—	2	2	no	—
84	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
85	#97	—	106	6	no	—
86	C₂²×C₄	—	2	6	no	—
87	C₂²×C₄	—	212	1	no	—
88	C₂²×C₄	C₂×C₂²⋊D₅₃	1	3	yes	—
89	C₂×C₈	—	106	6	no	—
90	C₂×C₈	—	212	3	no	—
91	C₂×C₈	—	212	3	no	—
92	C₄⋊C₄	—	2	24	no	—
93	C₄⋊C₄	—	2	12	no	—
94	C₄⋊C₄	—	2	12	no	—
95	C₄⋊C₄	—	2	6	no	—
96	C₄⋊C₄	—	212	6	no	—
97	C₄⋊C₄	C₂²⋊Dic₅₃	1	12	yes	—
98	C₄²	—	2	8	no	—
99	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
100	C₄²	C₂×C₂²⋊D₅₃	1	3	yes	—
101	C₂³	#13133186	1	1	yes	characteristic
102	C₂×C₄	—	2	3	no	—
103	C₂×C₄	—	2	3	no	—
104	C₂×C₄	—	2	12	no	—
105	C₂×C₄	—	4	6	no	—
106	C₂×C₄	—	4	6	no	—
107	C₂×C₄	—	424	3	no	—
108	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	3	yes	—
109	C₂×C₄	#13133186	1	3	yes	—
110	C₈	—	212	6	no	—
111	C₂²	#16810479	1	1	yes	characteristic
112	C₂²	#16810796	1	3	yes	—
113	C₂²	#16810806	1	3	yes	—
114	C₄	—	2	3	no	—
115	C₄	—	2	3	no	—
116	C₄	—	4	12	no	—
117	C₄	—	848	1	no	—
118	C₂	#20274017	1	1	yes	characteristic
119	C₂	#20274043	1	3	yes	—
120	C₂	#20274674	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20275685

C₂

yes

maximal

C₂×#16810365

C₂

yes

maximal

C₂×C₅₃×#564

C₂

yes

maximal, characteristic

C₂×#13133128

C₂²

yes

—

#16810493

C₂²

yes

characteristic

#16810385

—

#16810365

—

C₅₃×#564

—

C₅₃×#564

C₄

yes

complemented

C₂×#125×C₅₃

—

C₂×#125×C₅₃

C₂²

yes

—

C₂²×#88322

—

C₂×C₄⋊Dic₅₃

—

#13133128

—

#125×C₅₃

—

#125×C₅₃

—

#125×C₅₃

—

#125×C₅₃

C₂×C₄

yes

—

#125×C₅₃

D₄

yes

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

D₄

yes

—

C₂×C₄²×C₅₃

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂×D₄

yes

—

C₂²×Dic₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

C₂²⋊C₄

yes

—

C₄²×C₅₃

—

C₄²×C₅₃

C₂²×C₄

yes

characteristic

C₄²×C₅₃

C₂×D₄

yes

—

C₄⋊Dic₅₃

—

C₂×#88322

—

C₂×#88322

—

C₂×#88322

—

C₂³×C₅₃

#120

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×C₂²⋊C₄

yes

—

C₂×C₄×C₅₃

#120

yes

—

C₂×Dic₅₃

—

—

—

C₂²×C₅₃

yes

C₂²×C₅₃

yes

—

C₂²×C₅₃

#456

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

Dic₅₃

—

C₂×#564

D₅₃

yes

characteristic

#1992

—

C₂×#330

—

C₂×C₅₃

#1658

yes

characteristic

C₂×C₅₃

#1663

yes

—

C₂×C₅₃

#1793

yes

—

#564

—

#564

Dic₅₃

yes

complemented

C₂×#125

—

C₂×#125

C₂×D₅₃

yes

—

C₂×#97

—

#384

—

#340

—

106

—

#330

—

106

—

C₅₃

#8509

yes

characteristic, complemented, Sylow

C₂²×C₈

—

106

—

—

—

—

C₂×Dic₅₃

yes

—

#125

C₂²⋊D₅₃

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

106

—

C₂×C₄⋊C₄

C₂²⋊D₅₃

yes

—

C₂×C₄²

—

C₂×C₄²

C₂²×D₅₃

yes

characteristic

#97

—

106

—

C₂²×C₄

—

C₂²×C₄

—

212

—

C₂²×C₄

C₂×C₂²⋊D₅₃

yes

—

C₂×C₈

—

106

—

C₂×C₈

—

212

—

C₂×C₈

—

212

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

212

—

C₄⋊C₄

C₂²⋊Dic₅₃

yes

—

C₄²

—

C₄²

C₂²×Dic₅₃

yes

characteristic

100

C₄²

C₂×C₂²⋊D₅₃

yes

—

101

C₂³

#13133186

yes

characteristic

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

424

—

108

C₂×C₄

C₂×C₂²⋊Dic₅₃

yes

—

109

C₂×C₄

#13133186

yes

—

110

C₈

—

212

—

111

C₂²

#16810479

yes

characteristic

112

C₂²

#16810796

yes

—

113

C₂²

#16810806

yes

—

114

C₄

—

115

C₄

—

116

C₄

—

117

C₄

—

848

—

118

C₂

#20274017

yes

characteristic

119

C₂

#20274043

yes

—

120

C₂

#20274674

yes

—

	subgroup	G/H
center	C₂³	#13133186
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₂×C₅₃×#564	C₂
socle, prime radical	C₂³×C₅₃	#120
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₂×#564	D₅₃
hypocenter	C₅₃	#8509
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#564

subgroup

G/H

center

C₂³

#13133186

commutator

C₄²×C₅₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅₃

Fitting

C₂×C₅₃×#564

C₂

socle, prime radical

C₂³×C₅₃

#120

cosocle

C₂×C₄²×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#564

N	H ≅ G/N	type
9. C₅₃×#564	C₄	semidirect
65. #564	Dic₅₃	semidirect
72. C₅₃	#8509	semidirect

H ≅ G/N

type

9. C₅₃×#564

C₄

semidirect

65. #564

Dic₅₃

semidirect

72. C₅₃

#8509

semidirect

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

subgroup	index
#20275685	2
C₂×#16810365	2
C₂×C₅₃×#564	2
#8509	53

subgroup

index

#20275685

C₂×#16810365

C₂×C₅₃×#564

#8509

Subgroups — #27276548

Conjugacy classes of subgroups

Complements

Maximal subgroups