Group #20273429 — subgroups

Subgroups — #20273429

Order 6784.

332 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 85 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	#16810680	C₂	1	2	yes	maximal, complemented
2	#16810367	C₂	1	4	yes	maximal, complemented
3	C₂×#127×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₄⋊#88322	C₂²	1	4	yes	complemented
5	C₂²⋊#88322	—	2	2	no	—
6	#13133128	—	2	2	no	—
7	C₂²×D₄×C₅₃	—	2	2	no	—
8	C₂²×D₄×C₅₃	C₄	1	2	yes	—
9	#127×C₅₃	—	2	2	no	—
10	#127×C₅₃	C₄	1	2	yes	—
11	#127×C₅₃	C₂²	1	2	yes	—
12	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
13	C₂⁴×C₅₃	—	2	2	no	—
14	C₂×D₄×C₅₃	—	2	2	no	—
15	C₂×D₄×C₅₃	—	2	4	no	—
16	C₂×D₄×C₅₃	—	2	4	no	—
17	C₂×D₄×C₅₃	—	4	8	no	—
18	C₂×D₄×C₅₃	—	4	4	no	—
19	C₂×D₄×C₅₃	C₂×C₄	1	4	yes	—
20	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₄×C₅₃	D₄	1	2	yes	—
22	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
23	C₄²×C₅₃	D₄	1	2	yes	—
24	C₄²×C₅₃	C₂³	1	1	yes	characteristic
25	C₂×#88322	—	2	4	no	—
26	C₂³×C₅₃	—	2	4	no	—
27	C₂³×C₅₃	—	4	8	no	—
28	C₂³×C₅₃	—	4	4	no	—
29	C₂³×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
30	D₄×C₅₃	—	4	4	no	—
31	D₄×C₅₃	—	4	4	no	—
32	D₄×C₅₃	—	4	8	no	—
33	D₄×C₅₃	—	4	8	no	—
34	C₂×C₄×C₅₃	—	2	2	no	—
35	C₂×C₄×C₅₃	—	2	4	no	—
36	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
37	C₂×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
38	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
40	#88322	—	4	4	no	—
41	C₂²×C₅₃	—	2	2	no	—
42	C₂²×C₅₃	—	4	4	no	—
43	C₂²×C₅₃	—	4	8	no	—
44	C₂²×C₅₃	—	8	8	no	—
45	C₂²×C₅₃	#100	1	2	yes	—
46	C₂²×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
47	C₄×C₅₃	—	2	1	no	—
48	C₄×C₅₃	—	2	1	no	—
49	C₄×C₅₃	—	2	2	no	—
50	C₄×C₅₃	AGL(1,ℤ/8)	1	4	yes	—
51	#1555	—	53	1	no	maximal, Sylow
52	C₂×C₅₃	—	2	2	no	—
53	C₂×C₅₃	—	8	4	no	—
54	C₂×C₅₃	C₂×#100	1	1	yes	characteristic
55	C₂×C₅₃	#418	1	2	yes	—
56	C₂×#127	D₅₃	1	1	yes	characteristic
57	#432	—	53	2	no	—
58	#331	—	53	4	no	—
59	C₅₃	#1555	1	1	yes	characteristic, complemented, Sylow
60	C₂²×D₄	—	2	2	no	—
61	C₂²×D₄	Dic₅₃	1	2	yes	—
62	#127	—	2	2	no	—
63	#127	Dic₅₃	1	2	yes	—
64	#127	C₂×D₅₃	1	2	yes	—
65	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
66	C₄⋊C₈	—	53	4	no	—
67	C₂²⋊C₈	—	106	2	no	—
68	#97	—	106	2	no	—
69	C₂⁴	—	2	2	no	—
70	C₂×D₄	—	2	4	no	—
71	C₂×D₄	—	2	2	no	—
72	C₂×D₄	—	2	4	no	—
73	C₂×D₄	—	4	4	no	—
74	C₂×D₄	—	4	8	no	—
75	C₂×D₄	C₂×Dic₅₃	1	4	yes	—
76	C₂²×C₄	C₂×Dic₅₃	1	1	yes	characteristic
77	C₂²×C₄	C₂²⋊D₅₃	1	2	yes	—
78	C₂×C₈	—	106	4	no	—
79	C₄²	C₂×Dic₅₃	1	1	yes	characteristic
80	C₄²	C₂²⋊D₅₃	1	2	yes	—
81	C₄²	C₂²×D₅₃	1	1	yes	characteristic
82	C₂³	—	2	4	no	—
83	C₂³	—	4	8	no	—
84	C₂³	—	4	4	no	—
85	C₂³	C₂²⋊Dic₅₃	1	1	yes	characteristic
86	D₄	—	4	4	no	—
87	D₄	—	4	4	no	—
88	D₄	—	4	8	no	—
89	D₄	—	4	8	no	—
90	C₂×C₄	—	2	2	no	—
91	C₂×C₄	—	2	4	no	—
92	C₂×C₄	C₂²⋊Dic₅₃	1	2	yes	—
93	C₂×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
94	C₂×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
95	C₂×C₄	C₂×C₂²⋊D₅₃	1	2	yes	—
96	C₈	—	212	4	no	—
97	C₂²	—	2	2	no	—
98	C₂²	—	4	4	no	—
99	C₂²	—	4	8	no	—
100	C₂²	—	8	8	no	—
101	C₂²	#13133134	1	2	yes	—
102	C₂²	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
103	C₄	—	2	1	no	—
104	C₄	—	2	1	no	—
105	C₄	—	2	2	no	—
106	C₄	#13133243	1	4	yes	—
107	C₂	—	2	2	no	—
108	C₂	—	8	4	no	—
109	C₂	C₂×#13133134	1	1	yes	characteristic
110	C₂	#16810628	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16810680

C₂

yes

maximal, complemented

#16810367

C₂

yes

maximal, complemented

C₂×#127×C₅₃

C₂

yes

maximal, characteristic

C₄⋊#88322

C₂²

yes

complemented

C₂²⋊#88322

—

#13133128

—

C₂²×D₄×C₅₃

—

C₂²×D₄×C₅₃

C₄

yes

—

#127×C₅₃

—

#127×C₅₃

C₄

yes

—

#127×C₅₃

C₂²

yes

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₂⁴×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

D₄

yes

—

C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄²×C₅₃

D₄

yes

—

C₄²×C₅₃

C₂³

yes

characteristic

C₂×#88322

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

C₂²⋊C₄

yes

characteristic

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²⋊C₄

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#100

yes

—

C₂²×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

AGL(1,ℤ/8)

yes

—

#1555

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

C₂×#100

yes

characteristic

C₂×C₅₃

#418

yes

—

C₂×#127

D₅₃

yes

characteristic

#432

—

#331

—

C₅₃

#1555

yes

characteristic, complemented, Sylow

C₂²×D₄

—

C₂²×D₄

Dic₅₃

yes

—

#127

—

#127

Dic₅₃

yes

—

#127

C₂×D₅₃

yes

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

C₄⋊C₈

—

C₂²⋊C₈

—

106

—

#97

—

106

—

C₂⁴

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×Dic₅₃

yes

—

C₂²×C₄

C₂×Dic₅₃

yes

characteristic

C₂²×C₄

C₂²⋊D₅₃

yes

—

C₂×C₈

—

106

—

C₄²

C₂×Dic₅₃

yes

characteristic

C₄²

C₂²⋊D₅₃

yes

—

C₄²

C₂²×D₅₃

yes

characteristic

C₂³

—

C₂³

—

C₂³

—

C₂³

C₂²⋊Dic₅₃

yes

characteristic

D₄

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²⋊Dic₅₃

yes

—

C₂×C₄

C₂²⋊Dic₅₃

yes

characteristic

C₂×C₄

C₂²×Dic₅₃

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₅₃

yes

—

C₈

—

212

—

C₂²

—

C₂²

—

C₂²

—

100

C₂²

—

101

C₂²

#13133134

yes

—

102

C₂²

C₂×C₂²⋊Dic₅₃

yes

characteristic

103

C₄

—

104

C₄

—

105

C₄

—

106

C₄

#13133243

yes

—

107

C₂

—

108

C₂

—

109

C₂

C₂×#13133134

yes

characteristic

110

C₂

#16810628

yes

—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊Dic₅₃
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₄²	C₂²×D₅₃
Fitting	C₂×#127×C₅₃	C₂
socle	C₂²×C₅₃	C₂×C₂²⋊C₄
hypercenter	C₂×#127	D₅₃
hypocenter	C₅₃	#1555
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#127

subgroup

G/H

center

C₂²

C₂×C₂²⋊Dic₅₃

commutator

C₂×C₄×C₅₃

C₂²×C₄

Frattini

C₄²

C₂²×D₅₃

Fitting

C₂×#127×C₅₃

C₂

socle

C₂²×C₅₃

C₂×C₂²⋊C₄

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#127

N	H ≅ G/N	type
1. #16810680	C₂	semidirect
2. #16810367	C₂	semidirect
4. C₄⋊#88322	C₂²	semidirect
59. C₅₃	#1555	semidirect

H ≅ G/N

type

1. #16810680

C₂

semidirect

2. #16810367

C₂

semidirect

4. C₄⋊#88322

C₂²

semidirect

59. C₅₃

#1555

semidirect

Subgroups — #20273429

Conjugacy classes of subgroups

Complements