Group #20273539 — subgroups

Subgroups — #20273539

Order 6784.

192 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 192 conjugacy classes of subgroups — too many to draw. Only its 87 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	D₄×#88322	C₂	1	2	yes	maximal, complemented
2	#16810374	C₂	1	1	yes	maximal, characteristic, complemented
3	D₄⋊#88322	C₂	1	2	yes	maximal, complemented
4	#16810344	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×D₈×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×#88322	—	2	2	no	—
7	C₄⋊#88322	C₂²	1	2	yes	—
8	C₂²⋊#88322	—	2	2	no	—
9	C₄×#88322	C₂²	1	1	yes	characteristic
10	C₄×#88322	C₂²	1	1	yes	characteristic
11	C₂×D₈×C₅₃	C₄	1	1	yes	characteristic
12	C₄×D₄×C₅₃	C₂²	1	2	yes	—
13	#107×C₅₃	C₄	1	1	yes	characteristic
14	#102×C₅₃	C₄	1	2	yes	—
15	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×D₄×C₅₃	C₂×C₄	1	2	yes	—
17	C₂²×C₄×C₅₃	—	2	2	no	—
18	D₈×C₅₃	C₈	1	4	yes	complemented
19	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
20	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
21	C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
22	C₂²⋊C₄×C₅₃	—	2	2	no	—
23	C₄²×C₅₃	C₂³	1	1	yes	characteristic
24	C₂×#88322	—	2	2	no	—
25	C₂×#88322	—	2	1	no	—
26	C₂×#88322	—	4	4	no	—
27	C₂×#88322	D₄	1	2	yes	—
28	C₂³×C₅₃	—	2	2	no	—
29	D₄×C₅₃	—	2	2	no	—
30	D₄×C₅₃	C₂×C₈	1	4	yes	complemented
31	C₂×C₄×C₅₃	—	2	2	no	—
32	C₂×C₄×C₅₃	—	4	2	no	—
33	C₂×C₄×C₅₃	—	4	2	no	—
34	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
37	C₈×C₅₃	—	2	1	no	—
38	C₈×C₅₃	C₂×C₈	1	2	yes	complemented
39	#88322	—	2	2	no	—
40	#88322	—	4	1	no	—
41	#88322	—	4	2	no	—
42	C₂²×C₅₃	—	2	4	no	—
43	C₂²×C₅₃	—	4	2	no	—
44	C₂²×C₅₃	—	4	2	no	—
45	C₂²×C₅₃	C₄×D₄	1	1	yes	characteristic
46	C₄×C₅₃	—	2	1	no	—
47	C₄×C₅₃	—	4	2	no	—
48	C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
49	C₄×C₅₃	#131	1	1	yes	characteristic
50	C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
51	#1580	—	53	1	no	maximal, Sylow
52	C₂×C₅₃	—	4	4	no	—
53	C₂×C₅₃	C₈×D₄	1	1	yes	characteristic
54	C₂×C₅₃	#442	1	1	yes	characteristic
55	C₂×C₅₃	#444	1	1	yes	characteristic
56	C₄×D₈	D₅₃	1	1	yes	characteristic
57	C₈×D₄	—	53	2	no	—
58	#334	—	53	1	no	—
59	D₄⋊C₈	—	53	2	no	—
60	#322	—	53	1	no	—
61	C₅₃	#1580	1	1	yes	characteristic, complemented, Sylow
62	C₂×D₈	Dic₅₃	1	1	yes	characteristic
63	C₂²×C₈	—	106	2	no	—
64	C₄×D₄	C₂×D₅₃	1	2	yes	—
65	#107	Dic₅₃	1	1	yes	characteristic
66	C₄⋊C₈	—	53	2	no	—
67	#102	Dic₅₃	1	2	yes	—
68	C₂²⋊C₈	—	106	2	no	—
69	C₄×C₈	—	53	1	no	—
70	C₄×C₈	—	53	1	no	—
71	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
72	C₂×D₄	C₂×Dic₅₃	1	2	yes	—
73	C₂²×C₄	—	2	2	no	—
74	D₈	#88322	1	4	yes	complemented
75	C₂×C₈	—	53	2	no	—
76	C₂×C₈	—	106	2	no	—
77	C₂×C₈	—	106	1	no	—
78	C₂×C₈	—	212	4	no	—
79	C₂×C₈	C₂×Dic₅₃	1	1	yes	characteristic
80	C₂×C₈	C₂×Dic₅₃	1	1	yes	characteristic
81	C₄⋊C₄	C₂×Dic₅₃	1	2	yes	—
82	C₂²⋊C₄	—	2	2	no	—
83	C₄²	C₂²×D₅₃	1	1	yes	characteristic
84	C₂³	—	2	2	no	—
85	D₄	—	2	2	no	—
86	D₄	C₂×#88322	1	4	yes	complemented
87	C₂×C₄	—	2	2	no	—
88	C₂×C₄	—	4	2	no	—
89	C₂×C₄	—	4	2	no	—
90	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
91	C₂×C₄	#11719947	1	1	yes	characteristic
92	C₂×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
93	C₈	—	2	1	no	—
94	C₈	—	106	2	no	—
95	C₈	—	212	1	no	—
96	C₈	—	212	2	no	—
97	C₈	C₂×#88322	1	2	yes	complemented
98	C₂²	—	2	4	no	—
99	C₂²	—	4	2	no	—
100	C₂²	—	4	2	no	—
101	C₂²	D₄×Dic₅₃	1	1	yes	characteristic
102	C₄	—	2	1	no	—
103	C₄	—	4	2	no	—
104	C₄	C₂²×#88322	1	1	yes	characteristic
105	C₄	#13133224	1	1	yes	characteristic
106	C₄	#13133241	1	2	yes	—
107	C₂	—	4	4	no	—
108	C₂	D₄×#88322	1	1	yes	characteristic
109	C₂	#16810735	1	1	yes	characteristic
110	C₂	#16810745	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

D₄×#88322

C₂

yes

maximal, complemented

#16810374

C₂

yes

maximal, characteristic, complemented

D₄⋊#88322

C₂

yes

maximal, complemented

#16810344

C₂

yes

maximal, characteristic, complemented

C₄×D₈×C₅₃

C₂

yes

maximal, characteristic

C₂²×#88322

—

C₄⋊#88322

C₂²

yes

—

C₂²⋊#88322

—

C₄×#88322

C₂²

yes

characteristic

C₄×#88322

C₂²

yes

characteristic

C₂×D₈×C₅₃

C₄

yes

characteristic

C₄×D₄×C₅₃

C₂²

yes

—

#107×C₅₃

C₄

yes

characteristic

#102×C₅₃

C₄

yes

—

C₄×C₈×C₅₃

C₂²

yes

characteristic

C₂×D₄×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

—

D₈×C₅₃

C₈

yes

complemented

C₂×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂×C₈×C₅₃

C₂×C₄

yes

characteristic

C₄⋊C₄×C₅₃

C₂×C₄

yes

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

C₂³

yes

—

—

—

D₄

yes

—

C₂³×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

C₂×C₈

yes

complemented

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

#41

yes

characteristic

C₈×C₅₃

—

C₈×C₅₃

C₂×C₈

yes

—

—

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₄×D₄

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂²×C₈

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

—

#1580

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

C₈×D₄

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₄×D₈

D₅₃

yes

characteristic

C₈×D₄

—

#334

—

D₄⋊C₈

—

#322

—

C₅₃

#1580

yes

characteristic, complemented, Sylow

C₂×D₈

Dic₅₃

yes

characteristic

C₂²×C₈

—

106

—

C₄×D₄

C₂×D₅₃

yes

—

#107

Dic₅₃

yes

characteristic

C₄⋊C₈

—

#102

Dic₅₃

yes

—

C₂²⋊C₈

—

106

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₅₃

yes

characteristic

C₂×D₄

C₂×Dic₅₃

yes

—

C₂²×C₄

—

D₈

#88322

yes

complemented

C₂×C₈

—

C₂×C₈

—

106

—

C₂×C₈

—

106

—

C₂×C₈

—

212

—

C₂×C₈

C₂×Dic₅₃

yes

characteristic

C₂×C₈

C₂×Dic₅₃

yes

characteristic

C₄⋊C₄

C₂×Dic₅₃

yes

—

C₂²⋊C₄

—

C₄²

C₂²×D₅₃

yes

characteristic

C₂³

—

D₄

—

D₄

C₂×#88322

yes

complemented

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

D₄×D₅₃

yes

characteristic

C₂×C₄

#11719947

yes

characteristic

C₂×C₄

C₂²×Dic₅₃

yes

characteristic

C₈

—

C₈

—

106

—

C₈

—

212

—

C₈

—

212

—

C₈

C₂×#88322

yes

complemented

C₂²

—

C₂²

—

100

C₂²

—

101

C₂²

D₄×Dic₅₃

yes

characteristic

102

C₄

—

103

C₄

—

104

C₄

C₂²×#88322

yes

characteristic

105

C₄

#13133224

yes

characteristic

106

C₄

#13133241

yes

—

107

C₂

—

108

C₂

D₄×#88322

yes

characteristic

109

C₂

#16810735

yes

characteristic

110

C₂

#16810745

yes

characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₅₃
commutator	C₄×C₅₃	C₂²×C₈
Frattini	C₄²	C₂²×D₅₃
Fitting	C₄×D₈×C₅₃	C₂
socle	C₂²×C₅₃	C₄×D₄
hypercenter	C₄×D₈	D₅₃
hypocenter	C₅₃	#1580
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×D₈

subgroup

G/H

center

C₂×C₄

D₄×D₅₃

commutator

C₄×C₅₃

C₂²×C₈

Frattini

C₄²

C₂²×D₅₃

Fitting

C₄×D₈×C₅₃

C₂

socle

C₂²×C₅₃

C₄×D₄

hypercenter

C₄×D₈

D₅₃

hypocenter

C₅₃

#1580

LCS

G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×D₈

N	H ≅ G/N	type
1. D₄×#88322	C₂	semidirect
2. #16810374	C₂	semidirect
3. D₄⋊#88322	C₂	semidirect
4. #16810344	C₂	semidirect
18. D₈×C₅₃	C₈	semidirect
30. D₄×C₅₃	C₂×C₈	semidirect
38. C₈×C₅₃	C₂×C₈	semidirect
61. C₅₃	#1580	semidirect
74. D₈	#88322	semidirect
86. D₄	C₂×#88322	semidirect
97. C₈	C₂×#88322	semidirect

H ≅ G/N

type

1. D₄×#88322

C₂

semidirect

2. #16810374

C₂

semidirect

3. D₄⋊#88322

C₂

semidirect

4. #16810344

C₂

semidirect

18. D₈×C₅₃

C₈

semidirect

30. D₄×C₅₃

C₂×C₈

semidirect

38. C₈×C₅₃

C₂×C₈

semidirect

61. C₅₃

#1580

semidirect

74. D₈

#88322

semidirect

86. D₄

C₂×#88322

semidirect

97. C₈

C₂×#88322

semidirect

Subgroups — #20273539

Conjugacy classes of subgroups

Complements