Group #20277000 — subgroups

Subgroups — #20277000

Order 6784.

120 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 120 conjugacy classes of subgroups — too many to draw. Only its 53 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	#16811027	C₂	1	1	yes	maximal, characteristic
2	#16811024	C₂	1	1	yes	maximal, characteristic
3	#16810450	C₂	1	1	yes	maximal, characteristic
4	#16810449	C₂	1	1	yes	maximal, characteristic
5	#16810448	C₂	1	1	yes	maximal, characteristic
6	#16810398	C₂	1	1	yes	maximal, characteristic
7	C₅₃×#367	C₂	1	1	yes	maximal, characteristic
8	#13133216	—	2	1	no	—
9	#13133208	—	2	1	no	—
10	C₂×#11719908	C₂²	1	1	yes	characteristic
11	#13133151	C₂²	1	1	yes	characteristic
12	#13133151	C₂²	1	1	yes	characteristic
13	#13133150	—	2	1	no	—
14	#13133148	—	2	1	no	—
15	C₈×Dic₅₃	C₂²	1	1	yes	characteristic
16	C₂×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
17	#107×C₅₃	C₂²	1	1	yes	characteristic
18	#107×C₅₃	C₂²	1	1	yes	characteristic
19	C₂×Dic₁₀₆	—	4	1	no	—
20	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
21	C₄⋊C₄×C₅₃	—	2	1	no	—
22	C₄⋊C₄×C₅₃	—	2	1	no	—
23	C₄⋊Dic₅₃	—	2	1	no	—
24	C₄⋊Dic₅₃	—	2	1	no	—
25	C₄⋊Dic₅₃	—	4	1	no	—
26	#11719919	—	4	1	no	—
27	#11719919	—	4	1	no	—
28	C₄×Dic₅₃	D₄	1	1	yes	characteristic
29	C₂×#88322	D₄	1	1	yes	characteristic
30	C₁₆×C₅₃	Q₈	1	2	yes	complemented
31	#11719908	Q₈	1	2	yes	complemented
32	C₂×C₄×C₅₃	—	4	1	no	—
33	C₂×C₄×C₅₃	—	4	1	no	—
34	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
35	C₂×Dic₅₃	—	4	1	no	—
36	C₂×Dic₅₃	—	4	1	no	—
37	C₂×Dic₅₃	D₈	1	1	yes	characteristic
38	C₂×Dic₅₃	D₈	1	1	yes	characteristic
39	Dic₁₀₆	—	8	2	no	—
40	C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
41	C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
42	#88322	Q₁₆	1	2	yes	—
43	C₂²×C₅₃	C₂×D₈	1	1	yes	characteristic
44	C₄×C₅₃	—	8	1	no	—
45	C₄×C₅₃	—	8	1	no	—
46	C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
47	C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
48	Dic₅₃	—	2	1	no	—
49	Dic₅₃	—	2	1	no	—
50	Dic₅₃	—	8	1	no	—
51	Dic₅₃	—	8	1	no	—
52	#2255	—	53	1	no	maximal, Sylow
53	C₂×C₅₃	#500	1	1	yes	characteristic
54	C₂×C₅₃	#509	1	1	yes	characteristic
55	C₂×C₅₃	#510	1	1	yes	characteristic
56	#500	—	53	1	no	—
57	#499	—	53	1	no	—
58	#367	—	53	1	no	—
59	#367	D₅₃	1	1	yes	characteristic
60	#366	—	53	1	no	—
61	#366	—	53	1	no	—
62	#346	—	53	1	no	—
63	C₅₃	#2255	1	1	yes	characteristic, complemented, Sylow
64	C₄⋊Q₈	—	106	1	no	—
65	#125	—	106	1	no	—
66	C₂×C₁₆	—	53	1	no	—
67	C₂×C₁₆	C₂×D₅₃	1	1	yes	characteristic
68	#107	—	53	1	no	—
69	#107	—	53	1	no	—
70	#107	—	106	1	no	—
71	#107	C₂×D₅₃	1	1	yes	characteristic
72	#107	C₂×D₅₃	1	1	yes	characteristic
73	#106	—	106	1	no	—
74	C₄×C₈	—	53	1	no	—
75	C₂×Q₈	—	212	1	no	—
76	C₂×C₈	—	53	1	no	—
77	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
78	C₄⋊C₄	—	2	1	no	—
79	C₄⋊C₄	—	2	1	no	—
80	C₄⋊C₄	—	106	1	no	—
81	C₄⋊C₄	—	106	1	no	—
82	C₄⋊C₄	—	212	1	no	—
83	C₄⋊C₄	—	212	1	no	—
84	C₄⋊C₄	—	212	1	no	—
85	C₄²	—	53	1	no	—
86	C₁₆	—	53	2	no	—
87	C₁₆	Dic₁₀₆	1	2	yes	complemented
88	Q₈	—	424	2	no	—
89	C₂×C₄	—	4	1	no	—
90	C₂×C₄	—	4	1	no	—
91	C₂×C₄	—	53	1	no	—
92	C₂×C₄	—	53	1	no	—
93	C₂×C₄	—	212	1	no	—
94	C₂×C₄	—	212	1	no	—
95	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
96	C₈	—	53	2	no	—
97	C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
98	C₈	Q₈×D₅₃	1	1	yes	characteristic
99	C₂²	D₈×D₅₃	1	1	yes	characteristic
100	C₄	—	8	1	no	—
101	C₄	—	8	1	no	—
102	C₄	—	106	1	no	—
103	C₄	—	106	1	no	—
104	C₄	—	424	1	no	—
105	C₄	—	424	1	no	—
106	C₄	#13133216	1	1	yes	characteristic
107	C₄	Q₁₆×D₅₃	1	1	yes	characteristic
108	C₂	#16811027	1	1	yes	characteristic
109	C₂	#16811060	1	1	yes	characteristic
110	C₂	#16811062	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811027

C₂

yes

maximal, characteristic

#16811024

C₂

yes

maximal, characteristic

#16810450

C₂

yes

maximal, characteristic

#16810449

C₂

yes

maximal, characteristic

#16810448

C₂

yes

maximal, characteristic

#16810398

C₂

yes

maximal, characteristic

C₅₃×#367

C₂

yes

maximal, characteristic

#13133216

—

#13133208

—

C₂×#11719908

C₂²

yes

characteristic

#13133151

C₂²

yes

characteristic

#13133151

C₂²

yes

characteristic

#13133150

—

#13133148

—

C₈×Dic₅₃

C₂²

yes

characteristic

C₂×C₁₆×C₅₃

C₂²

yes

characteristic

#107×C₅₃

C₂²

yes

characteristic

#107×C₅₃

C₂²

yes

characteristic

C₂×Dic₁₀₆

—

C₂×C₈×C₅₃

C₂³

yes

characteristic

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

#11719919

—

#11719919

—

C₄×Dic₅₃

D₄

yes

characteristic

C₂×#88322

D₄

yes

characteristic

C₁₆×C₅₃

Q₈

yes

complemented

#11719908

Q₈

yes

complemented

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

D₈

yes

characteristic

C₂×Dic₅₃

D₈

yes

characteristic

Dic₁₀₆

—

C₈×C₅₃

C₂×Q₈

yes

characteristic

C₈×C₅₃

C₂×Q₈

yes

characteristic

#88322

Q₁₆

yes

—

C₂²×C₅₃

C₂×D₈

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₄⋊Q₈

yes

characteristic

C₄×C₅₃

C₂×Q₁₆

yes

characteristic

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

—

—

D₅₃

yes

—

—

—

C₅₃

#2255

yes

characteristic, complemented, Sylow

C₄⋊Q₈

—

106

—

#125

—

106

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₅₃

yes

—

—

—

106

—

#107

C₂×D₅₃

yes

characteristic

#107

C₂×D₅₃

yes

characteristic

#106

—

106

—

C₄×C₈

—

C₂×Q₈

—

212

—

C₂×C₈

—

C₂×C₈

C₂²×D₅₃

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

106

—

C₄⋊C₄

—

106

—

C₄⋊C₄

—

212

—

C₄⋊C₄

—

212

—

C₄⋊C₄

—

212

—

C₄²

—

C₁₆

—

C₁₆

Dic₁₀₆

yes

complemented

Q₈

—

424

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

212

—

C₂×C₄

—

212

—

C₂×C₄

D₄×D₅₃

yes

characteristic

C₈

—

C₈

C₂×Dic₁₀₆

yes

characteristic

C₈

Q₈×D₅₃

yes

characteristic

C₂²

D₈×D₅₃

yes

characteristic

100

C₄

—

101

C₄

—

102

C₄

—

106

—

103

C₄

—

106

—

104

C₄

—

424

—

105

C₄

—

424

—

106

C₄

#13133216

yes

characteristic

107

C₄

Q₁₆×D₅₃

yes

characteristic

108

C₂

#16811027

yes

characteristic

109

C₂

#16811060

yes

characteristic

110

C₂

#16811062

yes

characteristic

	subgroup	G/H
center	C₂²	D₈×D₅₃
commutator	C₂×C₈×C₅₃	C₂³
Frattini	C₂×C₈	C₂²×D₅₃
Fitting	C₅₃×#367	C₂
socle	C₂²×C₅₃	C₂×D₈
hypercenter	#367	D₅₃
hypocenter	C₅₃	#2255
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #367

subgroup

G/H

center

C₂²

D₈×D₅₃

commutator

C₂×C₈×C₅₃

C₂³

Frattini

C₂×C₈

C₂²×D₅₃

Fitting

C₅₃×#367

C₂

socle

C₂²×C₅₃

C₂×D₈

D₅₃

C₅₃

G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #367

N	H ≅ G/N	type
30. C₁₆×C₅₃	Q₈	semidirect
31. #11719908	Q₈	semidirect
63. C₅₃	#2255	semidirect
87. C₁₆	Dic₁₀₆	semidirect

H ≅ G/N

type

30. C₁₆×C₅₃

Q₈

semidirect

31. #11719908

Q₈

semidirect

63. C₅₃

#2255

semidirect

87. C₁₆

Dic₁₀₆

semidirect

Subgroups — #20277000

Conjugacy classes of subgroups

Complements