Group #20276891 — subgroups

Subgroups — #20276891

Order 6784.

120 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 120 conjugacy classes of subgroups — too many to draw. Only its 49 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	#16811025	C₂	1	1	yes	maximal, characteristic, complemented
2	#16810447	C₂	1	2	yes	maximal, complemented
3	#16810444	C₂	1	1	yes	maximal, characteristic, complemented
4	#16810433	C₂	1	2	yes	maximal
5	C₅₃×#443	C₂	1	1	yes	maximal, characteristic
6	#13133209	—	2	1	no	—
7	#13133157	C₂²	1	2	yes	—
8	#13133151	—	2	1	no	—
9	#13133149	C₂²	1	2	yes	—
10	#135×C₅₃	C₂²	1	1	yes	characteristic
11	#131×C₅₃	—	2	1	no	—
12	#108×C₅₃	C₂²	1	1	yes	characteristic
13	#104×C₅₃	—	2	1	no	—
14	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
15	#41×C₅₃	—	2	1	no	—
16	Q₁₆×C₅₃	Q₈	1	1	yes	characteristic
17	SD₁₆×C₅₃	—	2	1	no	—
18	D₈×C₅₃	Q₈	1	1	yes	characteristic
19	M₁₆×C₅₃	—	2	1	no	—
20	M₁₆×C₅₃	—	4	1	no	—
21	C₂×C₈×C₅₃	—	4	1	no	—
22	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
23	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
24	C₄²×C₅₃	D₄	1	1	yes	characteristic
25	C₄⋊Dic₅₃	—	2	2	no	—
26	#11719919	—	4	2	no	—
27	#11719908	—	2	2	no	—
28	Q₈×C₅₃	—	2	1	no	—
29	D₄×C₅₃	—	2	1	no	—
30	D₄×C₅₃	—	4	1	no	—
31	C₂×C₄×C₅₃	—	2	1	no	—
32	C₂×C₄×C₅₃	—	4	1	no	—
33	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
34	C₂×Dic₅₃	—	4	2	no	—
35	C₈×C₅₃	—	4	1	no	—
36	C₈×C₅₃	SD₁₆	1	1	yes	characteristic
37	C₈×C₅₃	SD₁₆	1	1	yes	characteristic
38	C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
39	C₈×C₅₃	#41	1	1	yes	characteristic
40	C₂²×C₅₃	—	4	1	no	—
41	C₂²×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
42	C₄×C₅₃	—	4	1	no	—
43	C₄×C₅₃	—	4	1	no	—
44	C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
45	C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
46	Dic₅₃	—	8	2	no	—
47	#2231	—	53	1	no	maximal, Sylow
48	C₂×C₅₃	—	2	1	no	—
49	C₂×C₅₃	—	8	1	no	—
50	C₂×C₅₃	#476	1	1	yes	characteristic
51	#499	—	53	1	no	—
52	#443	D₅₃	1	1	yes	characteristic
53	#365	—	53	2	no	—
54	#364	—	53	1	no	—
55	#360	—	53	2	no	—
56	C₅₃	#2231	1	1	yes	characteristic, complemented, Sylow
57	#135	C₂×D₅₃	1	1	yes	characteristic
58	#131	—	2	1	no	—
59	#125	—	106	1	no	—
60	M₃₂	—	53	2	no	—
61	#108	C₂×D₅₃	1	1	yes	characteristic
62	#107	—	106	1	no	—
63	#106	—	53	2	no	—
64	#104	—	2	1	no	—
65	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
66	#41	—	2	1	no	—
67	Q₁₆	Dic₁₀₆	1	1	yes	characteristic
68	SD₁₆	—	2	1	no	—
69	D₈	Dic₁₀₆	1	1	yes	characteristic
70	M₁₆	—	2	1	no	—
71	M₁₆	—	4	1	no	—
72	C₂×C₈	—	4	1	no	—
73	C₂×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
74	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
75	C₄⋊C₄	—	106	2	no	—
76	C₄⋊C₄	—	212	2	no	—
77	C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
78	C₁₆	—	106	2	no	—
79	Q₈	—	2	1	no	—
80	D₄	—	2	1	no	—
81	D₄	—	4	1	no	—
82	C₂×C₄	—	2	1	no	—
83	C₂×C₄	—	4	1	no	—
84	C₂×C₄	—	212	2	no	—
85	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
86	C₈	—	4	1	no	—
87	C₈	#11719923	1	1	yes	characteristic
88	C₈	#11719923	1	1	yes	characteristic
89	C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
90	C₈	#11719945	1	1	yes	characteristic
91	C₂²	—	4	1	no	—
92	C₂²	#13133242	1	1	yes	characteristic
93	C₄	—	4	1	no	—
94	C₄	—	4	1	no	—
95	C₄	—	424	2	no	—
96	C₄	#13133196	1	1	yes	characteristic
97	C₄	C₂×#11719923	1	1	yes	characteristic
98	C₂	—	2	1	no	—
99	C₂	—	8	1	no	—
100	C₂	#16810913	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811025

C₂

yes

maximal, characteristic, complemented

#16810447

C₂

yes

maximal, complemented

#16810444

C₂

yes

maximal, characteristic, complemented

#16810433

C₂

yes

maximal

C₅₃×#443

C₂

yes

maximal, characteristic

#13133209

—

#13133157

C₂²

yes

—

#13133151

—

#13133149

C₂²

yes

—

#135×C₅₃

C₂²

yes

characteristic

#131×C₅₃

—

#108×C₅₃

C₂²

yes

characteristic

#104×C₅₃

—

C₄×C₈×C₅₃

C₂²

yes

characteristic

#41×C₅₃

—

Q₁₆×C₅₃

Q₈

yes

characteristic

SD₁₆×C₅₃

—

D₈×C₅₃

Q₈

yes

characteristic

M₁₆×C₅₃

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

D₄

yes

characteristic

C₂×C₈×C₅₃

C₂³

yes

characteristic

C₄²×C₅₃

D₄

yes

characteristic

C₄⋊Dic₅₃

—

#11719919

—

#11719908

—

Q₈×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×Dic₅₃

—

C₈×C₅₃

—

C₈×C₅₃

SD₁₆

yes

characteristic

C₈×C₅₃

SD₁₆

yes

characteristic

C₈×C₅₃

C₂×Q₈

yes

characteristic

C₈×C₅₃

#41

yes

characteristic

C₂²×C₅₃

—

C₂²×C₅₃

AGL(1,ℤ/8)

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂²⋊Q₈

yes

characteristic

C₄×C₅₃

C₂×SD₁₆

yes

characteristic

Dic₅₃

—

#2231

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

yes

—

D₅₃

yes

—

—

—

C₅₃

#2231

yes

characteristic, complemented, Sylow

#135

C₂×D₅₃

yes

characteristic

#131

—

#125

—

106

—

M₃₂

—

#108

C₂×D₅₃

yes

characteristic

#107

—

106

—

#106

—

#104

—

C₄×C₈

C₂×D₅₃

yes

characteristic

#41

—

Q₁₆

Dic₁₀₆

yes

characteristic

SD₁₆

—

D₈

Dic₁₀₆

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

C₂²⋊D₅₃

yes

characteristic

C₂×C₈

C₂²×D₅₃

yes

characteristic

C₄⋊C₄

—

106

—

C₄⋊C₄

—

212

—

C₄²

C₂²⋊D₅₃

yes

characteristic

C₁₆

—

106

—

Q₈

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

212

—

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

C₈

—

C₈

yes

C₈

yes

C₈

C₂×Dic₁₀₆

yes

characteristic

C₈

#11719945

yes

characteristic

C₂²

—

C₂²

#13133242

yes

characteristic

C₄

—

C₄

—

C₄

—

424

—

C₄

yes

C₄

yes

C₂

—

C₂

—

100

C₂

#16810913

yes

characteristic

	subgroup	G/H
center	C₂	#16810913
commutator	C₂×C₈×C₅₃	C₂³
Frattini	C₂×C₈	C₂²×D₅₃
Fitting	C₅₃×#443	C₂
socle	C₂×C₅₃	#476
hypercenter	#443	D₅₃
hypocenter	C₅₃	#2231
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #443

subgroup

G/H

center

C₂

#16810913

commutator

C₂×C₈×C₅₃

C₂³

Frattini

C₂×C₈

C₂²×D₅₃

Fitting

C₅₃×#443

C₂

C₂×C₅₃

D₅₃

C₅₃

G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #443

N	H ≅ G/N	type
1. #16811025	C₂	semidirect
2. #16810447	C₂	semidirect
3. #16810444	C₂	semidirect
56. C₅₃	#2231	semidirect

H ≅ G/N

type

1. #16811025

C₂

semidirect

2. #16810447

C₂

semidirect

3. #16810444

C₂

semidirect

56. C₅₃

#2231

semidirect

Subgroups — #20276891

Conjugacy classes of subgroups

Complements