Group #20276883 — subgroups

Subgroups — #20276883

Order 6784.

126 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 126 conjugacy classes of subgroups — too many to draw. Only its 47 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	#16810741	C₂	1	1	yes	maximal, characteristic
2	#16810445	C₂	1	2	yes	maximal, complemented
3	#16810444	C₂	1	1	yes	maximal, characteristic, complemented
4	#16810432	C₂	1	2	yes	maximal
5	C₅₃×#498	C₂	1	1	yes	maximal, characteristic, complemented
6	#13133239	C₂²	1	1	yes	characteristic
7	#13133223	—	2	1	no	—
8	#13133157	C₂²	1	2	yes	—
9	#13133153	C₂²	1	1	yes	characteristic
10	#13133145	—	2	1	no	—
11	C₄⋊Q₈×C₅₃	—	2	1	no	—
12	#106×C₅₃	—	2	1	no	—
13	#106×C₅₃	C₂²	1	2	yes	—
14	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×Q₈×C₅₃	—	4	1	no	—
16	#11719945	—	2	1	no	—
17	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
18	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
19	C₄⋊C₄×C₅₃	—	2	2	no	—
20	C₄⋊C₄×C₅₃	—	4	1	no	—
21	C₄²×C₅₃	D₄	1	1	yes	characteristic
22	#11719917	—	2	1	no	—
23	Dic₂₁₂	Q₈	1	1	yes	characteristic, complemented
24	D₄₂₄	Q₈	1	1	yes	characteristic, complemented
25	#11719913	—	2	1	no	—
26	#11719912	—	4	1	no	—
27	C₈×D₅₃	—	4	1	no	—
28	#11719908	—	2	2	no	—
29	Q₈×C₅₃	—	8	2	no	—
30	C₂×C₄×C₅₃	—	2	1	no	—
31	C₂×C₄×C₅₃	—	4	2	no	—
32	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
33	C₂²⋊D₅₃	—	4	1	no	—
34	D₂₁₂	—	2	1	no	—
35	C₄×D₅₃	—	4	1	no	—
36	Dic₁₀₆	—	2	1	no	—
37	C₈×C₅₃	SD₁₆	1	1	yes	characteristic
38	C₈×C₅₃	SD₁₆	1	1	yes	characteristic
39	C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
40	C₈×C₅₃	#41	1	1	yes	characteristic
41	#88322	—	4	1	no	—
42	C₂²×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
43	C₂×D₅₃	—	4	1	no	—
44	C₄×C₅₃	—	2	2	no	—
45	C₄×C₅₃	—	8	2	no	—
46	C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
47	C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
48	Dic₅₃	—	4	1	no	—
49	#2230	—	53	1	no	maximal, Sylow
50	C₂×C₅₃	—	2	1	no	—
51	C₂×C₅₃	#476	1	1	yes	characteristic
52	D₅₃	—	8	1	no	—
53	#498	D₅₃	1	1	yes	characteristic, complemented
54	#443	—	53	1	no	—
55	#365	—	53	2	no	—
56	#364	—	53	1	no	—
57	#360	—	53	2	no	—
58	C₅₃	#2230	1	1	yes	characteristic, complemented, Sylow
59	#135	—	53	1	no	—
60	#131	—	106	1	no	—
61	C₄⋊Q₈	—	2	1	no	—
62	M₃₂	—	53	2	no	—
63	#108	—	53	1	no	—
64	#106	—	2	1	no	—
65	#106	C₂×D₅₃	1	2	yes	—
66	#104	—	106	1	no	—
67	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
68	#41	—	106	1	no	—
69	C₂×Q₈	—	4	1	no	—
70	Q₁₆	—	53	1	no	—
71	SD₁₆	—	106	1	no	—
72	D₈	—	53	1	no	—
73	M₁₆	—	106	1	no	—
74	M₁₆	—	212	1	no	—
75	C₂×C₈	—	212	1	no	—
76	C₂×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
77	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
78	C₄⋊C₄	—	2	2	no	—
79	C₄⋊C₄	—	4	1	no	—
80	C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
81	C₁₆	—	106	2	no	—
82	Q₈	—	8	2	no	—
83	Q₈	—	106	1	no	—
84	D₄	—	106	1	no	—
85	D₄	—	212	1	no	—
86	C₂×C₄	—	2	1	no	—
87	C₂×C₄	—	4	2	no	—
88	C₂×C₄	—	212	1	no	—
89	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
90	C₈	—	212	1	no	—
91	C₈	#11719924	1	1	yes	characteristic
92	C₈	#11719924	1	1	yes	characteristic
93	C₈	Q₈×D₅₃	1	1	yes	characteristic
94	C₈	#11719949	1	1	yes	characteristic
95	C₂²	—	212	1	no	—
96	C₂²	#13133243	1	1	yes	characteristic
97	C₄	—	2	2	no	—
98	C₄	—	8	2	no	—
99	C₄	—	212	1	no	—
100	C₄	#13133193	1	1	yes	characteristic
101	C₄	C₂×#11719924	1	1	yes	characteristic
102	C₂	—	2	1	no	—
103	C₂	—	424	1	no	—
104	C₂	#16810910	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16810741

C₂

yes

maximal, characteristic

#16810445

C₂

yes

maximal, complemented

#16810444

C₂

yes

maximal, characteristic, complemented

#16810432

C₂

yes

maximal

C₅₃×#498

C₂

yes

maximal, characteristic, complemented

#13133239

C₂²

yes

characteristic

#13133223

—

#13133157

C₂²

yes

—

#13133153

C₂²

yes

characteristic

#13133145

—

C₄⋊Q₈×C₅₃

—

#106×C₅₃

—

#106×C₅₃

C₂²

yes

—

C₄×C₈×C₅₃

C₂²

yes

characteristic

C₂×Q₈×C₅₃

—

#11719945

—

C₂×C₈×C₅₃

D₄

yes

characteristic

C₂×C₈×C₅₃

C₂³

yes

characteristic

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

D₄

yes

characteristic

#11719917

—

Dic₂₁₂

Q₈

yes

characteristic, complemented

D₄₂₄

Q₈

yes

characteristic, complemented

#11719913

—

#11719912

—

C₈×D₅₃

—

#11719908

—

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²⋊D₅₃

—

D₂₁₂

—

C₄×D₅₃

—

Dic₁₀₆

—

C₈×C₅₃

SD₁₆

yes

characteristic

C₈×C₅₃

SD₁₆

yes

characteristic

C₈×C₅₃

C₂×Q₈

yes

C₈×C₅₃

yes

—

C₂²×C₅₃

AGL(1,ℤ/8)

yes

characteristic

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂²⋊Q₈

yes

characteristic

C₄×C₅₃

C₂×SD₁₆

yes

characteristic

Dic₅₃

—

#2230

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

#476

yes

characteristic

D₅₃

—

#498

D₅₃

yes

characteristic, complemented

—

—

—

—

C₅₃

#2230

yes

characteristic, complemented, Sylow

#135

—

#131

—

106

—

C₄⋊Q₈

—

M₃₂

—

#108

—

#106

—

#106

C₂×D₅₃

yes

—

#104

—

106

—

C₄×C₈

C₂×D₅₃

yes

characteristic

#41

—

106

—

C₂×Q₈

—

Q₁₆

—

SD₁₆

—

106

—

D₈

—

M₁₆

—

106

—

M₁₆

—

212

—

C₂×C₈

—

212

—

C₂×C₈

C₂²⋊D₅₃

yes

characteristic

C₂×C₈

C₂²×D₅₃

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂²⋊D₅₃

yes

characteristic

C₁₆

—

106

—

Q₈

—

Q₈

—

106

—

D₄

—

106

—

D₄

—

212

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

212

—

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

C₈

—

212

—

C₈

yes

C₈

yes

C₈

Q₈×D₅₃

yes

characteristic

C₈

#11719949

yes

characteristic

C₂²

—

212

—

C₂²

#13133243

yes

characteristic

C₄

—

C₄

—

C₄

—

212

—

100

C₄

#13133193

yes

characteristic

101

C₄

C₂×#11719924

yes

characteristic

102

C₂

—

103

C₂

—

424

—

104

C₂

#16810910

yes

characteristic

	subgroup	G/H
center	C₂	#16810910
commutator	C₂×C₈×C₅₃	C₂³
Frattini	C₂×C₈	C₂²×D₅₃
Fitting	C₅₃×#498	C₂
socle	C₂×C₅₃	#476
hypercenter	#498	D₅₃
hypocenter	C₅₃	#2230
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #498

subgroup

G/H

center

C₂

#16810910

commutator

C₂×C₈×C₅₃

C₂³

Frattini

C₂×C₈

C₂²×D₅₃

Fitting

C₅₃×#498

C₂

C₂×C₅₃

D₅₃

C₅₃

G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #498

N	H ≅ G/N	type
2. #16810445	C₂	semidirect
3. #16810444	C₂	semidirect
5. C₅₃×#498	C₂	semidirect
23. Dic₂₁₂	Q₈	semidirect
24. D₄₂₄	Q₈	semidirect
53. #498	D₅₃	semidirect
58. C₅₃	#2230	semidirect

H ≅ G/N

type

2. #16810445

C₂

semidirect

3. #16810444

C₂

semidirect

5. C₅₃×#498

C₂

semidirect

23. Dic₂₁₂

Q₈

semidirect

24. D₄₂₄

Q₈

semidirect

53. #498

D₅₃

semidirect

58. C₅₃

#2230

semidirect

Subgroups — #20276883

Conjugacy classes of subgroups

Complements