Group #20276557 — subgroups

Subgroups — #20276557

Order 6784.

144 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 144 conjugacy classes of subgroups — too many to draw. Only its 79 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	#16811040	C₂	1	2	yes	maximal, complemented
2	#16810444	C₂	1	1	yes	maximal, characteristic, complemented
3	#16810407	C₂	1	2	yes	maximal, complemented
4	#16810399	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#443	C₂	1	1	yes	maximal, characteristic
6	#13133157	—	2	2	no	—
7	#13133157	C₂²	1	2	yes	—
8	C₂×#11719908	—	2	2	no	—
9	C₂×#11719908	C₂²	1	1	yes	characteristic
10	C₂×#11719908	C₂²	1	1	yes	characteristic
11	#135×C₅₃	C₄	1	1	yes	characteristic
12	#131×C₅₃	C₂²	1	2	yes	—
13	#108×C₅₃	C₄	1	1	yes	characteristic
14	#104×C₅₃	C₄	1	2	yes	—
15	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	#41×C₅₃	C₂×C₄	1	2	yes	—
17	Q₁₆×C₅₃	C₈	1	1	yes	characteristic
18	SD₁₆×C₅₃	C₈	1	2	yes	—
19	D₈×C₅₃	C₈	1	1	yes	characteristic
20	M₁₆×C₅₃	—	2	2	no	—
21	M₁₆×C₅₃	C₂×C₄	1	2	yes	—
22	C₂×C₈×C₅₃	—	2	2	no	—
23	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
24	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
25	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
26	#11719908	—	2	2	no	—
27	#11719908	—	2	1	no	—
28	#11719908	D₄	1	2	yes	—
29	Q₈×C₅₃	C₂×C₈	1	2	yes	—
30	D₄×C₅₃	—	2	2	no	—
31	D₄×C₅₃	C₂×C₈	1	2	yes	—
32	C₂×C₄×C₅₃	—	2	2	no	—
33	C₂×C₄×C₅₃	—	2	1	no	—
34	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
35	C₈×C₅₃	—	2	2	no	—
36	C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
37	C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
38	C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
39	C₈×C₅₃	#41	1	1	yes	characteristic
40	C₂²×C₅₃	—	2	2	no	—
41	C₂²×C₅₃	#131	1	1	yes	characteristic
42	C₄×C₅₃	—	2	2	no	—
43	C₄×C₅₃	—	4	1	no	—
44	C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
45	C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
46	#2171	—	53	1	no	maximal, Sylow
47	C₂×C₅₃	—	2	1	no	—
48	C₂×C₅₃	—	4	2	no	—
49	C₂×C₅₃	C₈×D₄	1	1	yes	characteristic
50	#504	—	53	2	no	—
51	#443	D₅₃	1	1	yes	characteristic
52	#364	—	53	1	no	—
53	#350	—	53	2	no	—
54	#346	—	53	1	no	—
55	C₅₃	#2171	1	1	yes	characteristic, complemented, Sylow
56	#135	Dic₅₃	1	1	yes	characteristic
57	#131	C₂×D₅₃	1	2	yes	—
58	M₃₂	—	53	2	no	—
59	M₃₂	—	106	2	no	—
60	C₂×C₁₆	—	53	1	no	—
61	C₂×C₁₆	—	53	1	no	—
62	C₂×C₁₆	—	106	2	no	—
63	#108	Dic₅₃	1	1	yes	characteristic
64	#104	Dic₅₃	1	2	yes	—
65	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
66	#41	C₂×Dic₅₃	1	2	yes	—
67	Q₁₆	#88322	1	1	yes	characteristic
68	SD₁₆	#88322	1	2	yes	—
69	D₈	#88322	1	1	yes	characteristic
70	M₁₆	—	2	2	no	—
71	M₁₆	C₂×Dic₅₃	1	2	yes	—
72	C₂×C₈	—	2	2	no	—
73	C₂×C₈	C₂×Dic₅₃	1	1	yes	characteristic
74	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
75	C₄²	C₂×Dic₅₃	1	1	yes	characteristic
76	C₁₆	—	53	2	no	—
77	C₁₆	—	106	2	no	—
78	C₁₆	—	106	1	no	—
79	Q₈	C₂×#88322	1	2	yes	—
80	D₄	—	2	2	no	—
81	D₄	C₂×#88322	1	2	yes	—
82	C₂×C₄	—	2	2	no	—
83	C₂×C₄	—	2	1	no	—
84	C₂×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
85	C₈	—	2	2	no	—
86	C₈	C₂×#88322	1	1	yes	characteristic
87	C₈	C₂×#88322	1	1	yes	characteristic
88	C₈	D₄×D₅₃	1	1	yes	characteristic
89	C₈	#11719947	1	1	yes	characteristic
90	C₂²	—	2	2	no	—
91	C₂²	#13133224	1	1	yes	characteristic
92	C₄	—	2	2	no	—
93	C₄	—	4	1	no	—
94	C₄	D₄×Dic₅₃	1	1	yes	characteristic
95	C₄	C₂²×#88322	1	1	yes	characteristic
96	C₂	—	2	1	no	—
97	C₂	—	4	2	no	—
98	C₂	D₄×#88322	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811040

C₂

yes

maximal, complemented

#16810444

C₂

yes

maximal, characteristic, complemented

#16810407

C₂

yes

maximal, complemented

#16810399

C₂

yes

maximal, characteristic, complemented

C₅₃×#443

C₂

yes

maximal, characteristic

#13133157

—

#13133157

C₂²

yes

—

C₂×#11719908

—

C₂×#11719908

C₂²

yes

characteristic

C₂×#11719908

C₂²

yes

characteristic

#135×C₅₃

C₄

yes

characteristic

#131×C₅₃

C₂²

yes

—

#108×C₅₃

C₄

yes

characteristic

#104×C₅₃

C₄

yes

—

C₄×C₈×C₅₃

C₂²

yes

characteristic

#41×C₅₃

C₂×C₄

yes

—

Q₁₆×C₅₃

C₈

yes

characteristic

SD₁₆×C₅₃

C₈

yes

—

D₈×C₅₃

C₈

yes

characteristic

M₁₆×C₅₃

—

M₁₆×C₅₃

C₂×C₄

yes

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂×C₈×C₅₃

C₂³

yes

characteristic

C₄²×C₅₃

C₂×C₄

yes

—

—

D₄

yes

—

Q₈×C₅₃

C₂×C₈

yes

—

D₄×C₅₃

—

D₄×C₅₃

C₂×C₈

yes

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₈×C₅₃

—

C₈×C₅₃

C₂×C₈

yes

characteristic

C₈×C₅₃

C₂×C₈

yes

characteristic

C₈×C₅₃

C₂×D₄

yes

characteristic

C₈×C₅₃

#41

yes

characteristic

C₂²×C₅₃

—

C₂²×C₅₃

#131

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₄×D₄

yes

characteristic

C₄×C₅₃

C₂²×C₈

yes

characteristic

#2171

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

C₈×D₄

yes

characteristic

#504

—

#443

D₅₃

yes

—

—

—

C₅₃

#2171

yes

characteristic, complemented, Sylow

#135

Dic₅₃

yes

characteristic

#131

C₂×D₅₃

yes

—

M₃₂

—

M₃₂

—

106

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

106

—

#108

Dic₅₃

yes

characteristic

#104

Dic₅₃

yes

—

C₄×C₈

C₂×D₅₃

yes

characteristic

#41

C₂×Dic₅₃

yes

—

Q₁₆

#88322

yes

characteristic

SD₁₆

#88322

yes

—

D₈

#88322

yes

characteristic

M₁₆

—

M₁₆

C₂×Dic₅₃

yes

—

C₂×C₈

—

C₂×C₈

C₂×Dic₅₃

yes

characteristic

C₂×C₈

C₂²×D₅₃

yes

characteristic

C₄²

C₂×Dic₅₃

yes

characteristic

C₁₆

—

C₁₆

—

106

—

C₁₆

—

106

—

Q₈

C₂×#88322

yes

—

D₄

—

D₄

C₂×#88322

yes

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²×Dic₅₃

yes

characteristic

C₈

—

C₈

yes

C₈

yes

C₈

D₄×D₅₃

yes

characteristic

C₈

#11719947

yes

characteristic

C₂²

—

C₂²

#13133224

yes

characteristic

C₄

—

C₄

—

C₄

D₄×Dic₅₃

yes

characteristic

C₄

C₂²×#88322

yes

characteristic

C₂

—

C₂

—

C₂

D₄×#88322

yes

characteristic

	subgroup	G/H
center	C₈	D₄×D₅₃
commutator	C₄×C₅₃	C₂²×C₈
Frattini	C₂×C₈	C₂²×D₅₃
Fitting	C₅₃×#443	C₂
socle	C₂×C₅₃	C₈×D₄
hypercenter	#443	D₅₃
hypocenter	C₅₃	#2171
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ #443

subgroup

G/H

center

C₈

D₄×D₅₃

commutator

C₄×C₅₃

C₂²×C₈

Frattini

C₂×C₈

C₂²×D₅₃

Fitting

C₅₃×#443

C₂

socle

C₂×C₅₃

C₈×D₄

D₅₃

C₅₃

G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₈ ◁ C₂×C₈ ◁ #443

N	H ≅ G/N	type
1. #16811040	C₂	semidirect
2. #16810444	C₂	semidirect
3. #16810407	C₂	semidirect
4. #16810399	C₂	semidirect
55. C₅₃	#2171	semidirect

H ≅ G/N

type

1. #16811040

C₂

semidirect

2. #16810444

C₂

semidirect

3. #16810407

C₂

semidirect

4. #16810399

C₂

semidirect

55. C₅₃

#2171

semidirect

Subgroups — #20276557

Conjugacy classes of subgroups

Complements