Group #20276940 — subgroups

Subgroups — #20276940

Order 6784.

160 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 61 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	#16811000	C₂	1	1	yes	maximal, characteristic
2	#16810423	C₂	1	4	yes	maximal, complemented
3	C₄×#11719908	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#500	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×D₄₂₄	—	2	1	no	—
6	C₂×D₄₂₄	C₂²	1	2	yes	—
7	#13133214	—	2	1	no	—
8	C₂×#11719908	C₂²	1	2	yes	—
9	C₄⋊Q₈×C₅₃	—	2	1	no	—
10	#107×C₅₃	—	2	1	no	—
11	#107×C₅₃	C₂²	1	2	yes	—
12	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
13	C₂×Q₈×C₅₃	—	4	1	no	—
14	C₂×D₂₁₂	—	2	2	no	—
15	C₂×D₂₁₂	—	4	1	no	—
16	C₂×D₂₁₂	—	4	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	D₄₂₄	—	2	4	no	—
23	D₄₂₄	—	4	2	no	—
24	#11719908	—	2	2	no	—
25	C₂²×D₅₃	—	4	2	no	—
26	Q₈×C₅₃	—	8	2	no	—
27	C₂×C₄×C₅₃	—	4	2	no	—
28	C₂×C₄×C₅₃	D₈	1	1	yes	characteristic
29	C₂×C₄×C₅₃	D₈	1	1	yes	characteristic
30	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
31	D₂₁₂	—	4	2	no	—
32	D₂₁₂	—	4	2	no	—
33	D₂₁₂	—	8	2	no	—
34	D₂₁₂	—	8	2	no	—
35	C₈×C₅₃	SD₁₆	1	2	yes	—
36	C₈×C₅₃	#41	1	2	yes	—
37	C₂²×C₅₃	C₂×D₈	1	1	yes	characteristic
38	C₂×D₅₃	—	8	2	no	—
39	C₂×D₅₃	—	8	2	no	—
40	C₂×D₅₃	—	8	2	no	—
41	C₄×C₅₃	—	8	2	no	—
42	C₄×C₅₃	D₁₆	1	2	yes	—
43	C₄×C₅₃	SD₃₂	1	2	yes	—
44	C₄×C₅₃	#124	1	1	yes	characteristic
45	C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
46	#2240	—	53	1	no	maximal, Sylow
47	C₂×C₅₃	#486	1	1	yes	characteristic
48	C₂×C₅₃	C₂×D₁₆	1	1	yes	characteristic
49	C₂×C₅₃	C₂×SD₃₂	1	1	yes	characteristic
50	D₅₃	—	16	2	no	—
51	#500	D₅₃	1	1	yes	characteristic, complemented
52	#493	—	53	1	no	—
53	#357	—	53	4	no	—
54	C₄×C₁₆	—	53	1	no	—
55	C₅₃	#2240	1	1	yes	characteristic, complemented, Sylow
56	C₂×D₈	—	53	2	no	—
57	C₂×D₈	—	106	1	no	—
58	C₄⋊Q₈	—	2	1	no	—
59	#127	—	106	1	no	—
60	C₂×C₁₆	—	53	2	no	—
61	#107	—	2	1	no	—
62	#107	C₂×D₅₃	1	2	yes	—
63	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
64	C₂×Q₈	—	4	1	no	—
65	C₂×D₄	—	106	2	no	—
66	C₂×D₄	—	212	1	no	—
67	C₂×D₄	—	212	1	no	—
68	D₈	—	106	4	no	—
69	D₈	—	212	2	no	—
70	C₂×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
71	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
72	C₄⋊C₄	—	2	2	no	—
73	C₄⋊C₄	—	4	1	no	—
74	C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
75	C₁₆	—	106	2	no	—
76	C₂³	—	212	2	no	—
77	Q₈	—	8	2	no	—
78	D₄	—	424	2	no	—
79	D₄	—	424	2	no	—
80	D₄	—	212	2	no	—
81	D₄	—	212	2	no	—
82	C₂×C₄	—	4	2	no	—
83	C₂×C₄	#11719922	1	1	yes	characteristic
84	C₂×C₄	#11719922	1	1	yes	characteristic
85	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
86	C₈	#11719924	1	2	yes	—
87	C₈	#11719949	1	2	yes	—
88	C₂²	—	424	2	no	—
89	C₂²	—	424	2	no	—
90	C₂²	—	424	2	no	—
91	C₂²	C₂×#11719922	1	1	yes	characteristic
92	C₄	—	8	2	no	—
93	C₄	#13133158	1	2	yes	—
94	C₄	#13133160	1	2	yes	—
95	C₄	#13133206	1	1	yes	characteristic
96	C₄	C₂×#11719924	1	1	yes	characteristic
97	C₂	—	848	2	no	—
98	C₂	#16810971	1	1	yes	characteristic
99	C₂	C₂×#13133158	1	1	yes	characteristic
100	C₂	C₂×#13133160	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811000

C₂

yes

maximal, characteristic

#16810423

C₂

yes

maximal, complemented

C₄×#11719908

C₂

yes

maximal, characteristic, complemented

C₅₃×#500

C₂

yes

maximal, characteristic, complemented

C₂×D₄₂₄

—

C₂×D₄₂₄

C₂²

yes

—

#13133214

—

C₂×#11719908

C₂²

yes

—

C₄⋊Q₈×C₅₃

—

#107×C₅₃

—

#107×C₅₃

C₂²

yes

—

C₄×C₈×C₅₃

C₂²

yes

characteristic

C₂×Q₈×C₅₃

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×C₈×C₅₃

D₄

yes

characteristic

C₂×C₈×C₅₃

C₂³

yes

characteristic

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

D₄

yes

characteristic

D₄₂₄

—

D₄₂₄

—

#11719908

—

C₂²×D₅₃

—

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

D₈

yes

characteristic

C₂×C₄×C₅₃

D₈

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

D₂₁₂

—

D₂₁₂

—

D₂₁₂

—

D₂₁₂

—

C₈×C₅₃

SD₁₆

yes

—

C₈×C₅₃

#41

yes

—

C₂²×C₅₃

C₂×D₈

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

D₁₆

yes

—

C₄×C₅₃

SD₃₂

yes

—

C₄×C₅₃

#124

yes

characteristic

C₄×C₅₃

C₂×SD₁₆

yes

—

C₂×C₅₃

yes

C₂×C₅₃

C₂×D₁₆

yes

characteristic

C₂×C₅₃

C₂×SD₃₂

yes

characteristic

D₅₃

—

#500

D₅₃

yes

characteristic, complemented

#493

—

#357

—

C₄×C₁₆

—

C₅₃

#2240

yes

characteristic, complemented, Sylow

C₂×D₈

—

C₂×D₈

—

106

—

C₄⋊Q₈

—

#127

—

106

—

C₂×C₁₆

—

#107

—

#107

C₂×D₅₃

yes

—

C₄×C₈

C₂×D₅₃

yes

characteristic

C₂×Q₈

—

C₂×D₄

—

106

—

C₂×D₄

—

212

—

C₂×D₄

—

212

—

D₈

—

106

—

D₈

—

212

—

C₂×C₈

C₂²⋊D₅₃

yes

characteristic

C₂×C₈

C₂²×D₅₃

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂²⋊D₅₃

yes

characteristic

C₁₆

—

106

—

C₂³

—

212

—

Q₈

—

D₄

—

424

—

D₄

—

424

—

D₄

—

212

—

D₄

—

212

—

C₂×C₄

—

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

C₈

#11719924

yes

—

C₈

#11719949

yes

—

C₂²

—

424

—

C₂²

—

424

—

C₂²

—

424

—

C₂²

C₂×#11719922

yes

characteristic

C₄

—

C₄

#13133158

yes

—

C₄

#13133160

yes

—

C₄

yes

C₄

yes

C₂

—

848

—

C₂

yes

C₂

yes

100

C₂

C₂×#13133160

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×#11719922
commutator	C₂×C₈×C₅₃	C₂³
Frattini	C₂×C₈	C₂²×D₅₃
Fitting	C₅₃×#500	C₂
socle	C₂²×C₅₃	C₂×D₈
hypercenter	#500	D₅₃
hypocenter	C₅₃	#2240
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #500

subgroup

G/H

center

C₂²

C₂×#11719922

commutator

C₂×C₈×C₅₃

C₂³

Frattini

C₂×C₈

C₂²×D₅₃

Fitting

C₅₃×#500

C₂

socle

C₂²×C₅₃

C₂×D₈

D₅₃

C₅₃

G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #500

N	H ≅ G/N	type
2. #16810423	C₂	semidirect
3. C₄×#11719908	C₂	semidirect
4. C₅₃×#500	C₂	semidirect
51. #500	D₅₃	semidirect
55. C₅₃	#2240	semidirect

H ≅ G/N

type

2. #16810423

C₂

semidirect

3. C₄×#11719908

C₂

semidirect

4. C₅₃×#500

C₂

semidirect

51. #500

D₅₃

semidirect

55. C₅₃

#2240

semidirect

Subgroups — #20276940

Conjugacy classes of subgroups

Complements