Group #20282984 — subgroups

Subgroups — #20282984

Order 6784.

348 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 133 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	#16810562	C₂	1	3	yes	maximal
2	#16810556	C₂	1	6	yes	maximal
3	#16810543	C₂	1	3	yes	maximal
4	#16810491	C₂	1	2	yes	maximal
5	C₅₃×#384	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₀₆	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊Dic₅₃	C₂²	1	3	yes	—
8	C₂×#11719919	C₂²	1	6	yes	—
9	#13133124	C₂²	1	6	yes	—
10	#13133124	C₂²	1	3	yes	—
11	#13133124	C₂²	1	3	yes	—
12	#13133124	C₂²	1	6	yes	—
13	C₂×C₄⋊C₄×C₅₃	C₂²	1	3	yes	—
14	C₂×C₄⋊C₄×C₅₃	C₂²	1	3	yes	—
15	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
16	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
17	C₂²×C₄×C₅₃	C₂³	1	3	yes	—
18	C₂²×C₄×C₅₃	C₂³	1	3	yes	—
19	C₂²×Dic₅₃	C₂³	1	6	yes	—
20	C₂²×Dic₅₃	C₂³	1	2	yes	—
21	C₂×Dic₁₀₆	—	4	3	no	—
22	C₄⋊C₄×C₅₃	—	4	3	no	—
23	C₄⋊C₄×C₅₃	—	4	3	no	—
24	C₄²×C₅₃	Q₈	1	4	yes	complemented
25	C₄⋊Dic₅₃	—	4	3	no	—
26	#11719919	—	4	6	no	—
27	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
28	C₂×C₄×C₅₃	—	2	3	no	—
29	C₂×C₄×C₅₃	—	2	3	no	—
30	C₂×C₄×C₅₃	—	2	3	no	—
31	C₂×C₄×C₅₃	—	2	6	no	—
32	C₂×C₄×C₅₃	—	2	3	no	—
33	C₂×C₄×C₅₃	C₂×Q₈	1	6	yes	—
34	C₂×Dic₅₃	—	2	6	no	—
35	C₂×Dic₅₃	—	2	6	no	—
36	C₂×Dic₅₃	—	2	12	no	—
37	Dic₁₀₆	—	4	4	no	—
38	C₂²×C₅₃	C₂²×Q₈	1	1	yes	characteristic
39	C₂²×C₅₃	H(5,𝔽₂)	1	3	yes	—
40	C₂²×C₅₃	2₋¹⁺⁴	1	3	yes	—
41	C₄×C₅₃	—	2	6	no	—
42	C₄×C₅₃	—	4	1	no	—
43	C₄×C₅₃	—	4	3	no	—
44	Dic₅₃	—	4	2	no	—
45	Dic₅₃	—	4	6	no	—
46	#2843	—	53	1	no	maximal, Sylow
47	C₂×C₅₃	#544	1	3	yes	—
48	C₂×C₅₃	#560	1	1	yes	characteristic
49	C₂×C₅₃	#563	1	3	yes	—
50	#400	—	53	3	no	—
51	#398	—	53	6	no	—
52	#395	—	53	3	no	—
53	#384	D₅₃	1	1	yes	characteristic
54	#383	—	53	2	no	—
55	C₅₃	#2843	1	1	yes	characteristic, complemented, Sylow
56	C₂²×Q₈	—	53	1	no	—
57	C₂×C₄⋊C₄	—	53	6	no	—
58	C₂×C₄⋊C₄	—	53	3	no	—
59	C₂×C₄⋊C₄	C₂×D₅₃	1	3	yes	—
60	C₂×C₄⋊C₄	C₂×D₅₃	1	3	yes	—
61	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
62	#95	—	53	6	no	—
63	#95	—	53	6	no	—
64	#95	—	53	3	no	—
65	#95	—	53	3	no	—
66	C₂×Q₈	—	212	3	no	—
67	C₂²×C₄	—	53	6	no	—
68	C₂²×C₄	—	53	2	no	—
69	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
70	C₂²×C₄	C₂²×D₅₃	1	3	yes	—
71	C₂²×C₄	C₂²×D₅₃	1	3	yes	—
72	C₄⋊C₄	—	4	3	no	—
73	C₄⋊C₄	—	4	3	no	—
74	C₄⋊C₄	—	212	6	no	—
75	C₄⋊C₄	—	212	3	no	—
76	C₄²	Dic₁₀₆	1	4	yes	complemented
77	C₂³	C₂³×D₅₃	1	1	yes	characteristic
78	Q₈	—	212	4	no	—
79	C₂×C₄	—	2	3	no	—
80	C₂×C₄	—	2	3	no	—
81	C₂×C₄	—	2	6	no	—
82	C₂×C₄	—	2	3	no	—
83	C₂×C₄	—	2	3	no	—
84	C₂×C₄	—	106	6	no	—
85	C₂×C₄	—	106	6	no	—
86	C₂×C₄	—	106	12	no	—
87	C₂×C₄	C₂×Dic₁₀₆	1	6	yes	—
88	C₂²	C₂²×Dic₁₀₆	1	1	yes	characteristic
89	C₂²	#13133262	1	3	yes	—
90	C₂²	#13133265	1	3	yes	—
91	C₄	—	2	6	no	—
92	C₄	—	4	3	no	—
93	C₄	—	4	1	no	—
94	C₄	—	212	2	no	—
95	C₄	—	212	6	no	—
96	C₂	#16811254	1	3	yes	—
97	C₂	#16811360	1	1	yes	characteristic
98	C₂	#16811372	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16810562

C₂

yes

maximal

#16810556

C₂

yes

maximal

#16810543

C₂

yes

maximal

#16810491

C₂

yes

maximal

C₅₃×#384

C₂

yes

maximal, characteristic

C₂²×Dic₁₀₆

C₂²

yes

characteristic

C₂×C₄⋊Dic₅₃

C₂²

yes

—

C₂×#11719919

C₂²

yes

—

#13133124

C₂²

yes

—

#13133124

C₂²

yes

—

#13133124

C₂²

yes

—

#13133124

C₂²

yes

—

C₂×C₄⋊C₄×C₅₃

C₂²

yes

—

C₂×C₄⋊C₄×C₅₃

C₂²

yes

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×C₄×C₅₃

C₂³

yes

—

C₂²×C₄×C₅₃

C₂³

yes

—

C₂²×Dic₅₃

C₂³

yes

—

C₂²×Dic₅₃

C₂³

yes

—

C₂×Dic₁₀₆

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

Q₈

yes

complemented

C₄⋊Dic₅₃

—

#11719919

—

C₂³×C₅₃

C₂⁴

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×Q₈

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

Dic₁₀₆

—

C₂²×C₅₃

C₂²×Q₈

yes

characteristic

C₂²×C₅₃

H(5,𝔽₂)

yes

—

C₂²×C₅₃

2₋¹⁺⁴

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

Dic₅₃

—

Dic₅₃

—

#2843

—

maximal, Sylow

C₂×C₅₃

#544

yes

—

C₂×C₅₃

#560

yes

characteristic

C₂×C₅₃

#563

yes

—

—

—

—

D₅₃

yes

characteristic

#383

—

C₅₃

#2843

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₅₃

yes

—

C₂×C₄⋊C₄

C₂×D₅₃

yes

—

C₂×C₄²

C₂×D₅₃

yes

—

—

—

—

C₂×Q₈

—

212

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₅₃

yes

characteristic

C₂²×C₄

C₂²×D₅₃

yes

—

C₂²×C₄

C₂²×D₅₃

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

212

—

C₄⋊C₄

—

212

—

C₄²

Dic₁₀₆

yes

complemented

C₂³

C₂³×D₅₃

yes

characteristic

Q₈

—

212

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

106

—

C₂×C₄

—

106

—

C₂×C₄

—

106

—

C₂×C₄

C₂×Dic₁₀₆

yes

—

C₂²

C₂²×Dic₁₀₆

yes

characteristic

C₂²

#13133262

yes

—

C₂²

#13133265

yes

—

C₄

—

C₄

—

C₄

—

C₄

—

212

—

C₄

—

212

—

C₂

#16811254

yes

—

C₂

#16811360

yes

characteristic

C₂

#16811372

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₅₃
commutator, socle	C₂³×C₅₃	C₂⁴
Fitting	C₅₃×#384	C₂
hypercenter	#384	D₅₃
hypocenter	C₅₃	#2843
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ #384

subgroup

G/H

center, Frattini

C₂³

C₂³×D₅₃

commutator, socle

C₂³×C₅₃

C₂⁴

Fitting

C₅₃×#384

C₂

D₅₃

C₅₃

G ▷ C₂³×C₅₃ ▷ C₅₃

C₁ ◁ C₂³ ◁ #384

N	H ≅ G/N	type
24. C₄²×C₅₃	Q₈	semidirect
55. C₅₃	#2843	semidirect
76. C₄²	Dic₁₀₆	semidirect

H ≅ G/N

type

24. C₄²×C₅₃

Q₈

semidirect

55. C₅₃

#2843

semidirect

76. C₄²

Dic₁₀₆

semidirect

Subgroups — #20282984

Conjugacy classes of subgroups

Complements