Group #20276732 — subgroups

Subgroups — #20276732

Order 6784.

248 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 45 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	#16810795	C₂	1	3	yes	maximal, complemented
2	#16810420	C₂	1	3	yes	maximal, complemented
3	C₅₃×H(3,𝔽₄)	C₂	1	1	yes	maximal, characteristic, complemented
4	#13133265	C₂²	1	1	yes	characteristic, complemented
5	#13133243	—	2	3	no	—
6	#13133145	—	2	3	no	—
7	#13133136	C₂²	1	3	yes	complemented
8	#124×C₅₃	—	2	3	no	—
9	#124×C₅₃	C₂²	1	3	yes	—
10	#120×C₅₃	—	2	3	no	—
11	C₂⁴×C₅₃	—	2	1	no	—
12	C₂×Q₈×C₅₃	—	2	1	no	—
13	C₂×Q₈×C₅₃	C₂³	1	1	yes	characteristic
14	C₂×D₄×C₅₃	—	2	3	no	—
15	C₂×D₄×C₅₃	D₄	1	3	yes	—
16	#11719949	—	4	1	no	—
17	Q₈×D₅₃	—	4	1	no	—
18	#11719945	—	2	3	no	—
19	C₂²⋊C₄×C₅₃	—	2	3	no	—
20	C₂²⋊C₄×C₅₃	—	2	6	no	—
21	C₄²×C₅₃	D₄	1	3	yes	—
22	#11719923	—	4	3	no	—
23	#11719922	—	4	3	no	—
24	#11719917	—	2	3	no	—
25	C₂³×C₅₃	—	2	3	no	—
26	C₂³×C₅₃	—	8	2	no	—
27	C₂³×C₅₃	—	8	1	no	—
28	Q₈×C₅₃	—	4	1	no	—
29	Q₈×C₅₃	—	8	1	no	—
30	D₄×C₅₃	—	4	3	no	—
31	D₄×C₅₃	—	8	3	no	—
32	C₂×C₄×C₅₃	—	2	3	no	—
33	C₂×C₄×C₅₃	C₂×D₄	1	3	yes	—
34	C₂²⋊D₅₃	—	4	1	no	—
35	D₂₁₂	—	2	3	no	—
36	C₄×D₅₃	—	4	3	no	—
37	Dic₁₀₆	—	2	3	no	—
38	#88322	—	4	3	no	—
39	C₂²×C₅₃	—	4	6	no	—
40	C₂²×C₅₃	—	4	3	no	—
41	C₂²×C₅₃	—	8	4	no	—
42	C₂²×C₅₃	#120	1	1	yes	characteristic
43	C₂×D₅₃	—	4	1	no	—
44	C₄×C₅₃	—	2	3	no	—
45	C₄×C₅₃	—	4	3	no	—
46	Dic₅₃	—	4	1	no	—
47	#2202	—	53	1	no	maximal, Sylow
48	C₂×C₅₃	—	2	1	no	—
49	C₂×C₅₃	—	8	3	no	—
50	C₂×C₅₃	B(4,𝔽₂)	1	1	yes	characteristic
51	D₅₃	—	8	1	no	—
52	H(3,𝔽₄)	D₅₃	1	1	yes	characteristic, complemented
53	#454	—	53	3	no	—
54	#355	—	53	3	no	—
55	C₅₃	#2202	1	1	yes	characteristic, complemented, Sylow
56	2₋¹⁺⁴	—	53	1	no	—
57	AGL(1,ℤ/8)	—	106	3	no	—
58	#124	—	2	3	no	—
59	#124	C₂×D₅₃	1	3	yes	—
60	#120	—	2	3	no	—
61	#104	—	106	3	no	—
62	#101	—	53	3	no	—
63	C₂⁴	—	2	1	no	—
64	#41	—	106	3	no	—
65	#41	—	212	1	no	—
66	C₂×Q₈	—	2	1	no	—
67	C₂×Q₈	—	212	1	no	—
68	C₂×Q₈	C₂²×D₅₃	1	1	yes	characteristic
69	C₂×D₄	—	2	3	no	—
70	C₂×D₄	C₂²⋊D₅₃	1	3	yes	—
71	SD₁₆	—	212	3	no	—
72	D₈	—	212	3	no	—
73	M₁₆	—	106	3	no	—
74	C₂²⋊C₄	—	2	3	no	—
75	C₂²⋊C₄	—	2	6	no	—
76	C₄²	C₂²⋊D₅₃	1	3	yes	—
77	C₂³	—	2	3	no	—
78	C₂³	—	8	2	no	—
79	C₂³	—	8	1	no	—
80	Q₈	—	4	1	no	—
81	Q₈	—	8	1	no	—
82	Q₈	—	106	3	no	—
83	D₄	—	4	3	no	—
84	D₄	—	8	3	no	—
85	D₄	—	106	3	no	—
86	D₄	—	212	1	no	—
87	C₂×C₄	—	2	3	no	—
88	C₂×C₄	—	212	3	no	—
89	C₂×C₄	C₂×C₂²⋊D₅₃	1	3	yes	—
90	C₈	—	212	3	no	—
91	C₂²	—	4	6	no	—
92	C₂²	—	4	3	no	—
93	C₂²	—	8	4	no	—
94	C₂²	—	212	1	no	—
95	C₂²	#13133186	1	1	yes	characteristic
96	C₄	—	2	3	no	—
97	C₄	—	4	3	no	—
98	C₄	—	212	1	no	—
99	C₂	—	2	1	no	—
100	C₂	—	8	3	no	—
101	C₂	—	424	1	no	—
102	C₂	#16810810	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16810795

C₂

yes

maximal, complemented

#16810420

C₂

yes

maximal, complemented

C₅₃×H(3,𝔽₄)

C₂

yes

maximal, characteristic, complemented

#13133265

C₂²

yes

characteristic, complemented

#13133243

—

#13133145

—

#13133136

C₂²

yes

complemented

#124×C₅₃

—

#124×C₅₃

C₂²

yes

—

#120×C₅₃

—

C₂⁴×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

C₂³

yes

characteristic

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

D₄

yes

—

#11719949

—

Q₈×D₅₃

—

#11719945

—

C₂²⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

D₄

yes

—

#11719923

—

#11719922

—

#11719917

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

Q₈×C₅₃

—

Q₈×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂²⋊D₅₃

—

D₂₁₂

—

C₄×D₅₃

—

Dic₁₀₆

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#120

yes

characteristic

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

Dic₅₃

—

#2202

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

B(4,𝔽₂)

yes

characteristic

D₅₃

—

H(3,𝔽₄)

D₅₃

yes

characteristic, complemented

#454

—

#355

—

C₅₃

#2202

yes

characteristic, complemented, Sylow

2₋¹⁺⁴

—

AGL(1,ℤ/8)

—

106

—

#124

—

#124

C₂×D₅₃

yes

—

#120

—

#104

—

106

—

#101

—

C₂⁴

—

#41

—

106

—

#41

—

212

—

C₂×Q₈

—

C₂×Q₈

—

212

—

C₂×Q₈

C₂²×D₅₃

yes

characteristic

C₂×D₄

—

C₂×D₄

C₂²⋊D₅₃

yes

—

SD₁₆

—

212

—

D₈

—

212

—

M₁₆

—

106

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

C₂²⋊D₅₃

yes

—

C₂³

—

C₂³

—

C₂³

—

Q₈

—

Q₈

—

Q₈

—

106

—

D₄

—

D₄

—

D₄

—

106

—

D₄

—

212

—

C₂×C₄

—

C₂×C₄

—

212

—

C₂×C₄

C₂×C₂²⋊D₅₃

yes

—

C₈

—

212

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

212

—

C₂²

#13133186

yes

characteristic

C₄

—

C₄

—

C₄

—

212

—

C₂

—

100

C₂

—

101

C₂

—

424

—

102

C₂

#16810810

yes

characteristic

	subgroup	G/H
center	C₂	#16810810
commutator	C₂×Q₈×C₅₃	C₂³
Frattini	C₂×Q₈	C₂²×D₅₃
Fitting	C₅₃×H(3,𝔽₄)	C₂
socle	C₂×C₅₃	B(4,𝔽₂)
hypercenter	H(3,𝔽₄)	D₅₃
hypocenter	C₅₃	#2202
LCS	G ▷ C₂×Q₈×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×Q₈ ◁ H(3,𝔽₄)

subgroup

G/H

center

C₂

#16810810

commutator

C₂×Q₈×C₅₃

C₂³

Frattini

C₂×Q₈

C₂²×D₅₃

Fitting

C₅₃×H(3,𝔽₄)

C₂

C₂×C₅₃

D₅₃

C₅₃

G ▷ C₂×Q₈×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂ ◁ C₂² ◁ C₂×Q₈ ◁ H(3,𝔽₄)

N	H ≅ G/N	type
1. #16810795	C₂	semidirect
2. #16810420	C₂	semidirect
3. C₅₃×H(3,𝔽₄)	C₂	semidirect
4. #13133265	C₂²	semidirect
7. #13133136	C₂²	semidirect
52. H(3,𝔽₄)	D₅₃	semidirect
55. C₅₃	#2202	semidirect

H ≅ G/N

type

1. #16810795

C₂

semidirect

2. #16810420

C₂

semidirect

3. C₅₃×H(3,𝔽₄)

C₂

semidirect

4. #13133265

C₂²

semidirect

7. #13133136

C₂²

semidirect

52. H(3,𝔽₄)

D₅₃

semidirect

55. C₅₃

#2202

semidirect

Subgroups — #20276732

Conjugacy classes of subgroups

Complements