Group #20272863 — subgroups

Subgroups — #20272863

Order 6784.

112 conjugacy classes of subgroups; only the 25 normal subgroups are shown.

normal subgroups only

This group has 112 conjugacy classes of subgroups — too many to draw. Only its 25 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	#16810420	C₂	1	1	yes	maximal, characteristic, complemented
2	#16810363	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#455	C₂	1	1	yes	maximal, characteristic
4	#13133136	—	2	1	no	—
5	#13133128	—	2	1	no	—
6	H(5,𝔽₂)×C₅₃	C₄	1	1	yes	characteristic
7	#137×C₅₃	—	2	1	no	—
8	#124×C₅₃	C₂²	1	1	yes	characteristic
9	#104×C₅₃	—	2	1	no	—
10	#100×C₅₃	C₄	1	1	yes	characteristic
11	#41×C₅₃	—	2	1	no	—
12	#41×C₅₃	—	4	1	no	—
13	C₂×Q₈×C₅₃	D₄	1	1	yes	characteristic
14	C₂×D₄×C₅₃	—	4	1	no	—
15	C₂×D₄×C₅₃	—	4	1	no	—
16	C₂×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
17	Q₁₆×C₅₃	—	4	1	no	—
18	SD₁₆×C₅₃	—	4	1	no	—
19	M₁₆×C₅₃	—	2	1	no	—
20	C₂²⋊C₄×C₅₃	—	4	1	no	—
21	C₄²×C₅₃	D₄	1	1	yes	characteristic
22	#11719917	—	4	1	no	—
23	C₂×#88322	—	4	1	no	—
24	C₂³×C₅₃	—	2	1	no	—
25	C₂³×C₅₃	—	4	1	no	—
26	Q₈×C₅₃	—	2	1	no	—
27	Q₈×C₅₃	—	4	1	no	—
28	D₄×C₅₃	—	2	1	no	—
29	D₄×C₅₃	—	4	1	no	—
30	D₄×C₅₃	—	4	2	no	—
31	D₄×C₅₃	—	4	1	no	—
32	C₂×C₄×C₅₃	—	2	1	no	—
33	C₂×C₄×C₅₃	—	2	1	no	—
34	C₂×C₄×C₅₃	—	4	1	no	—
35	C₂×C₄×C₅₃	—	4	1	no	—
36	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
37	C₈×C₅₃	—	4	1	no	—
38	#88322	—	4	2	no	—
39	#88322	—	8	1	no	—
40	C₂²×C₅₃	—	4	1	no	—
41	C₂²×C₅₃	—	4	1	no	—
42	C₂²×C₅₃	—	8	1	no	—
43	C₂²×C₅₃	—	16	1	no	—
44	C₂²×C₅₃	#99	1	1	yes	characteristic
45	C₄×C₅₃	—	2	1	no	—
46	C₄×C₅₃	—	4	1	no	—
47	C₄×C₅₃	—	4	1	no	—
48	C₄×C₅₃	—	4	1	no	—
49	#1403	—	53	1	no	maximal, Sylow
50	C₂×C₅₃	—	2	1	no	—
51	C₂×C₅₃	—	8	1	no	—
52	C₂×C₅₃	—	8	1	no	—
53	C₂×C₅₃	AΣL(1,𝔽₁₆)	1	1	yes	characteristic
54	#455	D₅₃	1	1	yes	characteristic
55	#355	—	53	1	no	—
56	#329	—	53	1	no	—
57	C₅₃	#1403	1	1	yes	characteristic, complemented, Sylow
58	H(5,𝔽₂)	Dic₅₃	1	1	yes	characteristic
59	#137	—	2	1	no	—
60	#124	C₂×D₅₃	1	1	yes	characteristic
61	#104	—	2	1	no	—
62	#101	—	106	1	no	—
63	#100	Dic₅₃	1	1	yes	characteristic
64	#97	—	106	1	no	—
65	#41	—	2	1	no	—
66	#41	—	4	1	no	—
67	C₂×Q₈	C₂²⋊D₅₃	1	1	yes	characteristic
68	C₂×D₄	—	4	1	no	—
69	C₂×D₄	—	4	1	no	—
70	C₂×D₄	C₂×Dic₅₃	1	1	yes	characteristic
71	Q₁₆	—	4	1	no	—
72	SD₁₆	—	4	1	no	—
73	M₁₆	—	2	1	no	—
74	M₁₆	—	212	1	no	—
75	C₂×C₈	—	212	1	no	—
76	C₂²⋊C₄	—	4	1	no	—
77	C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
78	C₂³	—	2	1	no	—
79	C₂³	—	4	1	no	—
80	Q₈	—	2	1	no	—
81	Q₈	—	4	1	no	—
82	D₄	—	2	1	no	—
83	D₄	—	4	1	no	—
84	D₄	—	4	2	no	—
85	D₄	—	4	1	no	—
86	C₂×C₄	—	2	1	no	—
87	C₂×C₄	—	2	1	no	—
88	C₂×C₄	—	4	1	no	—
89	C₂×C₄	—	4	1	no	—
90	C₂×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
91	C₈	—	4	1	no	—
92	C₈	—	424	1	no	—
93	C₈	—	212	2	no	—
94	C₂²	—	4	1	no	—
95	C₂²	—	4	1	no	—
96	C₂²	—	8	1	no	—
97	C₂²	—	16	1	no	—
98	C₂²	#13133132	1	1	yes	characteristic
99	C₄	—	2	1	no	—
100	C₄	—	4	1	no	—
101	C₄	—	4	1	no	—
102	C₄	—	4	1	no	—
103	C₂	—	2	1	no	—
104	C₂	—	8	1	no	—
105	C₂	—	8	1	no	—
106	C₂	#16810410	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16810420

C₂

yes

maximal, characteristic, complemented

#16810363

C₂

yes

maximal, characteristic, complemented

C₅₃×#455

C₂

yes

maximal, characteristic

#13133136

—

#13133128

—

H(5,𝔽₂)×C₅₃

C₄

yes

characteristic

#137×C₅₃

—

#124×C₅₃

C₂²

yes

characteristic

#104×C₅₃

—

#100×C₅₃

C₄

yes

characteristic

#41×C₅₃

—

#41×C₅₃

—

C₂×Q₈×C₅₃

D₄

yes

characteristic

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

C₂×C₄

yes

characteristic

Q₁₆×C₅₃

—

SD₁₆×C₅₃

—

M₁₆×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

D₄

yes

characteristic

#11719917

—

C₂×#88322

—

C₂³×C₅₃

—

C₂³×C₅₃

—

Q₈×C₅₃

—

Q₈×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₈×C₅₃

—

#88322

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#99

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

#1403

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

AΣL(1,𝔽₁₆)

yes

characteristic

#455

D₅₃

yes

characteristic

#355

—

#329

—

C₅₃

#1403

yes

characteristic, complemented, Sylow

H(5,𝔽₂)

Dic₅₃

yes

characteristic

#137

—

#124

C₂×D₅₃

yes

characteristic

#104

—

#101

—

106

—

#100

Dic₅₃

yes

characteristic

#97

—

106

—

#41

—

#41

—

C₂×Q₈

C₂²⋊D₅₃

yes

characteristic

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×Dic₅₃

yes

characteristic

Q₁₆

—

SD₁₆

—

M₁₆

—

M₁₆

—

212

—

C₂×C₈

—

212

—

C₂²⋊C₄

—

C₄²

C₂²⋊D₅₃

yes

characteristic

C₂³

—

C₂³

—

Q₈

—

Q₈

—

D₄

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²⋊Dic₅₃

yes

characteristic

C₈

—

C₈

—

424

—

C₈

—

212

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

#13133132

yes

characteristic

C₄

—

100

C₄

—

101

C₄

—

102

C₄

—

103

C₂

—

104

C₂

—

105

C₂

—

106

C₂

#16810410

yes

characteristic

	subgroup	G/H
center	C₂	#16810410
commutator	C₂×D₄×C₅₃	C₂×C₄
Frattini	#124	C₂×D₅₃
Fitting	C₅₃×#455	C₂
socle	C₂×C₅₃	AΣL(1,𝔽₁₆)
hypercenter	#455	D₅₃
hypocenter	C₅₃	#1403
LCS	G ▷ C₂×D₄×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ #124 ◁ #455

subgroup

G/H

center

C₂

#16810410

commutator

C₂×D₄×C₅₃

C₂×C₄

Frattini

#124

C₂×D₅₃

Fitting

C₅₃×#455

C₂

socle

C₂×C₅₃

AΣL(1,𝔽₁₆)

D₅₃

C₅₃

G ▷ C₂×D₄×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ #124 ◁ #455

N	H ≅ G/N	type
1. #16810420	C₂	semidirect
2. #16810363	C₂	semidirect
57. C₅₃	#1403	semidirect

H ≅ G/N

type

1. #16810420

C₂

semidirect

2. #16810363

C₂

semidirect

57. C₅₃

#1403

semidirect

Subgroups — #20272863

Conjugacy classes of subgroups

Complements