Group #20273453 — subgroups

Subgroups — #20273453

Order 6784.

210 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 85 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	#16810647	C₂	1	2	yes	maximal
2	#16810369	C₂	1	4	yes	maximal, complemented
3	C₅₃×#533	C₂	1	1	yes	maximal, characteristic
4	C₂×#11719917	—	2	2	no	—
5	C₄⋊#88322	—	2	2	no	—
6	C₄⋊#88322	C₂²	1	4	yes	—
7	C₄⋊Q₈×C₅₃	C₂²	1	2	yes	—
8	#125×C₅₃	C₄	1	2	yes	—
9	C₂²⋊Q₈×C₅₃	—	2	2	no	—
10	C₄×Q₈×C₅₃	—	2	2	no	—
11	#117×C₅₃	C₄	1	2	yes	—
12	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
13	C₂×Q₈×C₅₃	—	2	2	no	—
14	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₄×C₅₃	D₄	1	2	yes	—
16	C₄⋊C₄×C₅₃	—	2	4	no	—
17	C₄⋊C₄×C₅₃	—	2	2	no	—
18	C₄⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
19	C₂²⋊C₄×C₅₃	—	2	2	no	—
20	C₄²×C₅₃	—	2	2	no	—
21	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
22	C₄²×C₅₃	D₄	1	2	yes	—
23	C₄²×C₅₃	C₂³	1	1	yes	characteristic
24	#11719917	—	4	4	no	—
25	C₂×#88322	—	2	4	no	—
26	C₂³×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
27	Q₈×C₅₃	—	4	2	no	—
28	Q₈×C₅₃	—	4	2	no	—
29	C₂×C₄×C₅₃	—	2	2	no	—
30	C₂×C₄×C₅₃	—	2	2	no	—
31	C₂×C₄×C₅₃	—	2	2	no	—
32	C₂×C₄×C₅₃	—	2	4	no	—
33	C₂×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
34	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
35	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
37	#88322	—	4	4	no	—
38	C₂²×C₅₃	—	2	1	no	—
39	C₂²×C₅₃	—	2	2	no	—
40	C₂²×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
41	C₄×C₅₃	—	2	1	no	—
42	C₄×C₅₃	—	2	2	no	—
43	C₄×C₅₃	—	4	4	no	—
44	C₄×C₅₃	#101	1	2	yes	—
45	C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
46	C₄×C₅₃	#137	1	2	yes	—
47	#1561	—	53	1	no	maximal, Sylow
48	C₂×C₅₃	—	4	1	no	—
49	C₂×C₅₃	C₂×#101	1	1	yes	characteristic
50	C₂×C₅₃	#417	1	2	yes	—
51	#533	D₅₃	1	1	yes	characteristic
52	#423	—	53	2	no	—
53	#332	—	53	4	no	—
54	C₅₃	#1561	1	1	yes	characteristic, complemented, Sylow
55	C₂×M₁₆	—	106	2	no	—
56	C₄⋊Q₈	C₂×D₅₃	1	2	yes	—
57	#125	Dic₅₃	1	2	yes	—
58	C₂²⋊Q₈	—	2	2	no	—
59	C₄×Q₈	—	2	2	no	—
60	#117	Dic₅₃	1	2	yes	—
61	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
62	C₄⋊C₈	—	53	4	no	—
63	C₄⋊C₈	—	106	2	no	—
64	C₂×Q₈	—	2	2	no	—
65	C₂²×C₄	C₂×Dic₅₃	1	1	yes	characteristic
66	C₂²×C₄	C₂²⋊D₅₃	1	2	yes	—
67	M₁₆	—	212	4	no	—
68	C₂×C₈	—	106	4	no	—
69	C₄⋊C₄	—	2	2	no	—
70	C₄⋊C₄	—	2	4	no	—
71	C₄⋊C₄	C₂×Dic₅₃	1	4	yes	—
72	C₂²⋊C₄	—	2	2	no	—
73	C₄²	—	2	2	no	—
74	C₄²	C₂×Dic₅₃	1	1	yes	characteristic
75	C₄²	C₂²⋊D₅₃	1	2	yes	—
76	C₄²	C₂²×D₅₃	1	1	yes	characteristic
77	C₂³	C₂²⋊Dic₅₃	1	1	yes	characteristic
78	Q₈	—	4	2	no	—
79	Q₈	—	4	2	no	—
80	C₂×C₄	—	2	2	no	—
81	C₂×C₄	—	2	2	no	—
82	C₂×C₄	—	2	2	no	—
83	C₂×C₄	—	2	4	no	—
84	C₂×C₄	C₂²⋊Dic₅₃	1	2	yes	—
85	C₂×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
86	C₂×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
87	C₂×C₄	C₂×C₂²⋊D₅₃	1	2	yes	—
88	C₈	—	212	4	no	—
89	C₂²	—	2	1	no	—
90	C₂²	—	2	2	no	—
91	C₂²	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
92	C₄	—	2	1	no	—
93	C₄	—	2	2	no	—
94	C₄	—	4	4	no	—
95	C₄	#13133136	1	2	yes	—
96	C₄	#13133242	1	2	yes	—
97	C₄	#13133247	1	2	yes	—
98	C₂	—	4	1	no	—
99	C₂	C₂×#13133136	1	1	yes	characteristic
100	C₂	#16810624	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16810647

C₂

yes

maximal

#16810369

C₂

yes

maximal, complemented

C₅₃×#533

C₂

yes

maximal, characteristic

C₂×#11719917

—

C₄⋊#88322

—

C₄⋊#88322

C₂²

yes

—

C₄⋊Q₈×C₅₃

C₂²

yes

—

#125×C₅₃

C₄

yes

—

C₂²⋊Q₈×C₅₃

—

C₄×Q₈×C₅₃

—

#117×C₅₃

C₄

yes

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₂×Q₈×C₅₃

—

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

D₄

yes

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

C₂×C₄

yes

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄²×C₅₃

D₄

yes

—

C₄²×C₅₃

C₂³

yes

characteristic

#11719917

—

C₂×#88322

—

C₂³×C₅₃

C₂²⋊C₄

yes

characteristic

Q₈×C₅₃

—

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²⋊C₄

yes

—

C₂×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#101

yes

—

C₄×C₅₃

AGL(1,ℤ/8)

yes

—

C₄×C₅₃

#137

yes

—

#1561

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

C₂×#101

yes

characteristic

C₂×C₅₃

#417

yes

—

#533

D₅₃

yes

characteristic

#423

—

#332

—

C₅₃

#1561

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

106

—

C₄⋊Q₈

C₂×D₅₃

yes

—

#125

Dic₅₃

yes

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

Dic₅₃

yes

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

106

—

C₂×Q₈

—

C₂²×C₄

C₂×Dic₅₃

yes

characteristic

C₂²×C₄

C₂²⋊D₅₃

yes

—

M₁₆

—

212

—

C₂×C₈

—

106

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×Dic₅₃

yes

—

C₂²⋊C₄

—

C₄²

—

C₄²

C₂×Dic₅₃

yes

characteristic

C₄²

C₂²⋊D₅₃

yes

—

C₄²

C₂²×D₅₃

yes

characteristic

C₂³

C₂²⋊Dic₅₃

yes

characteristic

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²⋊Dic₅₃

yes

—

C₂×C₄

C₂²⋊Dic₅₃

yes

characteristic

C₂×C₄

C₂²×Dic₅₃

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₅₃

yes

—

C₈

—

212

—

C₂²

—

C₂²

—

C₂²

C₂×C₂²⋊Dic₅₃

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

#13133136

yes

—

C₄

#13133242

yes

—

C₄

#13133247

yes

—

C₂

—

C₂

C₂×#13133136

yes

characteristic

100

C₂

#16810624

yes

—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊Dic₅₃
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₄²	C₂²×D₅₃
Fitting	C₅₃×#533	C₂
socle	C₂²×C₅₃	C₂×C₂²⋊C₄
hypercenter	#533	D₅₃
hypocenter	C₅₃	#1561
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #533

subgroup

G/H

center

C₂²

C₂×C₂²⋊Dic₅₃

commutator

C₂×C₄×C₅₃

C₂²×C₄

Frattini

C₄²

C₂²×D₅₃

Fitting

C₅₃×#533

C₂

socle

C₂²×C₅₃

C₂×C₂²⋊C₄

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #533

N	H ≅ G/N	type
2. #16810369	C₂	semidirect
54. C₅₃	#1561	semidirect

H ≅ G/N

type

2. #16810369

C₂

semidirect

54. C₅₃

#1561

semidirect

Subgroups — #20273453

Conjugacy classes of subgroups

Complements