Group #20274209 — subgroups

Subgroups — #20274209

Order 6784.

152 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 152 conjugacy classes of subgroups — too many to draw. Only its 57 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	#16811003	C₂	1	1	yes	maximal, characteristic
2	#16810908	C₂	1	2	yes	maximal
3	#16810374	C₂	1	1	yes	maximal, characteristic
4	#16810370	C₂	1	2	yes	maximal
5	C₅₃×#446	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₂₁₂	—	2	1	no	—
7	C₂×Dic₂₁₂	—	2	1	no	—
8	#13133217	C₂²	1	2	yes	—
9	#13133148	—	2	2	no	—
10	C₄⋊#88322	C₂²	1	2	yes	—
11	#13133141	—	2	2	no	—
12	C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
13	C₄⋊C₈×C₅₃	—	2	1	no	—
14	C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
15	#97×C₅₃	—	2	1	no	—
16	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×Q₈×C₅₃	D₄	1	1	yes	characteristic
18	C₂×Dic₁₀₆	—	2	2	no	—
19	C₂×C₈×C₅₃	—	2	1	no	—
20	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
21	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
22	C₄⋊C₄×C₅₃	—	2	1	no	—
23	C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
24	C₄²×C₅₃	—	2	1	no	—
25	C₄²×C₅₃	C₂³	1	1	yes	characteristic
26	C₄⋊Dic₅₃	—	2	2	no	—
27	C₄⋊Dic₅₃	—	2	2	no	—
28	C₂×#88322	—	2	2	no	—
29	Dic₂₁₂	—	4	2	no	—
30	Dic₂₁₂	—	4	2	no	—
31	Q₈×C₅₃	—	2	2	no	—
32	C₂×C₄×C₅₃	—	2	1	no	—
33	C₂×C₄×C₅₃	—	2	1	no	—
34	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
35	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
37	C₂×Dic₅₃	—	4	2	no	—
38	Dic₁₀₆	—	4	2	no	—
39	Dic₁₀₆	—	4	2	no	—
40	C₈×C₅₃	—	2	1	no	—
41	C₈×C₅₃	—	4	1	no	—
42	C₈×C₅₃	SD₁₆	1	2	yes	—
43	#88322	—	4	2	no	—
44	C₂²×C₅₃	#121	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	C₂×SD₁₆	1	1	yes	characteristic
49	C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
50	C₄×C₅₃	#137	1	2	yes	—
51	Dic₅₃	—	8	2	no	—
52	#1690	—	53	1	no	maximal, Sylow
53	C₂×C₅₃	#460	1	1	yes	characteristic
54	C₂×C₅₃	#470	1	1	yes	characteristic
55	C₂×C₅₃	#473	1	1	yes	characteristic
56	#494	—	53	1	no	—
57	#475	—	53	2	no	—
58	#446	D₅₃	1	1	yes	characteristic
59	#334	—	53	1	no	—
60	#332	—	53	2	no	—
61	C₅₃	#1690	1	1	yes	characteristic, complemented, Sylow
62	C₂×Q₁₆	—	106	1	no	—
63	C₂×Q₁₆	—	106	1	no	—
64	C₄⋊Q₈	—	53	2	no	—
65	C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic
66	#106	—	106	2	no	—
67	C₄⋊C₈	—	2	1	no	—
68	C₄⋊C₈	—	53	2	no	—
69	C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
70	#103	—	106	2	no	—
71	#97	—	2	1	no	—
72	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
73	C₂×Q₈	—	106	2	no	—
74	C₂×Q₈	D₂₁₂	1	1	yes	characteristic
75	Q₁₆	—	212	2	no	—
76	Q₁₆	—	212	2	no	—
77	C₂×C₈	—	2	1	no	—
78	C₂×C₈	—	106	2	no	—
79	C₂×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
80	C₂×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
81	C₄⋊C₄	—	2	1	no	—
82	C₄⋊C₄	—	106	2	no	—
83	C₄⋊C₄	—	106	2	no	—
84	C₄⋊C₄	D₂₁₂	1	1	yes	characteristic
85	C₄²	—	2	1	no	—
86	C₄²	C₂²×D₅₃	1	1	yes	characteristic
87	Q₈	—	2	2	no	—
88	Q₈	—	212	2	no	—
89	Q₈	—	212	2	no	—
90	C₂×C₄	—	2	1	no	—
91	C₂×C₄	—	2	1	no	—
92	C₂×C₄	—	212	2	no	—
93	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
94	C₂×C₄	#11719945	1	1	yes	characteristic
95	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
96	C₈	—	2	1	no	—
97	C₈	—	4	1	no	—
98	C₈	—	212	2	no	—
99	C₈	#11719924	1	2	yes	—
100	C₂²	#13133191	1	1	yes	characteristic
101	C₄	—	2	1	no	—
102	C₄	—	4	1	no	—
103	C₄	—	4	1	no	—
104	C₄	—	424	2	no	—
105	C₄	C₂×#11719924	1	1	yes	characteristic
106	C₄	#13133242	1	1	yes	characteristic
107	C₄	#13133250	1	2	yes	—
108	C₂	#16810823	1	1	yes	characteristic
109	C₂	#16810881	1	1	yes	characteristic
110	C₂	#16810898	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811003

C₂

yes

maximal, characteristic

#16810908

C₂

yes

maximal

#16810374

C₂

yes

maximal, characteristic

#16810370

C₂

yes

maximal

C₅₃×#446

C₂

yes

maximal, characteristic

C₂×Dic₂₁₂

—

C₂×Dic₂₁₂

—

#13133217

C₂²

yes

—

#13133148

—

C₄⋊#88322

C₂²

yes

—

#13133141

—

C₄×Q₈×C₅₃

C₂²

yes

characteristic

C₄⋊C₈×C₅₃

—

C₄⋊C₈×C₅₃

C₂²

yes

characteristic

#97×C₅₃

—

C₄×C₈×C₅₃

C₂²

yes

characteristic

C₂×Q₈×C₅₃

D₄

yes

characteristic

C₂×Dic₁₀₆

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

D₄

yes

characteristic

C₂×C₈×C₅₃

D₄

yes

characteristic

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

D₄

yes

characteristic

C₄²×C₅₃

—

C₄²×C₅₃

C₂³

yes

characteristic

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₂×#88322

—

Dic₂₁₂

—

Dic₂₁₂

—

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

#41

yes

characteristic

C₂×Dic₅₃

—

Dic₁₀₆

—

Dic₁₀₆

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

SD₁₆

yes

—

#88322

—

C₂²×C₅₃

#121

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂×SD₁₆

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

—

Dic₅₃

—

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

—

D₅₃

yes

characteristic

#334

—

#332

—

C₅₃

#1690

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

106

—

C₂×Q₁₆

—

106

—

C₄⋊Q₈

—

C₄×Q₈

C₂×D₅₃

yes

characteristic

#106

—

106

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₅₃

yes

characteristic

#103

—

106

—

#97

—

C₄×C₈

C₂×D₅₃

yes

characteristic

C₂×Q₈

—

106

—

C₂×Q₈

D₂₁₂

yes

characteristic

Q₁₆

—

212

—

Q₁₆

—

212

—

C₂×C₈

—

C₂×C₈

—

106

—

C₂×C₈

C₂²⋊D₅₃

yes

characteristic

C₂×C₈

C₂²⋊D₅₃

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

106

—

C₄⋊C₄

—

106

—

C₄⋊C₄

D₂₁₂

yes

characteristic

C₄²

—

C₄²

C₂²×D₅₃

yes

characteristic

Q₈

—

Q₈

—

212

—

Q₈

—

212

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

212

—

C₂×C₄

C₂×D₂₁₂

yes

characteristic

C₂×C₄

#11719945

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

C₈

—

C₈

—

C₈

—

212

—

C₈

#11719924

yes

—

100

C₂²

#13133191

yes

characteristic

101

C₄

—

102

C₄

—

103

C₄

—

104

C₄

—

424

—

105

C₄

C₂×#11719924

yes

characteristic

106

C₄

#13133242

yes

characteristic

107

C₄

#13133250

yes

—

108

C₂

#16810823

yes

characteristic

109

C₂

#16810881

yes

characteristic

110

C₂

#16810898

yes

characteristic

	subgroup	G/H
center	C₂²	#13133191
commutator	C₄²×C₅₃	C₂³
Frattini	C₄²	C₂²×D₅₃
Fitting	C₅₃×#446	C₂
socle	C₂²×C₅₃	#121
hypercenter	#446	D₅₃
hypocenter	C₅₃	#1690
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #446

subgroup

G/H

center

C₂²

#13133191

commutator

C₄²×C₅₃

C₂³

Frattini

C₄²

C₂²×D₅₃

Fitting

C₅₃×#446

C₂

C₂²×C₅₃

D₅₃

C₅₃

G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #446

N	H ≅ G/N	type
61. C₅₃	#1690	semidirect

H ≅ G/N

type

61. C₅₃

#1690

semidirect

Subgroups — #20274209

Conjugacy classes of subgroups

Complements