Group #20286519 — subgroups

Subgroups — #20286519

Order 6784.

444 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 117 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	#16811436	C₂	1	4	yes	maximal, complemented
2	#16810895	C₂	1	4	yes	maximal, complemented
3	#16810805	C₂	1	4	yes	maximal, complemented
4	#16810741	C₂	1	2	yes	maximal, complemented
5	C₅₃×#494	C₂	1	1	yes	maximal, characteristic, complemented
6	#13133265	C₂²	1	4	yes	—
7	#13133249	—	2	8	no	—
8	#13133248	C₂²	1	8	yes	—
9	#13133240	—	2	4	no	—
10	#13133239	C₂²	1	2	yes	—
11	Q₁₆×D₅₃	—	2	4	no	—
12	#13133223	C₂²	1	4	yes	—
13	#13133153	C₂²	1	2	yes	—
14	#13133145	C₂²	1	4	yes	—
15	#13133136	C₂²	1	4	yes	—
16	C₂×Q₁₆×C₅₃	C₂²	1	4	yes	—
17	C₄⋊Q₈×C₅₃	C₂²	1	2	yes	—
18	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
19	C₂×Q₈×C₅₃	C₂³	1	4	yes	—
20	#11719949	—	2	8	no	—
21	Q₈×D₅₃	—	2	8	no	—
22	#11719945	—	4	4	no	—
23	#11719945	C₂³	1	4	yes	—
24	Q₁₆×C₅₃	—	2	4	no	—
25	Q₁₆×C₅₃	—	2	4	no	—
26	C₂×C₈×C₅₃	C₂³	1	2	yes	—
27	C₄⋊C₄×C₅₃	—	4	2	no	—
28	C₄²×C₅₃	C₂³	1	1	yes	characteristic
29	#11719925	—	2	8	no	—
30	#11719924	—	2	8	no	—
31	#11719917	C₂³	1	4	yes	—
32	Dic₂₁₂	D₄	1	2	yes	—
33	D₄₂₄	D₄	1	2	yes	—
34	#11719913	D₄	1	4	yes	—
35	#11719912	—	2	4	no	—
36	C₈×D₅₃	—	2	4	no	—
37	Q₈×C₅₃	—	2	8	no	—
38	C₂×C₄×C₅₃	—	2	1	no	—
39	C₂×C₄×C₅₃	—	2	4	no	—
40	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
41	C₂²⋊D₅₃	—	2	4	no	—
42	D₂₁₂	—	4	4	no	—
43	D₂₁₂	C₂×D₄	1	4	yes	—
44	C₄×D₅₃	—	2	4	no	—
45	C₄×D₅₃	—	4	8	no	—
46	Dic₁₀₆	—	4	4	no	—
47	Dic₁₀₆	C₂×D₄	1	4	yes	—
48	C₈×C₅₃	C₂×D₄	1	2	yes	—
49	C₈×C₅₃	C₂×D₄	1	2	yes	—
50	#88322	—	2	4	no	—
51	C₂²×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
52	C₂×D₅₃	—	2	4	no	—
53	C₄×C₅₃	—	2	2	no	—
54	C₄×C₅₃	—	4	4	no	—
55	C₄×C₅₃	C₂²×D₄	1	2	yes	—
56	Dic₅₃	—	2	4	no	—
57	#3293	—	53	1	no	maximal, Sylow
58	C₂×C₅₃	—	2	1	no	—
59	C₂×C₅₃	D₄²	1	1	yes	characteristic
60	D₅₃	—	4	4	no	—
61	#578	—	53	4	no	—
62	#494	D₅₃	1	1	yes	characteristic, complemented
63	#473	—	53	4	no	—
64	#456	—	53	4	no	—
65	#443	—	53	2	no	—
66	C₅₃	#3293	1	1	yes	characteristic, complemented, Sylow
67	2₋¹⁺⁴	—	53	4	no	—
68	#137	—	53	8	no	—
69	#137	—	106	8	no	—
70	#135	—	53	2	no	—
71	#135	—	106	4	no	—
72	C₂×Q₁₆	—	106	4	no	—
73	C₂×Q₁₆	C₂×D₅₃	1	4	yes	—
74	#131	—	53	4	no	—
75	C₄⋊Q₈	C₂×D₅₃	1	2	yes	—
76	#108	—	53	2	no	—
77	#104	—	53	4	no	—
78	#101	—	53	4	no	—
79	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
80	#41	—	53	4	no	—
81	#41	—	106	8	no	—
82	#41	—	212	4	no	—
83	C₂×Q₈	—	106	8	no	—
84	C₂×Q₈	C₂²×D₅₃	1	4	yes	—
85	Q₁₆	—	2	4	no	—
86	Q₁₆	—	2	4	no	—
87	Q₁₆	—	53	2	no	—
88	Q₁₆	—	106	8	no	—
89	SD₁₆	—	53	4	no	—
90	SD₁₆	—	106	8	no	—
91	D₈	—	53	2	no	—
92	M₁₆	—	53	4	no	—
93	M₁₆	—	106	4	no	—
94	C₂×C₈	—	106	4	no	—
95	C₂×C₈	C₂²×D₅₃	1	2	yes	—
96	C₄⋊C₄	—	4	2	no	—
97	C₄²	C₂²×D₅₃	1	1	yes	characteristic
98	Q₈	—	2	8	no	—
99	Q₈	—	53	4	no	—
100	Q₈	—	212	4	no	—
101	D₄	—	53	4	no	—
102	D₄	—	106	4	no	—
103	D₄	—	212	4	no	—
104	C₂×C₄	—	2	1	no	—
105	C₂×C₄	—	2	4	no	—
106	C₂×C₄	—	106	4	no	—
107	C₂×C₄	—	212	8	no	—
108	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
109	C₈	—	106	4	no	—
110	C₈	D₄×D₅₃	1	2	yes	—
111	C₈	D₄×D₅₃	1	2	yes	—
112	C₂²	—	106	4	no	—
113	C₂²	#13133262	1	1	yes	characteristic
114	C₄	—	2	2	no	—
115	C₄	—	4	4	no	—
116	C₄	—	106	4	no	—
117	C₄	C₂×D₄×D₅₃	1	2	yes	—
118	C₂	—	2	1	no	—
119	C₂	—	212	4	no	—
120	C₂	#16811258	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811436

C₂

yes

maximal, complemented

#16810895

C₂

yes

maximal, complemented

#16810805

C₂

yes

maximal, complemented

#16810741

C₂

yes

maximal, complemented

C₅₃×#494

C₂

yes

maximal, characteristic, complemented

#13133265

C₂²

yes

—

#13133249

—

#13133248

C₂²

yes

—

#13133240

—

#13133239

C₂²

yes

—

Q₁₆×D₅₃

—

#13133223

C₂²

yes

—

#13133153

C₂²

yes

—

#13133145

C₂²

yes

—

#13133136

C₂²

yes

—

C₂×Q₁₆×C₅₃

C₂²

yes

—

C₄⋊Q₈×C₅₃

C₂²

yes

—

C₄×C₈×C₅₃

C₂²

yes

characteristic

C₂×Q₈×C₅₃

C₂³

yes

—

#11719949

—

Q₈×D₅₃

—

#11719945

—

#11719945

C₂³

yes

—

Q₁₆×C₅₃

—

Q₁₆×C₅₃

—

C₂×C₈×C₅₃

C₂³

yes

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

C₂³

yes

—

—

C₂³

yes

—

Dic₂₁₂

D₄

yes

—

D₄₂₄

D₄

yes

—

#11719913

D₄

yes

—

#11719912

—

C₈×D₅₃

—

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂⁴

yes

characteristic

C₂²⋊D₅₃

—

D₂₁₂

—

D₂₁₂

C₂×D₄

yes

—

C₄×D₅₃

—

C₄×D₅₃

—

Dic₁₀₆

—

Dic₁₀₆

C₂×D₄

yes

—

C₈×C₅₃

C₂×D₄

yes

—

C₈×C₅₃

C₂×D₄

yes

—

#88322

—

C₂²×C₅₃

H(5,𝔽₂)

yes

characteristic

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂²×D₄

yes

—

Dic₅₃

—

#3293

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

D₄²

yes

characteristic

D₅₃

—

#578

—

#494

D₅₃

yes

characteristic, complemented

#473

—

#456

—

#443

—

C₅₃

#3293

yes

characteristic, complemented, Sylow

2₋¹⁺⁴

—

#137

—

#137

—

106

—

#135

—

#135

—

106

—

C₂×Q₁₆

—

106

—

C₂×Q₁₆

C₂×D₅₃

yes

—

#131

—

C₄⋊Q₈

C₂×D₅₃

yes

—

#108

—

#104

—

#101

—

C₄×C₈

C₂×D₅₃

yes

characteristic

#41

—

#41

—

106

—

#41

—

212

—

C₂×Q₈

—

106

—

C₂×Q₈

C₂²×D₅₃

yes

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

106

—

SD₁₆

—

SD₁₆

—

106

—

D₈

—

M₁₆

—

M₁₆

—

106

—

C₂×C₈

—

106

—

C₂×C₈

C₂²×D₅₃

yes

—

C₄⋊C₄

—

C₄²

C₂²×D₅₃

yes

characteristic

Q₈

—

Q₈

—

100

Q₈

—

212

—

101

D₄

—

102

D₄

—

106

—

103

D₄

—

212

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

106

—

107

C₂×C₄

—

212

—

108

C₂×C₄

C₂³×D₅₃

yes

characteristic

109

C₈

—

106

—

110

C₈

D₄×D₅₃

yes

—

111

C₈

D₄×D₅₃

yes

—

112

C₂²

—

106

—

113

C₂²

#13133262

yes

characteristic

114

C₄

—

115

C₄

—

116

C₄

—

106

—

117

C₄

C₂×D₄×D₅₃

yes

—

118

C₂

—

119

C₂

—

212

—

120

C₂

#16811258

yes

characteristic

	subgroup	G/H
center	C₂	#16811258
commutator	C₂×C₄×C₅₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₃
Fitting	C₅₃×#494	C₂
socle	C₂×C₅₃	D₄²
hypercenter	#494	D₅₃
hypocenter	C₅₃	#3293
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #494

subgroup

G/H

center

C₂

#16811258

commutator

C₂×C₄×C₅₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₅₃

Fitting

C₅₃×#494

C₂

socle

C₂×C₅₃

D₄²

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ #494

N	H ≅ G/N	type
1. #16811436	C₂	semidirect
2. #16810895	C₂	semidirect
3. #16810805	C₂	semidirect
4. #16810741	C₂	semidirect
5. C₅₃×#494	C₂	semidirect
62. #494	D₅₃	semidirect
66. C₅₃	#3293	semidirect

H ≅ G/N

type

1. #16811436

C₂

semidirect

2. #16810895

C₂

semidirect

3. #16810805

C₂

semidirect

4. #16810741

C₂

semidirect

5. C₅₃×#494

C₂

semidirect

62. #494

D₅₃

semidirect

66. C₅₃

#3293

semidirect

Subgroups — #20286519

Conjugacy classes of subgroups

Complements