Group #20286514 — subgroups

Subgroups — #20286514

Order 6784.

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

normal subgroups only

This group has 476 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	#16811424	C₂	1	4	yes	maximal, complemented
2	#16810890	C₂	1	4	yes	maximal, complemented
3	#16810790	C₂	1	4	yes	maximal, complemented
4	#16810741	C₂	1	2	yes	maximal, complemented
5	C₅₃×#493	C₂	1	1	yes	maximal, characteristic, complemented
6	#13133262	C₂²	1	4	yes	complemented
7	#13133243	C₂²	1	8	yes	complemented
8	#13133241	—	2	8	no	—
9	#13133239	C₂²	1	2	yes	—
10	#13133236	—	2	4	no	—
11	D₈×D₅₃	—	2	4	no	—
12	#13133223	C₂²	1	4	yes	complemented
13	#13133153	C₂²	1	2	yes	complemented
14	#13133145	C₂²	1	4	yes	complemented
15	#13133134	C₂²	1	4	yes	complemented
16	C₂×D₈×C₅₃	C₂²	1	4	yes	complemented
17	#127×C₅₃	C₂²	1	2	yes	complemented
18	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic, complemented
19	C₂×D₄×C₅₃	—	4	2	no	—
20	C₂×D₄×C₅₃	C₂³	1	4	yes	—
21	C₂×C₂²⋊D₅₃	—	4	4	no	—
22	#11719947	—	2	8	no	—
23	D₄×D₅₃	—	2	8	no	—
24	#11719945	C₂³	1	4	yes	—
25	D₈×C₅₃	—	2	4	no	—
26	D₈×C₅₃	—	2	4	no	—
27	C₂×C₈×C₅₃	C₂³	1	2	yes	—
28	C₄²×C₅₃	C₂³	1	1	yes	characteristic
29	#11719923	—	2	8	no	—
30	#11719922	—	2	8	no	—
31	#11719917	C₂³	1	4	yes	—
32	Dic₂₁₂	D₄	1	2	yes	complemented
33	D₄₂₄	D₄	1	2	yes	complemented
34	#11719913	D₄	1	4	yes	complemented
35	#11719912	—	2	4	no	—
36	C₈×D₅₃	—	2	4	no	—
37	C₂³×C₅₃	—	2	4	no	—
38	C₂²×D₅₃	—	4	4	no	—
39	D₄×C₅₃	—	2	8	no	—
40	D₄×C₅₃	—	8	4	no	—
41	C₂×C₄×C₅₃	—	2	1	no	—
42	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
43	C₂²⋊D₅₃	—	2	4	no	—
44	C₂²⋊D₅₃	—	4	8	no	—
45	C₂×Dic₅₃	—	4	4	no	—
46	D₂₁₂	C₂×D₄	1	4	yes	—
47	C₄×D₅₃	—	2	4	no	—
48	Dic₁₀₆	C₂×D₄	1	4	yes	—
49	C₈×C₅₃	C₂×D₄	1	2	yes	—
50	C₈×C₅₃	C₂×D₄	1	2	yes	—
51	#88322	—	2	4	no	—
52	C₂²×C₅₃	—	4	4	no	—
53	C₂²×C₅₃	—	8	4	no	—
54	C₂²×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
55	C₂×D₅₃	—	2	4	no	—
56	C₂×D₅₃	—	16	4	no	—
57	C₄×C₅₃	—	2	2	no	—
58	C₄×C₅₃	C₂²×D₄	1	2	yes	—
59	Dic₅₃	—	2	4	no	—
60	#3292	—	53	1	no	maximal, Sylow
61	C₂×C₅₃	—	2	1	no	—
62	C₂×C₅₃	—	8	4	no	—
63	C₂×C₅₃	D₄²	1	1	yes	characteristic
64	D₅₃	—	4	4	no	—
65	#576	—	53	4	no	—
66	#493	D₅₃	1	1	yes	characteristic, complemented
67	#472	—	53	4	no	—
68	#453	—	53	4	no	—
69	#443	—	53	2	no	—
70	C₅₃	#3292	1	1	yes	characteristic, complemented, Sylow
71	H(5,𝔽₂)	—	53	4	no	—
72	AGL(1,ℤ/8)	—	53	8	no	—
73	AGL(1,ℤ/8)	—	106	8	no	—
74	#135	—	53	2	no	—
75	#135	—	106	4	no	—
76	C₂×D₈	—	106	4	no	—
77	C₂×D₈	C₂×D₅₃	1	4	yes	complemented
78	#131	—	53	4	no	—
79	#127	C₂×D₅₃	1	2	yes	complemented
80	#108	—	53	2	no	—
81	#104	—	53	4	no	—
82	#100	—	53	4	no	—
83	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic, complemented
84	#41	—	53	4	no	—
85	#41	—	106	8	no	—
86	C₂×D₄	—	4	2	no	—
87	C₂×D₄	—	106	8	no	—
88	C₂×D₄	—	212	4	no	—
89	C₂×D₄	C₂²×D₅₃	1	4	yes	—
90	Q₁₆	—	53	2	no	—
91	SD₁₆	—	53	4	no	—
92	SD₁₆	—	106	8	no	—
93	D₈	—	2	4	no	—
94	D₈	—	2	4	no	—
95	D₈	—	53	2	no	—
96	D₈	—	106	8	no	—
97	M₁₆	—	53	4	no	—
98	M₁₆	—	106	4	no	—
99	C₂×C₈	—	106	4	no	—
100	C₂×C₈	C₂²×D₅₃	1	2	yes	—
101	C₄²	C₂²×D₅₃	1	1	yes	characteristic
102	C₂³	—	2	4	no	—
103	C₂³	—	212	4	no	—
104	Q₈	—	53	4	no	—
105	D₄	—	2	8	no	—
106	D₄	—	8	4	no	—
107	D₄	—	53	4	no	—
108	D₄	—	106	4	no	—
109	D₄	—	212	8	no	—
110	C₂×C₄	—	2	1	no	—
111	C₂×C₄	—	106	4	no	—
112	C₂×C₄	—	212	4	no	—
113	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
114	C₈	—	106	4	no	—
115	C₈	D₄×D₅₃	1	2	yes	—
116	C₈	D₄×D₅₃	1	2	yes	—
117	C₂²	—	4	4	no	—
118	C₂²	—	8	4	no	—
119	C₂²	—	848	4	no	—
120	C₂²	—	106	4	no	—
121	C₂²	#13133262	1	1	yes	characteristic
122	C₄	—	2	2	no	—
123	C₄	—	106	4	no	—
124	C₄	C₂×D₄×D₅₃	1	2	yes	—
125	C₂	—	2	1	no	—
126	C₂	—	8	4	no	—
127	C₂	—	212	4	no	—
128	C₂	#16811258	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811424

C₂

yes

maximal, complemented

#16810890

C₂

yes

maximal, complemented

#16810790

C₂

yes

maximal, complemented

#16810741

C₂

yes

maximal, complemented

C₅₃×#493

C₂

yes

maximal, characteristic, complemented

#13133262

C₂²

yes

complemented

#13133243

C₂²

yes

complemented

#13133241

—

#13133239

C₂²

yes

—

#13133236

—

D₈×D₅₃

—

#13133223

C₂²

yes

complemented

#13133153

C₂²

yes

complemented

#13133145

C₂²

yes

complemented

#13133134

C₂²

yes

complemented

C₂×D₈×C₅₃

C₂²

yes

complemented

#127×C₅₃

C₂²

yes

complemented

C₄×C₈×C₅₃

C₂²

yes

characteristic, complemented

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

C₂³

yes

—

C₂×C₂²⋊D₅₃

—

#11719947

—

D₄×D₅₃

—

#11719945

C₂³

yes

—

D₈×C₅₃

—

D₈×C₅₃

—

C₂×C₈×C₅₃

C₂³

yes

—

C₄²×C₅₃

C₂³

yes

—

—

C₂³

yes

—

Dic₂₁₂

D₄

yes

complemented

D₄₂₄

D₄

yes

complemented

#11719913

D₄

yes

complemented

#11719912

—

C₈×D₅₃

—

C₂³×C₅₃

—

C₂²×D₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂⁴

yes

characteristic

C₂²⋊D₅₃

—

C₂²⋊D₅₃

—

C₂×Dic₅₃

—

D₂₁₂

C₂×D₄

yes

—

C₄×D₅₃

—

Dic₁₀₆

C₂×D₄

yes

—

C₈×C₅₃

C₂×D₄

yes

—

C₈×C₅₃

C₂×D₄

yes

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

H(5,𝔽₂)

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂²×D₄

yes

—

Dic₅₃

—

#3292

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

D₄²

yes

characteristic

D₅₃

—

#576

—

#493

D₅₃

yes

characteristic, complemented

#472

—

#453

—

#443

—

C₅₃

#3292

yes

characteristic, complemented, Sylow

H(5,𝔽₂)

—

AGL(1,ℤ/8)

—

AGL(1,ℤ/8)

—

106

—

#135

—

#135

—

106

—

C₂×D₈

—

106

—

C₂×D₈

C₂×D₅₃

yes

complemented

#131

—

#127

C₂×D₅₃

yes

—

—

—

C₄×C₈

C₂×D₅₃

yes

characteristic, complemented

#41

—

#41

—

106

—

C₂×D₄

—

C₂×D₄

—

106

—

C₂×D₄

—

212

—

C₂×D₄

C₂²×D₅₃

yes

—

Q₁₆

—

SD₁₆

—

SD₁₆

—

106

—

D₈

—

D₈

—

D₈

—

D₈

—

106

—

M₁₆

—

M₁₆

—

106

—

C₂×C₈

—

106

—

100

C₂×C₈

C₂²×D₅₃

yes

—

101

C₄²

C₂²×D₅₃

yes

characteristic

102

C₂³

—

103

C₂³

—

212

—

104

Q₈

—

105

D₄

—

106

D₄

—

107

D₄

—

108

D₄

—

106

—

109

D₄

—

212

—

110

C₂×C₄

—

111

C₂×C₄

—

106

—

112

C₂×C₄

—

212

—

113

C₂×C₄

C₂³×D₅₃

yes

characteristic

114

C₈

—

106

—

115

C₈

D₄×D₅₃

yes

—

116

C₈

D₄×D₅₃

yes

—

117

C₂²

—

118

C₂²

—

119

C₂²

—

848

—

120

C₂²

—

106

—

121

C₂²

#13133262

yes

characteristic

122

C₄

—

123

C₄

—

106

—

124

C₄

C₂×D₄×D₅₃

yes

—

125

C₂

—

126

C₂

—

127

C₂

—

212

—

128

C₂

#16811258

yes

characteristic

	subgroup	G/H
center	C₂	#16811258
commutator	C₂×C₄×C₅₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₃
Fitting	C₅₃×#493	C₂
socle	C₂×C₅₃	D₄²
hypercenter	#493	D₅₃
hypocenter	C₅₃	#3292
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #493

subgroup

G/H

center

C₂

#16811258

commutator

C₂×C₄×C₅₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₅₃

Fitting

C₅₃×#493

C₂

socle

C₂×C₅₃

D₄²

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ #493

N	H ≅ G/N	type
1. #16811424	C₂	semidirect
2. #16810890	C₂	semidirect
3. #16810790	C₂	semidirect
4. #16810741	C₂	semidirect
5. C₅₃×#493	C₂	semidirect
6. #13133262	C₂²	semidirect
7. #13133243	C₂²	semidirect
12. #13133223	C₂²	semidirect
13. #13133153	C₂²	semidirect
14. #13133145	C₂²	semidirect
15. #13133134	C₂²	semidirect
16. C₂×D₈×C₅₃	C₂²	semidirect
17. #127×C₅₃	C₂²	semidirect
18. C₄×C₈×C₅₃	C₂²	semidirect
32. Dic₂₁₂	D₄	semidirect
33. D₄₂₄	D₄	semidirect
34. #11719913	D₄	semidirect
66. #493	D₅₃	semidirect
70. C₅₃	#3292	semidirect
77. C₂×D₈	C₂×D₅₃	semidirect
79. #127	C₂×D₅₃	semidirect
83. C₄×C₈	C₂×D₅₃	semidirect

H ≅ G/N

type

1. #16811424

C₂

semidirect

2. #16810890

C₂

semidirect

3. #16810790

C₂

semidirect

4. #16810741

C₂

semidirect

5. C₅₃×#493

C₂

semidirect

6. #13133262

C₂²

semidirect

7. #13133243

C₂²

semidirect

12. #13133223

C₂²

semidirect

13. #13133153

C₂²

semidirect

14. #13133145

C₂²

semidirect

15. #13133134

C₂²

semidirect

16. C₂×D₈×C₅₃

C₂²

semidirect

17. #127×C₅₃

C₂²

semidirect

18. C₄×C₈×C₅₃

C₂²

semidirect

32. Dic₂₁₂

D₄

semidirect

33. D₄₂₄

D₄

semidirect

34. #11719913

D₄

semidirect

66. #493

D₅₃

semidirect

70. C₅₃

#3292

semidirect

77. C₂×D₈

C₂×D₅₃

semidirect

79. #127

C₂×D₅₃

semidirect

83. C₄×C₈

C₂×D₅₃

semidirect

Subgroups — #20286514

Conjugacy classes of subgroups

Complements