Group #20276788 — subgroups

Subgroups — #20276788

Order 6784.

170 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 170 conjugacy classes of subgroups — too many to draw. Only its 47 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	#16811059	C₂	1	2	yes	maximal, complemented
2	#16810741	C₂	1	1	yes	maximal, characteristic, complemented
3	#16810444	C₂	1	1	yes	maximal, characteristic, complemented
4	#16810436	C₂	1	2	yes	maximal, complemented
5	C₅₃×#493	C₂	1	1	yes	maximal, characteristic, complemented
6	#13133239	C₂²	1	1	yes	characteristic
7	#13133223	—	2	1	no	—
8	#13133161	—	2	2	no	—
9	#13133158	—	2	2	no	—
10	#13133157	C₂²	1	2	yes	—
11	#13133153	C₂²	1	1	yes	characteristic
12	#13133145	—	2	1	no	—
13	C₂×D₈×C₅₃	—	2	1	no	—
14	C₂×D₈×C₅₃	C₂²	1	2	yes	—
15	#127×C₅₃	—	2	1	no	—
16	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×D₄×C₅₃	—	2	2	no	—
18	C₂×D₄×C₅₃	—	4	2	no	—
19	#11719945	—	2	1	no	—
20	D₈×C₅₃	—	2	2	no	—
21	D₈×C₅₃	—	2	2	no	—
22	D₈×C₅₃	—	4	1	no	—
23	D₈×C₅₃	—	4	1	no	—
24	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
25	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
26	C₄²×C₅₃	D₄	1	1	yes	characteristic
27	#11719917	—	2	1	no	—
28	Dic₂₁₂	D₄	1	1	yes	characteristic, complemented
29	D₄₂₄	D₄	1	1	yes	characteristic, complemented
30	#11719913	—	2	1	no	—
31	#11719912	—	4	1	no	—
32	C₈×D₅₃	—	4	1	no	—
33	#11719908	—	2	2	no	—
34	C₂³×C₅₃	—	4	2	no	—
35	D₄×C₅₃	—	4	2	no	—
36	D₄×C₅₃	—	4	2	no	—
37	D₄×C₅₃	—	8	4	no	—
38	C₂×C₄×C₅₃	—	2	1	no	—
39	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
40	C₂²⋊D₅₃	—	4	1	no	—
41	D₂₁₂	—	2	1	no	—
42	C₄×D₅₃	—	4	1	no	—
43	Dic₁₀₆	—	2	1	no	—
44	C₈×C₅₃	D₈	1	1	yes	characteristic
45	C₈×C₅₃	D₈	1	1	yes	characteristic
46	C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
47	C₈×C₅₃	#41	1	1	yes	characteristic
48	#88322	—	4	1	no	—
49	C₂²×C₅₃	—	8	4	no	—
50	C₂²×C₅₃	—	8	2	no	—
51	C₂²×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
52	C₂×D₅₃	—	4	1	no	—
53	C₄×C₅₃	—	2	2	no	—
54	C₄×C₅₃	#121	1	1	yes	characteristic
55	C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
56	Dic₅₃	—	4	1	no	—
57	#2213	—	53	1	no	maximal, Sylow
58	C₂×C₅₃	—	2	1	no	—
59	C₂×C₅₃	—	16	2	no	—
60	C₂×C₅₃	#459	1	1	yes	characteristic
61	D₅₃	—	8	1	no	—
62	#509	—	53	2	no	—
63	#493	D₅₃	1	1	yes	characteristic, complemented
64	#443	—	53	1	no	—
65	#364	—	53	1	no	—
66	#361	—	53	2	no	—
67	C₅₃	#2213	1	1	yes	characteristic, complemented, Sylow
68	#135	—	53	1	no	—
69	C₂×D₈	—	2	1	no	—
70	C₂×D₈	C₂×D₅₃	1	2	yes	—
71	#131	—	106	1	no	—
72	#127	—	2	1	no	—
73	SD₃₂	—	106	2	no	—
74	D₁₆	—	106	2	no	—
75	M₃₂	—	53	2	no	—
76	#108	—	53	1	no	—
77	#104	—	106	1	no	—
78	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
79	#41	—	106	1	no	—
80	C₂×D₄	—	2	2	no	—
81	C₂×D₄	—	4	2	no	—
82	Q₁₆	—	53	1	no	—
83	SD₁₆	—	106	1	no	—
84	D₈	—	2	2	no	—
85	D₈	—	2	2	no	—
86	D₈	—	4	1	no	—
87	D₈	—	4	1	no	—
88	D₈	—	53	1	no	—
89	M₁₆	—	106	1	no	—
90	M₁₆	—	212	1	no	—
91	C₂×C₈	—	212	1	no	—
92	C₂×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
93	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
94	C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
95	C₁₆	—	106	2	no	—
96	C₂³	—	4	2	no	—
97	Q₈	—	106	1	no	—
98	D₄	—	4	2	no	—
99	D₄	—	4	2	no	—
100	D₄	—	8	4	no	—
101	D₄	—	106	1	no	—
102	D₄	—	212	1	no	—
103	C₂×C₄	—	2	1	no	—
104	C₂×C₄	—	212	1	no	—
105	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
106	C₈	—	212	1	no	—
107	C₈	#11719922	1	1	yes	characteristic
108	C₈	#11719922	1	1	yes	characteristic
109	C₈	D₄×D₅₃	1	1	yes	characteristic
110	C₈	#11719947	1	1	yes	characteristic
111	C₂²	—	8	4	no	—
112	C₂²	—	8	2	no	—
113	C₂²	—	212	1	no	—
114	C₂²	#13133243	1	1	yes	characteristic
115	C₄	—	2	2	no	—
116	C₄	—	212	1	no	—
117	C₄	#13133188	1	1	yes	characteristic
118	C₄	C₂×#11719922	1	1	yes	characteristic
119	C₂	—	2	1	no	—
120	C₂	—	16	2	no	—
121	C₂	—	424	1	no	—
122	C₂	#16810815	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811059

C₂

yes

maximal, complemented

#16810741

C₂

yes

maximal, characteristic, complemented

#16810444

C₂

yes

maximal, characteristic, complemented

#16810436

C₂

yes

maximal, complemented

C₅₃×#493

C₂

yes

maximal, characteristic, complemented

#13133239

C₂²

yes

—

—

—

C₂²

yes

—

#13133153

C₂²

yes

characteristic

#13133145

—

C₂×D₈×C₅₃

—

C₂×D₈×C₅₃

C₂²

yes

—

#127×C₅₃

—

C₄×C₈×C₅₃

C₂²

yes

characteristic

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

#11719945

—

D₈×C₅₃

—

D₈×C₅₃

—

D₈×C₅₃

—

D₈×C₅₃

—

C₂×C₈×C₅₃

D₄

yes

characteristic

C₂×C₈×C₅₃

C₂³

yes

characteristic

C₄²×C₅₃

D₄

yes

characteristic

#11719917

—

Dic₂₁₂

D₄

yes

characteristic, complemented

D₄₂₄

D₄

yes

characteristic, complemented

#11719913

—

#11719912

—

C₈×D₅₃

—

#11719908

—

C₂³×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²⋊D₅₃

—

D₂₁₂

—

C₄×D₅₃

—

Dic₁₀₆

—

C₈×C₅₃

D₈

yes

characteristic

C₈×C₅₃

D₈

yes

characteristic

C₈×C₅₃

C₂×D₄

yes

C₈×C₅₃

yes

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

AGL(1,ℤ/8)

yes

characteristic

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#121

yes

characteristic

C₄×C₅₃

C₂×D₈

yes

characteristic

Dic₅₃

—

#2213

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

#459

yes

characteristic

D₅₃

—

#509

—

#493

D₅₃

yes

characteristic, complemented

#443

—

#364

—

#361

—

C₅₃

#2213

yes

characteristic, complemented, Sylow

#135

—

C₂×D₈

—

C₂×D₈

C₂×D₅₃

yes

—

#131

—

106

—

#127

—

SD₃₂

—

106

—

D₁₆

—

106

—

M₃₂

—

#108

—

#104

—

106

—

C₄×C₈

C₂×D₅₃

yes

characteristic

#41

—

106

—

C₂×D₄

—

C₂×D₄

—

Q₁₆

—

SD₁₆

—

106

—

D₈

—

D₈

—

D₈

—

D₈

—

D₈

—

M₁₆

—

106

—

M₁₆

—

212

—

C₂×C₈

—

212

—

C₂×C₈

C₂²⋊D₅₃

yes

characteristic

C₂×C₈

C₂²×D₅₃

yes

characteristic

C₄²

C₂²⋊D₅₃

yes

characteristic

C₁₆

—

106

—

C₂³

—

Q₈

—

106

—

D₄

—

D₄

—

100

D₄

—

101

D₄

—

106

—

102

D₄

—

212

—

103

C₂×C₄

—

104

C₂×C₄

—

212

—

105

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

106

C₈

—

212

—

107

C₈

#11719922

yes

characteristic

108

C₈

#11719922

yes

characteristic

109

C₈

D₄×D₅₃

yes

characteristic

110

C₈

#11719947

yes

characteristic

111

C₂²

—

112

C₂²

—

113

C₂²

—

212

—

114

C₂²

#13133243

yes

characteristic

115

C₄

—

116

C₄

—

212

—

117

C₄

#13133188

yes

characteristic

118

C₄

C₂×#11719922

yes

characteristic

119

C₂

—

120

C₂

—

121

C₂

—

424

—

122

C₂

#16810815

yes

characteristic

	subgroup	G/H
center	C₂	#16810815
commutator	C₂×C₈×C₅₃	C₂³
Frattini	C₂×C₈	C₂²×D₅₃
Fitting	C₅₃×#493	C₂
socle	C₂×C₅₃	#459
hypercenter	#493	D₅₃
hypocenter	C₅₃	#2213
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #493

subgroup

G/H

center

C₂

#16810815

commutator

C₂×C₈×C₅₃

C₂³

Frattini

C₂×C₈

C₂²×D₅₃

Fitting

C₅₃×#493

C₂

C₂×C₅₃

D₅₃

C₅₃

G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #493

N	H ≅ G/N	type
1. #16811059	C₂	semidirect
2. #16810741	C₂	semidirect
3. #16810444	C₂	semidirect
4. #16810436	C₂	semidirect
5. C₅₃×#493	C₂	semidirect
28. Dic₂₁₂	D₄	semidirect
29. D₄₂₄	D₄	semidirect
63. #493	D₅₃	semidirect
67. C₅₃	#2213	semidirect

H ≅ G/N

type

1. #16811059

C₂

semidirect

2. #16810741

C₂

semidirect

3. #16810444

C₂

semidirect

4. #16810436

C₂

semidirect

5. C₅₃×#493

C₂

semidirect

28. Dic₂₁₂

D₄

semidirect

29. D₄₂₄

D₄

semidirect

63. #493

D₅₃

semidirect

67. C₅₃

#2213

semidirect

Subgroups — #20276788

Conjugacy classes of subgroups

Complements