Group #20276885 — subgroups

Subgroups — #20276885

Order 6784.

126 conjugacy classes of subgroups; only the 43 normal subgroups are shown.

normal subgroups only

This group has 126 conjugacy classes of subgroups — too many to draw. Only its 43 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	#16811020	C₂	1	1	yes	maximal, characteristic, complemented
2	#16810739	C₂	1	1	yes	maximal, characteristic
3	#16810447	C₂	1	1	yes	maximal, characteristic, complemented
4	#16810443	C₂	1	1	yes	maximal, characteristic, complemented
5	#16810434	C₂	1	1	yes	maximal, characteristic
6	#16810432	C₂	1	1	yes	maximal, characteristic
7	C₅₃×#365	C₂	1	1	yes	maximal, characteristic, complemented
8	#13133239	C₂²	1	1	yes	characteristic
9	#13133225	—	2	1	no	—
10	#13133216	—	2	1	no	—
11	#13133157	C₂²	1	1	yes	characteristic
12	#13133149	C₂²	1	1	yes	characteristic
13	#13133148	—	2	1	no	—
14	#13133143	—	2	1	no	—
15	C₈×Dic₅₃	C₂²	1	1	yes	characteristic
16	M₃₂×C₅₃	C₂²	1	1	yes	characteristic
17	#108×C₅₃	C₂²	1	1	yes	characteristic
18	#106×C₅₃	C₂²	1	1	yes	characteristic
19	#11719945	—	2	1	no	—
20	C₂×Dic₁₀₆	—	4	1	no	—
21	M₁₆×C₅₃	—	2	1	no	—
22	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
23	C₄⋊C₄×C₅₃	—	2	1	no	—
24	C₄⋊Dic₅₃	—	2	1	no	—
25	#11719919	—	4	1	no	—
26	C₄×Dic₅₃	D₄	1	1	yes	characteristic
27	C₂×#88322	D₄	1	1	yes	characteristic
28	Dic₂₁₂	Q₈	1	1	yes	characteristic, complemented
29	D₄₂₄	Q₈	1	1	yes	characteristic, complemented
30	#11719913	—	2	1	no	—
31	#11719912	—	4	1	no	—
32	C₈×D₅₃	—	4	1	no	—
33	C₁₆×C₅₃	—	2	1	no	—
34	#11719908	—	2	1	no	—
35	C₂×C₄×C₅₃	—	4	1	no	—
36	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
37	C₂²⋊D₅₃	—	4	1	no	—
38	C₂×Dic₅₃	—	2	1	no	—
39	C₂×Dic₅₃	—	4	1	no	—
40	D₂₁₂	—	2	1	no	—
41	C₄×D₅₃	—	4	1	no	—
42	Dic₁₀₆	—	2	1	no	—
43	Dic₁₀₆	—	8	2	no	—
44	C₈×C₅₃	—	4	1	no	—
45	C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
46	C₈×C₅₃	#41	1	1	yes	characteristic
47	#88322	SD₁₆	1	1	yes	characteristic
48	#88322	SD₁₆	1	1	yes	characteristic
49	C₂²×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
50	C₂×D₅₃	—	4	1	no	—
51	C₄×C₅₃	—	8	1	no	—
52	C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
53	C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
54	Dic₅₃	—	2	2	no	—
55	Dic₅₃	—	4	1	no	—
56	Dic₅₃	—	8	1	no	—
57	#2230	—	53	1	no	maximal, Sylow
58	C₂×C₅₃	—	2	1	no	—
59	C₂×C₅₃	#476	1	1	yes	characteristic
60	D₅₃	—	8	1	no	—
61	#498	—	53	1	no	—
62	#443	—	53	1	no	—
63	#365	—	53	1	no	—
64	#365	D₅₃	1	1	yes	characteristic, complemented
65	#364	—	53	1	no	—
66	#360	—	53	1	no	—
67	#360	—	53	1	no	—
68	C₅₃	#2230	1	1	yes	characteristic, complemented, Sylow
69	#135	—	53	1	no	—
70	#131	—	106	1	no	—
71	C₄⋊Q₈	—	106	1	no	—
72	M₃₂	—	53	1	no	—
73	M₃₂	C₂×D₅₃	1	1	yes	characteristic
74	#108	C₂×D₅₃	1	1	yes	characteristic
75	#106	—	53	1	no	—
76	#106	—	106	1	no	—
77	#106	C₂×D₅₃	1	1	yes	characteristic
78	#104	—	106	1	no	—
79	C₄×C₈	—	53	1	no	—
80	#41	—	106	1	no	—
81	C₂×Q₈	—	212	1	no	—
82	Q₁₆	—	53	1	no	—
83	SD₁₆	—	106	1	no	—
84	D₈	—	53	1	no	—
85	M₁₆	—	2	1	no	—
86	M₁₆	—	212	1	no	—
87	C₂×C₈	—	53	1	no	—
88	C₂×C₈	—	212	1	no	—
89	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
90	C₄⋊C₄	—	2	1	no	—
91	C₄⋊C₄	—	106	1	no	—
92	C₄⋊C₄	—	212	1	no	—
93	C₄²	—	53	1	no	—
94	C₁₆	—	2	1	no	—
95	C₁₆	—	106	1	no	—
96	Q₈	—	106	1	no	—
97	Q₈	—	424	2	no	—
98	D₄	—	106	1	no	—
99	D₄	—	212	1	no	—
100	C₂×C₄	—	4	1	no	—
101	C₂×C₄	—	106	1	no	—
102	C₂×C₄	—	212	1	no	—
103	C₂×C₄	—	212	1	no	—
104	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
105	C₈	—	4	1	no	—
106	C₈	—	53	1	no	—
107	C₈	—	53	1	no	—
108	C₈	#11719945	1	1	yes	characteristic
109	C₈	Q₈×D₅₃	1	1	yes	characteristic
110	C₂²	—	212	1	no	—
111	C₂²	#13133244	1	1	yes	characteristic
112	C₄	—	8	1	no	—
113	C₄	—	106	2	no	—
114	C₄	—	424	1	no	—
115	C₄	—	212	1	no	—
116	C₄	#13133195	1	1	yes	characteristic
117	C₄	SD₁₆×D₅₃	1	1	yes	characteristic
118	C₂	—	2	1	no	—
119	C₂	—	424	1	no	—
120	C₂	#16810912	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811020

C₂

yes

maximal, characteristic, complemented

#16810739

C₂

yes

maximal, characteristic

#16810447

C₂

yes

maximal, characteristic, complemented

#16810443

C₂

yes

maximal, characteristic, complemented

#16810434

C₂

yes

maximal, characteristic

#16810432

C₂

yes

maximal, characteristic

C₅₃×#365

C₂

yes

maximal, characteristic, complemented

#13133239

C₂²

yes

—

—

C₂²

yes

characteristic

#13133149

C₂²

yes

characteristic

#13133148

—

#13133143

—

C₈×Dic₅₃

C₂²

yes

characteristic

M₃₂×C₅₃

C₂²

yes

characteristic

#108×C₅₃

C₂²

yes

characteristic

#106×C₅₃

C₂²

yes

characteristic

#11719945

—

C₂×Dic₁₀₆

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

C₂³

yes

characteristic

C₄⋊C₄×C₅₃

—

C₄⋊Dic₅₃

—

#11719919

—

C₄×Dic₅₃

D₄

yes

characteristic

C₂×#88322

D₄

yes

characteristic

Dic₂₁₂

Q₈

yes

characteristic, complemented

D₄₂₄

Q₈

yes

characteristic, complemented

#11719913

—

#11719912

—

C₈×D₅₃

—

C₁₆×C₅₃

—

#11719908

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²⋊D₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

D₂₁₂

—

C₄×D₅₃

—

Dic₁₀₆

—

Dic₁₀₆

—

C₈×C₅₃

—

C₈×C₅₃

C₂×Q₈

yes

C₈×C₅₃

yes

SD₁₆

yes

characteristic

#88322

SD₁₆

yes

characteristic

C₂²×C₅₃

AGL(1,ℤ/8)

yes

characteristic

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂²⋊Q₈

yes

characteristic

C₄×C₅₃

C₂×SD₁₆

yes

characteristic

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

#2230

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

#476

yes

characteristic

D₅₃

—

—

—

—

D₅₃

yes

characteristic, complemented

#364

—

#360

—

#360

—

C₅₃

#2230

yes

characteristic, complemented, Sylow

#135

—

#131

—

106

—

C₄⋊Q₈

—

106

—

M₃₂

—

M₃₂

C₂×D₅₃

yes

characteristic

#108

C₂×D₅₃

yes

characteristic

#106

—

#106

—

106

—

#106

C₂×D₅₃

yes

characteristic

#104

—

106

—

C₄×C₈

—

#41

—

106

—

C₂×Q₈

—

212

—

Q₁₆

—

SD₁₆

—

106

—

D₈

—

M₁₆

—

M₁₆

—

212

—

C₂×C₈

—

C₂×C₈

—

212

—

C₂×C₈

C₂²×D₅₃

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

106

—

C₄⋊C₄

—

212

—

C₄²

—

C₁₆

—

C₁₆

—

106

—

Q₈

—

106

—

Q₈

—

424

—

D₄

—

106

—

D₄

—

212

—

100

C₂×C₄

—

101

C₂×C₄

—

106

—

102

C₂×C₄

—

212

—

103

C₂×C₄

—

212

—

104

C₂×C₄

D₄×D₅₃

yes

characteristic

105

C₈

—

106

C₈

—

107

C₈

—

108

C₈

#11719945

yes

characteristic

109

C₈

Q₈×D₅₃

yes

characteristic

110

C₂²

—

212

—

111

C₂²

#13133244

yes

characteristic

112

C₄

—

113

C₄

—

106

—

114

C₄

—

424

—

115

C₄

—

212

—

116

C₄

#13133195

yes

characteristic

117

C₄

SD₁₆×D₅₃

yes

characteristic

118

C₂

—

119

C₂

—

424

—

120

C₂

#16810912

yes

characteristic

	subgroup	G/H
center	C₂	#16810912
commutator	C₂×C₈×C₅₃	C₂³
Frattini	C₂×C₈	C₂²×D₅₃
Fitting	C₅₃×#365	C₂
socle	C₂×C₅₃	#476
hypercenter	#365	D₅₃
hypocenter	C₅₃	#2230
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #365

subgroup

G/H

center

C₂

#16810912

commutator

C₂×C₈×C₅₃

C₂³

Frattini

C₂×C₈

C₂²×D₅₃

Fitting

C₅₃×#365

C₂

C₂×C₅₃

D₅₃

C₅₃

G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #365

N	H ≅ G/N	type
1. #16811020	C₂	semidirect
3. #16810447	C₂	semidirect
4. #16810443	C₂	semidirect
7. C₅₃×#365	C₂	semidirect
28. Dic₂₁₂	Q₈	semidirect
29. D₄₂₄	Q₈	semidirect
64. #365	D₅₃	semidirect
68. C₅₃	#2230	semidirect

H ≅ G/N

type

1. #16811020

C₂

semidirect

3. #16810447

C₂

semidirect

4. #16810443

C₂

semidirect

7. C₅₃×#365

C₂

semidirect

28. Dic₂₁₂

Q₈

semidirect

29. D₄₂₄

Q₈

semidirect

64. #365

D₅₃

semidirect

68. C₅₃

#2230

semidirect

Subgroups — #20276885

Conjugacy classes of subgroups

Complements