Group #20276982 — subgroups

Subgroups — #20276982

Order 6784.

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

normal subgroups only

This group has 198 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	C₂×#13133160	C₂	1	2	yes	maximal, complemented
2	C₂×#13133158	C₂	1	2	yes	maximal, complemented
3	#16811000	C₂	1	1	yes	maximal, characteristic, complemented
4	#16810399	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#495	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₄₂₄	—	2	1	no	—
7	C₂×D₄₂₄	C₂²	1	2	yes	—
8	#13133214	—	2	1	no	—
9	#13133160	—	2	4	no	—
10	#13133158	—	2	4	no	—
11	C₂×#11719908	C₂²	1	2	yes	—
12	C₂×Q₁₆×C₅₃	C₂²	1	1	yes	characteristic
13	C₂×SD₁₆×C₅₃	—	2	1	no	—
14	C₂×D₈×C₅₃	C₂²	1	1	yes	characteristic
15	#124×C₅₃	—	2	1	no	—
16	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×Q₈×C₅₃	—	2	1	no	—
18	C₂×D₄×C₅₃	—	2	1	no	—
19	C₂×D₂₁₂	—	2	2	no	—
20	C₂×D₂₁₂	—	4	2	no	—
21	Q₁₆×C₅₃	—	2	2	no	—
22	SD₁₆×C₅₃	—	4	2	no	—
23	D₈×C₅₃	—	2	2	no	—
24	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
25	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
26	C₂²⋊C₄×C₅₃	—	4	2	no	—
27	C₄²×C₅₃	D₄	1	1	yes	characteristic
28	D₄₂₄	—	2	2	no	—
29	D₄₂₄	—	2	2	no	—
30	D₄₂₄	—	4	2	no	—
31	#11719908	D₄	1	4	yes	complemented
32	C₂³×C₅₃	—	4	1	no	—
33	C₂²×D₅₃	—	4	2	no	—
34	Q₈×C₅₃	—	4	1	no	—
35	Q₈×C₅₃	—	4	1	no	—
36	D₄×C₅₃	—	4	1	no	—
37	D₄×C₅₃	—	4	1	no	—
38	C₂×C₄×C₅₃	—	4	1	no	—
39	C₂×C₄×C₅₃	D₈	1	2	yes	—
40	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
41	D₂₁₂	—	4	2	no	—
42	D₂₁₂	—	4	2	no	—
43	D₂₁₂	—	8	4	no	—
44	C₈×C₅₃	D₈	1	2	yes	—
45	C₈×C₅₃	C₂×D₄	1	2	yes	—
46	C₂²×C₅₃	—	8	1	no	—
47	C₂²×C₅₃	—	8	2	no	—
48	C₂²×C₅₃	C₂×D₈	1	1	yes	characteristic
49	C₂×D₅₃	—	8	2	no	—
50	C₂×D₅₃	—	8	4	no	—
51	C₄×C₅₃	—	2	2	no	—
52	C₄×C₅₃	—	8	1	no	—
53	C₄×C₅₃	#127	1	1	yes	characteristic
54	C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
55	#2250	—	53	1	no	maximal, Sylow
56	C₂×C₅₃	—	16	1	no	—
57	C₂×C₅₃	#493	1	1	yes	characteristic
58	C₂×C₅₃	#509	1	2	yes	—
59	D₅₃	—	16	2	no	—
60	C₂×SD₃₂	—	53	2	no	—
61	C₂×D₁₆	—	53	2	no	—
62	#495	D₅₃	1	1	yes	characteristic, complemented
63	#493	—	53	1	no	—
64	#346	—	53	1	no	—
65	C₅₃	#2250	1	1	yes	characteristic, complemented, Sylow
66	C₂×Q₁₆	C₂×D₅₃	1	1	yes	characteristic
67	C₂×SD₁₆	—	2	1	no	—
68	C₂×D₈	—	53	2	no	—
69	C₂×D₈	—	106	1	no	—
70	C₂×D₈	C₂×D₅₃	1	1	yes	characteristic
71	#127	—	106	1	no	—
72	#124	—	2	1	no	—
73	SD₃₂	—	106	4	no	—
74	D₁₆	—	106	4	no	—
75	C₂×C₁₆	—	53	2	no	—
76	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
77	C₂×Q₈	—	2	1	no	—
78	C₂×D₄	—	2	1	no	—
79	C₂×D₄	—	106	2	no	—
80	C₂×D₄	—	212	2	no	—
81	Q₁₆	—	2	2	no	—
82	SD₁₆	—	4	2	no	—
83	D₈	—	2	2	no	—
84	D₈	—	106	2	no	—
85	D₈	—	106	2	no	—
86	D₈	—	212	2	no	—
87	C₂×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
88	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
89	C₂²⋊C₄	—	4	2	no	—
90	C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
91	C₁₆	—	53	4	no	—
92	C₂³	—	4	1	no	—
93	C₂³	—	212	2	no	—
94	Q₈	—	4	1	no	—
95	Q₈	—	4	1	no	—
96	D₄	—	4	1	no	—
97	D₄	—	4	1	no	—
98	D₄	—	424	4	no	—
99	D₄	—	212	2	no	—
100	D₄	—	212	2	no	—
101	C₂×C₄	—	4	1	no	—
102	C₂×C₄	#11719922	1	2	yes	—
103	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
104	C₈	#11719922	1	2	yes	—
105	C₈	D₄×D₅₃	1	2	yes	—
106	C₂²	—	8	2	no	—
107	C₂²	—	8	1	no	—
108	C₂²	—	424	4	no	—
109	C₂²	—	424	2	no	—
110	C₂²	C₂×#11719922	1	1	yes	characteristic
111	C₄	—	2	2	no	—
112	C₄	—	8	1	no	—
113	C₄	#13133213	1	1	yes	characteristic
114	C₄	C₂×#11719922	1	1	yes	characteristic
115	C₂	—	16	1	no	—
116	C₂	—	848	2	no	—
117	C₂	#16810999	1	1	yes	characteristic
118	C₂	#16811058	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133160

C₂

yes

maximal, complemented

C₂×#13133158

C₂

yes

maximal, complemented

#16811000

C₂

yes

maximal, characteristic, complemented

#16810399

C₂

yes

maximal, characteristic, complemented

C₅₃×#495

C₂

yes

maximal, characteristic, complemented

C₂×D₄₂₄

—

C₂×D₄₂₄

C₂²

yes

—

—

—

—

C₂²

yes

—

C₂×Q₁₆×C₅₃

C₂²

yes

characteristic

C₂×SD₁₆×C₅₃

—

C₂×D₈×C₅₃

C₂²

yes

characteristic

#124×C₅₃

—

C₄×C₈×C₅₃

C₂²

yes

characteristic

C₂×Q₈×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

Q₁₆×C₅₃

—

SD₁₆×C₅₃

—

D₈×C₅₃

—

C₂×C₈×C₅₃

D₄

yes

characteristic

C₂×C₈×C₅₃

C₂³

yes

characteristic

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

D₄

yes

characteristic

D₄₂₄

—

D₄₂₄

—

D₄₂₄

—

#11719908

D₄

yes

complemented

C₂³×C₅₃

—

C₂²×D₅₃

—

Q₈×C₅₃

—

Q₈×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

D₈

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

D₂₁₂

—

D₂₁₂

—

D₂₁₂

—

C₈×C₅₃

D₈

yes

—

C₈×C₅₃

C₂×D₄

yes

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂×D₈

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#127

yes

characteristic

C₄×C₅₃

C₂×D₈

yes

characteristic

#2250

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

#493

yes

characteristic

C₂×C₅₃

#509

yes

—

D₅₃

—

C₂×SD₃₂

—

C₂×D₁₆

—

#495

D₅₃

yes

characteristic, complemented

#493

—

#346

—

C₅₃

#2250

yes

characteristic, complemented, Sylow

C₂×Q₁₆

C₂×D₅₃

yes

characteristic

C₂×SD₁₆

—

C₂×D₈

—

C₂×D₈

—

106

—

C₂×D₈

C₂×D₅₃

yes

characteristic

#127

—

106

—

#124

—

SD₃₂

—

106

—

D₁₆

—

106

—

C₂×C₁₆

—

C₄×C₈

C₂×D₅₃

yes

characteristic

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

106

—

C₂×D₄

—

212

—

Q₁₆

—

SD₁₆

—

D₈

—

D₈

—

106

—

D₈

—

106

—

D₈

—

212

—

C₂×C₈

C₂²⋊D₅₃

yes

characteristic

C₂×C₈

C₂²×D₅₃

yes

characteristic

C₂²⋊C₄

—

C₄²

C₂²⋊D₅₃

yes

characteristic

C₁₆

—

C₂³

—

C₂³

—

212

—

Q₈

—

Q₈

—

D₄

—

D₄

—

D₄

—

424

—

D₄

—

212

—

100

D₄

—

212

—

101

C₂×C₄

—

102

C₂×C₄

#11719922

yes

—

103

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

104

C₈

#11719922

yes

—

105

C₈

D₄×D₅₃

yes

—

106

C₂²

—

107

C₂²

—

108

C₂²

—

424

—

109

C₂²

—

424

—

110

C₂²

C₂×#11719922

yes

characteristic

111

C₄

—

112

C₄

—

113

C₄

#13133213

yes

characteristic

114

C₄

C₂×#11719922

yes

characteristic

115

C₂

—

116

C₂

—

848

—

117

C₂

#16810999

yes

characteristic

118

C₂

#16811058

yes

—

	subgroup	G/H
center	C₂²	C₂×#11719922
commutator	C₂×C₈×C₅₃	C₂³
Frattini	C₂×C₈	C₂²×D₅₃
Fitting	C₅₃×#495	C₂
socle	C₂²×C₅₃	C₂×D₈
hypercenter	#495	D₅₃
hypocenter	C₅₃	#2250
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #495

subgroup

G/H

center

C₂²

C₂×#11719922

commutator

C₂×C₈×C₅₃

C₂³

Frattini

C₂×C₈

C₂²×D₅₃

Fitting

C₅₃×#495

C₂

socle

C₂²×C₅₃

C₂×D₈

D₅₃

C₅₃

G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #495

N	H ≅ G/N	type
1. C₂×#13133160	C₂	semidirect
2. C₂×#13133158	C₂	semidirect
3. #16811000	C₂	semidirect
4. #16810399	C₂	semidirect
5. C₅₃×#495	C₂	semidirect
31. #11719908	D₄	semidirect
62. #495	D₅₃	semidirect
65. C₅₃	#2250	semidirect

H ≅ G/N

type

1. C₂×#13133160

C₂

semidirect

2. C₂×#13133158

C₂

semidirect

3. #16811000

C₂

semidirect

4. #16810399

C₂

semidirect

5. C₅₃×#495

C₂

semidirect

31. #11719908

D₄

semidirect

62. #495

D₅₃

semidirect

65. C₅₃

#2250

semidirect

Subgroups — #20276982

Conjugacy classes of subgroups

Complements