Group #20277026 — subgroups

Subgroups — #20277026

Order 6784.

112 conjugacy classes of subgroups; only the 35 normal subgroups are shown.

normal subgroups only

This group has 112 conjugacy classes of subgroups — too many to draw. Only its 35 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	#16811052	C₂	1	1	yes	maximal, characteristic, complemented
2	D₁₆×D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
3	#16810462	C₂	1	1	yes	maximal, characteristic, complemented
4	#16810461	C₂	1	1	yes	maximal, characteristic, complemented
5	#16810458	C₂	1	1	yes	maximal, characteristic, complemented
6	#16810456	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×D₃₂	C₂	1	1	yes	maximal, characteristic, complemented
8	#13133236	—	2	1	no	—
9	D₈×D₅₃	—	2	1	no	—
10	Dic₄₂₄	C₂²	1	1	yes	characteristic, complemented
11	#13133161	—	2	1	no	—
12	D₈₄₈	C₂²	1	1	yes	characteristic
13	#13133158	—	2	1	no	—
14	C₁₆×D₅₃	C₂²	1	1	yes	characteristic
15	#13133123	C₂²	1	1	yes	characteristic, complemented
16	D₁₆×C₅₃	C₂²	1	1	yes	characteristic
17	D₁₆×C₅₃	C₂²	1	1	yes	characteristic, complemented
18	C₃₂×C₅₃	C₂²	1	1	yes	characteristic, complemented
19	#11719947	—	4	1	no	—
20	D₄×D₅₃	—	4	1	no	—
21	D₈×C₅₃	—	2	1	no	—
22	D₈×C₅₃	—	2	1	no	—
23	#11719923	—	4	1	no	—
24	#11719922	—	4	1	no	—
25	Dic₂₁₂	—	2	1	no	—
26	D₄₂₄	—	2	1	no	—
27	C₈×D₅₃	D₄	1	1	yes	characteristic
28	C₁₆×C₅₃	C₂³	1	1	yes	characteristic
29	#11719908	D₄	1	1	yes	characteristic
30	C₂²×D₅₃	—	8	1	no	—
31	D₄×C₅₃	—	4	1	no	—
32	D₄×C₅₃	—	4	1	no	—
33	C₂²⋊D₅₃	—	8	1	no	—
34	C₂²⋊D₅₃	—	8	1	no	—
35	C₂×Dic₅₃	—	8	1	no	—
36	D₂₁₂	—	4	1	no	—
37	C₄×D₅₃	D₈	1	1	yes	characteristic
38	Dic₁₀₆	—	4	1	no	—
39	C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
40	#88322	D₈	1	1	yes	characteristic
41	C₂²×C₅₃	—	8	1	no	—
42	C₂²×C₅₃	—	8	1	no	—
43	C₂×D₅₃	—	8	1	no	—
44	C₂×D₅₃	—	16	2	no	—
45	C₂×D₅₃	D₁₆	1	1	yes	characteristic
46	C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
47	Dic₅₃	—	8	1	no	—
48	Dic₅₃	D₁₆	1	1	yes	characteristic
49	#2263	—	53	1	no	maximal, Sylow
50	C₂×C₅₃	—	16	1	no	—
51	C₂×C₅₃	—	16	1	no	—
52	C₂×C₅₃	C₂×D₁₆	1	1	yes	characteristic
53	D₅₃	—	2	1	no	—
54	D₅₃	—	16	1	no	—
55	#508	—	53	1	no	—
56	C₂×D₁₆	—	53	1	no	—
57	SD₆₄	—	53	1	no	—
58	SD₆₄	—	53	1	no	—
59	D₃₂	—	53	1	no	—
60	D₃₂	D₅₃	1	1	yes	characteristic, complemented
61	M₆₄	—	53	1	no	—
62	C₅₃	#2263	1	1	yes	characteristic, complemented, Sylow
63	#135	—	106	1	no	—
64	C₂×D₈	—	106	1	no	—
65	Q₃₂	—	53	1	no	—
66	SD₃₂	—	106	1	no	—
67	D₁₆	—	53	1	no	—
68	D₁₆	—	106	1	no	—
69	D₁₆	C₂×D₅₃	1	1	yes	characteristic
70	D₁₆	C₂×D₅₃	1	1	yes	characteristic, complemented
71	C₂×C₁₆	—	53	1	no	—
72	C₃₂	—	53	1	no	—
73	C₃₂	C₂×D₅₃	1	1	yes	characteristic, complemented
74	#41	—	212	1	no	—
75	C₂×D₄	—	212	1	no	—
76	Q₁₆	—	106	1	no	—
77	SD₁₆	—	212	1	no	—
78	D₈	—	2	1	no	—
79	D₈	—	2	1	no	—
80	D₈	—	106	1	no	—
81	D₈	—	212	1	no	—
82	C₂×C₈	—	53	1	no	—
83	C₁₆	—	53	1	no	—
84	C₁₆	C₂²×D₅₃	1	1	yes	characteristic
85	C₂³	—	424	1	no	—
86	Q₈	—	212	1	no	—
87	D₄	—	4	1	no	—
88	D₄	—	4	1	no	—
89	D₄	—	424	1	no	—
90	D₄	—	424	1	no	—
91	D₄	—	212	1	no	—
92	C₂×C₄	—	53	1	no	—
93	C₂×C₄	—	424	1	no	—
94	C₈	—	53	1	no	—
95	C₈	D₄×D₅₃	1	1	yes	characteristic
96	C₂²	—	8	1	no	—
97	C₂²	—	8	1	no	—
98	C₂²	—	53	1	no	—
99	C₂²	—	848	2	no	—
100	C₂²	—	424	1	no	—
101	C₄	—	53	1	no	—
102	C₄	—	424	1	no	—
103	C₄	D₈×D₅₃	1	1	yes	characteristic
104	C₂	—	16	1	no	—
105	C₂	—	16	1	no	—
106	C₂	—	848	1	no	—
107	C₂	—	106	1	no	—
108	C₂	D₁₆×D₅₃	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811052

C₂

yes

maximal, characteristic, complemented

D₁₆×D₅₃

C₂

yes

maximal, characteristic, complemented

#16810462

C₂

yes

maximal, characteristic, complemented

#16810461

C₂

yes

maximal, characteristic, complemented

#16810458

C₂

yes

maximal, characteristic, complemented

#16810456

C₂

yes

maximal, characteristic, complemented

C₅₃×D₃₂

C₂

yes

maximal, characteristic, complemented

#13133236

—

D₈×D₅₃

—

Dic₄₂₄

C₂²

yes

characteristic, complemented

#13133161

—

D₈₄₈

C₂²

yes

characteristic

#13133158

—

C₁₆×D₅₃

C₂²

yes

characteristic

#13133123

C₂²

yes

characteristic, complemented

D₁₆×C₅₃

C₂²

yes

characteristic

D₁₆×C₅₃

C₂²

yes

characteristic, complemented

C₃₂×C₅₃

C₂²

yes

characteristic, complemented

#11719947

—

D₄×D₅₃

—

D₈×C₅₃

—

D₈×C₅₃

—

#11719923

—

#11719922

—

Dic₂₁₂

—

D₄₂₄

—

C₈×D₅₃

D₄

yes

characteristic

C₁₆×C₅₃

C₂³

yes

characteristic

#11719908

D₄

yes

characteristic

C₂²×D₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

C₂²⋊D₅₃

—

C₂²⋊D₅₃

—

C₂×Dic₅₃

—

D₂₁₂

—

C₄×D₅₃

D₈

yes

characteristic

Dic₁₀₆

—

C₈×C₅₃

C₂×D₄

yes

characteristic

#88322

D₈

yes

characteristic

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

D₁₆

yes

characteristic

C₄×C₅₃

C₂×D₈

yes

characteristic

Dic₅₃

—

Dic₅₃

D₁₆

yes

characteristic

#2263

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

C₂×D₁₆

yes

characteristic

D₅₃

—

D₅₃

—

#508

—

C₂×D₁₆

—

SD₆₄

—

SD₆₄

—

D₃₂

—

D₃₂

D₅₃

yes

characteristic, complemented

M₆₄

—

C₅₃

#2263

yes

characteristic, complemented, Sylow

#135

—

106

—

C₂×D₈

—

106

—

Q₃₂

—

SD₃₂

—

106

—

D₁₆

—

D₁₆

—

106

—

D₁₆

C₂×D₅₃

yes

characteristic

D₁₆

C₂×D₅₃

yes

characteristic, complemented

C₂×C₁₆

—

C₃₂

—

C₃₂

C₂×D₅₃

yes

characteristic, complemented

#41

—

212

—

C₂×D₄

—

212

—

Q₁₆

—

106

—

SD₁₆

—

212

—

D₈

—

D₈

—

D₈

—

106

—

D₈

—

212

—

C₂×C₈

—

C₁₆

—

C₁₆

C₂²×D₅₃

yes

characteristic

C₂³

—

424

—

Q₈

—

212

—

D₄

—

D₄

—

D₄

—

424

—

D₄

—

424

—

D₄

—

212

—

C₂×C₄

—

C₂×C₄

—

424

—

C₈

—

C₈

D₄×D₅₃

yes

characteristic

C₂²

—

C₂²

—

C₂²

—

C₂²

—

848

—

100

C₂²

—

424

—

101

C₄

—

102

C₄

—

424

—

103

C₄

D₈×D₅₃

yes

characteristic

104

C₂

—

105

C₂

—

106

C₂

—

848

—

107

C₂

—

106

—

108

C₂

D₁₆×D₅₃

yes

characteristic

	subgroup	G/H
center	C₂	D₁₆×D₅₃
commutator	C₁₆×C₅₃	C₂³
Frattini	C₁₆	C₂²×D₅₃
Fitting	C₅₃×D₃₂	C₂
socle	C₂×C₅₃	C₂×D₁₆
hypercenter	D₃₂	D₅₃
hypocenter	C₅₃	#2263
LCS	G ▷ C₁₆×C₅₃ ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ D₃₂

subgroup

G/H

center

C₂

D₁₆×D₅₃

commutator

C₁₆×C₅₃

C₂³

Frattini

C₁₆

C₂²×D₅₃

Fitting

C₅₃×D₃₂

C₂

socle

C₂×C₅₃

C₂×D₁₆

hypercenter

D₃₂

D₅₃

hypocenter

C₅₃

#2263

LCS

G ▷ C₁₆×C₅₃ ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ D₃₂

N	H ≅ G/N	type
1. #16811052	C₂	semidirect
2. D₁₆×D₅₃	C₂	semidirect
3. #16810462	C₂	semidirect
4. #16810461	C₂	semidirect
5. #16810458	C₂	semidirect
6. #16810456	C₂	semidirect
7. C₅₃×D₃₂	C₂	semidirect
10. Dic₄₂₄	C₂²	semidirect
15. #13133123	C₂²	semidirect
17. D₁₆×C₅₃	C₂²	semidirect
18. C₃₂×C₅₃	C₂²	semidirect
60. D₃₂	D₅₃	semidirect
62. C₅₃	#2263	semidirect
70. D₁₆	C₂×D₅₃	semidirect
73. C₃₂	C₂×D₅₃	semidirect

H ≅ G/N

type

1. #16811052

C₂

semidirect

2. D₁₆×D₅₃

C₂

semidirect

3. #16810462

C₂

semidirect

4. #16810461

C₂

semidirect

5. #16810458

C₂

semidirect

6. #16810456

C₂

semidirect

7. C₅₃×D₃₂

C₂

semidirect

10. Dic₄₂₄

C₂²

semidirect

15. #13133123

C₂²

semidirect

17. D₁₆×C₅₃

C₂²

semidirect

18. C₃₂×C₅₃

C₂²

semidirect

60. D₃₂

D₅₃

semidirect

62. C₅₃

#2263

semidirect

70. D₁₆

C₂×D₅₃

semidirect

73. C₃₂

C₂×D₅₃

semidirect

Subgroups — #20277026

Conjugacy classes of subgroups

Complements