Group #20280817 — subgroups

Subgroups — #20280817

Order 6784.

804 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 804 conjugacy classes of subgroups — too many to draw. Only its 181 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×#13133191	C₂	1	6	yes	maximal, complemented
2	#16810560	C₂	1	4	yes	maximal, complemented
3	#16810533	C₂	1	4	yes	maximal, complemented
4	C₂×#95×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×D₂₁₂	C₂²	1	6	yes	—
6	C₂²×C₂²⋊D₅₃	C₂²	1	4	yes	—
7	#13133191	—	2	24	no	—
8	C₂×C₄⋊Dic₅₃	C₂²	1	6	yes	—
9	C₂×#11719921	C₂²	1	12	yes	—
10	C₂³×C₄×C₅₃	C₂²	1	3	yes	—
11	#95×C₅₃	C₂²	1	4	yes	—
12	C₂⁴×C₅₃	C₂³	1	1	yes	characteristic
13	C₂³×D₅₃	C₂³	1	4	yes	—
14	C₂²×C₄×C₅₃	—	4	6	no	—
15	C₂²×C₄×C₅₃	D₄	1	12	yes	complemented
16	C₂²×C₄×C₅₃	C₂³	1	6	yes	—
17	C₂×C₂²⋊D₅₃	—	2	24	no	—
18	C₂²×Dic₅₃	C₂³	1	4	yes	—
19	C₂×D₂₁₂	—	2	12	no	—
20	C₂×D₂₁₂	—	4	12	no	—
21	#11719921	—	2	24	no	—
22	C₄⋊Dic₅₃	—	2	12	no	—
23	C₂³×C₅₃	—	2	3	no	—
24	C₂³×C₅₃	—	2	1	no	—
25	C₂³×C₅₃	C₂×D₄	1	6	yes	—
26	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
27	C₂²×D₅₃	—	2	12	no	—
28	C₂²×D₅₃	—	2	12	no	—
29	C₂²×D₅₃	—	2	4	no	—
30	C₂×C₄×C₅₃	—	2	12	no	—
31	C₂×C₄×C₅₃	—	4	12	no	—
32	C₂×C₄×C₅₃	C₂×D₄	1	12	yes	—
33	C₂²⋊D₅₃	—	4	16	no	—
34	C₂×Dic₅₃	—	2	12	no	—
35	D₂₁₂	—	4	24	no	—
36	C₂²×C₅₃	—	2	6	no	—
37	C₂²×C₅₃	—	2	6	no	—
38	C₂²×C₅₃	#127	1	4	yes	—
39	C₂²×C₅₃	C₂²×D₄	1	3	yes	—
40	C₂²×C₅₃	H(5,𝔽₂)	1	3	yes	—
41	C₂²×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
42	C₂×D₅₃	—	4	4	no	—
43	C₂×D₅₃	—	4	12	no	—
44	C₂×D₅₃	—	4	12	no	—
45	C₄×C₅₃	—	2	12	no	—
46	Dic₅₃	—	4	4	no	—
47	#2656	—	53	1	no	maximal, Sylow
48	C₂×C₅₃	—	2	4	no	—
49	C₂×C₅₃	C₂×#127	1	1	yes	characteristic
50	C₂×C₅₃	#534	1	3	yes	—
51	C₂×C₅₃	#537	1	3	yes	—
52	D₅₃	—	8	4	no	—
53	C₂×#121	—	53	6	no	—
54	#399	—	53	4	no	—
55	#392	—	53	4	no	—
56	C₂×#95	D₅₃	1	1	yes	characteristic, complemented
57	C₅₃	#2656	1	1	yes	characteristic, complemented, Sylow
58	C₂²×D₄	—	53	4	no	—
59	C₂²×D₄	—	53	6	no	—
60	C₂³×C₄	C₂×D₅₃	1	3	yes	—
61	#121	—	106	24	no	—
62	C₂×C₄⋊C₄	—	53	6	no	—
63	C₂×C₂²⋊C₄	—	53	12	no	—
64	#95	C₂×D₅₃	1	4	yes	—
65	C₂⁴	—	53	4	no	—
66	C₂⁴	C₂²×D₅₃	1	1	yes	characteristic
67	C₂×D₄	—	106	24	no	—
68	C₂×D₄	—	106	12	no	—
69	C₂×D₄	—	212	12	no	—
70	C₂²×C₄	—	4	6	no	—
71	C₂²×C₄	—	53	4	no	—
72	C₂²×C₄	D₂₁₂	1	12	yes	complemented
73	C₂²×C₄	C₂²×D₅₃	1	6	yes	—
74	C₄⋊C₄	—	106	12	no	—
75	C₂²⋊C₄	—	106	24	no	—
76	C₂³	—	2	3	no	—
77	C₂³	—	2	1	no	—
78	C₂³	—	106	12	no	—
79	C₂³	—	106	12	no	—
80	C₂³	—	106	4	no	—
81	C₂³	C₂×D₂₁₂	1	6	yes	—
82	C₂³	C₂³×D₅₃	1	1	yes	characteristic
83	D₄	—	212	24	no	—
84	D₄	—	212	16	no	—
85	C₂×C₄	—	2	12	no	—
86	C₂×C₄	—	4	12	no	—
87	C₂×C₄	—	106	12	no	—
88	C₂×C₄	C₂×D₂₁₂	1	12	yes	—
89	C₂²	—	2	6	no	—
90	C₂²	—	2	6	no	—
91	C₂²	—	212	4	no	—
92	C₂²	—	212	12	no	—
93	C₂²	—	212	12	no	—
94	C₂²	#13133214	1	4	yes	—
95	C₂²	C₂²×D₂₁₂	1	3	yes	—
96	C₂²	#13133262	1	3	yes	—
97	C₂²	#13133265	1	1	yes	characteristic
98	C₄	—	2	12	no	—
99	C₄	—	212	4	no	—
100	C₂	—	2	4	no	—
101	C₂	—	424	4	no	—
102	C₂	C₂×#13133214	1	1	yes	characteristic
103	C₂	#16811185	1	3	yes	—
104	C₂	#16811204	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133191

C₂

yes

maximal, complemented

#16810560

C₂

yes

maximal, complemented

#16810533

C₂

yes

maximal, complemented

C₂×#95×C₅₃

C₂

yes

maximal, characteristic, complemented

C₂²×D₂₁₂

C₂²

yes

—

C₂²×C₂²⋊D₅₃

C₂²

yes

—

#13133191

—

C₂×C₄⋊Dic₅₃

C₂²

yes

—

C₂×#11719921

C₂²

yes

—

C₂³×C₄×C₅₃

C₂²

yes

—

#95×C₅₃

C₂²

yes

—

C₂⁴×C₅₃

C₂³

yes

characteristic

C₂³×D₅₃

C₂³

yes

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

D₄

yes

complemented

C₂²×C₄×C₅₃

C₂³

yes

—

C₂×C₂²⋊D₅₃

—

C₂²×Dic₅₃

C₂³

yes

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

#11719921

—

C₄⋊Dic₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

C₂×D₄

yes

—

C₂³×C₅₃

C₂⁴

yes

characteristic

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂²⋊D₅₃

—

C₂×Dic₅₃

—

D₂₁₂

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#127

yes

—

C₂²×C₅₃

C₂²×D₄

yes

—

C₂²×C₅₃

H(5,𝔽₂)

yes

—

C₂²×C₅₃

2₋¹⁺⁴

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

Dic₅₃

—

#2656

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

C₂×#127

yes

characteristic

C₂×C₅₃

#534

yes

—

C₂×C₅₃

#537

yes

—

D₅₃

—

C₂×#121

—

#399

—

#392

—

C₂×#95

D₅₃

yes

characteristic, complemented

C₅₃

#2656

yes

characteristic, complemented, Sylow

C₂²×D₄

—

C₂²×D₄

—

C₂³×C₄

C₂×D₅₃

yes

—

#121

—

106

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

#95

C₂×D₅₃

yes

—

C₂⁴

—

C₂⁴

C₂²×D₅₃

yes

characteristic

C₂×D₄

—

106

—

C₂×D₄

—

106

—

C₂×D₄

—

212

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

D₂₁₂

yes

complemented

C₂²×C₄

C₂²×D₅₃

yes

—

C₄⋊C₄

—

106

—

C₂²⋊C₄

—

106

—

C₂³

—

C₂³

—

C₂³

—

106

—

C₂³

—

106

—

C₂³

—

106

—

C₂³

C₂×D₂₁₂

yes

—

C₂³

C₂³×D₅₃

yes

characteristic

D₄

—

212

—

D₄

—

212

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

106

—

C₂×C₄

C₂×D₂₁₂

yes

—

C₂²

—

C₂²

—

C₂²

—

212

—

C₂²

—

212

—

C₂²

—

212

—

C₂²

#13133214

yes

—

C₂²

C₂²×D₂₁₂

yes

—

C₂²

#13133262

yes

—

C₂²

#13133265

yes

characteristic

C₄

—

C₄

—

212

—

100

C₂

—

101

C₂

—

424

—

102

C₂

C₂×#13133214

yes

characteristic

103

C₂

#16811185

yes

—

104

C₂

#16811204

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₅₃
commutator, socle	C₂³×C₅₃	C₂⁴
Fitting	C₂×#95×C₅₃	C₂
hypercenter	C₂×#95	D₅₃
hypocenter	C₅₃	#2656
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×#95

subgroup

G/H

center, Frattini

C₂³

C₂³×D₅₃

commutator, socle

C₂³×C₅₃

C₂⁴

Fitting

C₂×#95×C₅₃

C₂

hypercenter

C₂×#95

D₅₃

C₅₃

G ▷ C₂³×C₅₃ ▷ C₅₃

C₁ ◁ C₂³ ◁ C₂×#95

N	H ≅ G/N	type
1. C₂×#13133191	C₂	semidirect
2. #16810560	C₂	semidirect
3. #16810533	C₂	semidirect
4. C₂×#95×C₅₃	C₂	semidirect
15. C₂²×C₄×C₅₃	D₄	semidirect
56. C₂×#95	D₅₃	semidirect
57. C₅₃	#2656	semidirect
72. C₂²×C₄	D₂₁₂	semidirect

H ≅ G/N

type

1. C₂×#13133191

C₂

semidirect

2. #16810560

C₂

semidirect

3. #16810533

C₂

semidirect

4. C₂×#95×C₅₃

C₂

semidirect

15. C₂²×C₄×C₅₃

D₄

semidirect

56. C₂×#95

D₅₃

semidirect

57. C₅₃

#2656

semidirect

72. C₂²×C₄

D₂₁₂

semidirect

Subgroups — #20280817

Conjugacy classes of subgroups

Complements