Group #20291259 — subgroups

Subgroups — #20291259

Order 6784.

648 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 648 conjugacy classes of subgroups — too many to draw. Only its 152 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	#16811571	C₂	1	12	yes	maximal, complemented
2	#16811568	C₂	1	2	yes	maximal, complemented
3	#127×D₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂³×#6157	C₂²	1	4	yes	—
5	C₂×C₂²⋊#6157	C₂²	1	12	yes	—
6	(C₂×#6157)⋊C₄	C₂²	1	12	yes	—
7	C₂×D₄×D₅₃	C₂²	1	6	yes	—
8	#13133214	C₄	1	1	yes	characteristic, complemented
9	#13133213	C₄	1	6	yes	complemented
10	C₄²×D₅₃	C₂²	1	1	yes	characteristic
11	#127×C₅₃	C₄	1	1	yes	characteristic, complemented
12	C₂³×D₅₃	C₂³	1	4	yes	—
13	C₂²×#6157	—	4	12	no	—
14	C₂²×#6157	C₂³	1	8	yes	—
15	C₂×D₄×C₅₃	C₂×C₄	1	6	yes	complemented
16	C₂×C₂²⋊D₅₃	C₂×C₄	1	12	yes	complemented
17	D₄×D₅₃	—	4	6	no	—
18	D₄×D₅₃	—	4	6	no	—
19	C₂×D₂₁₂	C₂×C₄	1	6	yes	complemented
20	C₂×C₄×D₅₃	C₂³	1	3	yes	—
21	C₂²⋊#6157	—	4	12	no	—
22	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic, complemented
23	C₄×Dic₅₃	C₂×C₄	1	3	yes	complemented
24	C₂³×C₅₃	C₂²×C₄	1	4	yes	—
25	C₂²×D₅₃	—	2	12	no	—
26	C₂²×D₅₃	—	2	12	no	—
27	C₂²×D₅₃	C₂²×C₄	1	4	yes	—
28	C₂²×D₅₃	C₂⁴	1	1	yes	characteristic
29	C₂×#6157	—	2	24	no	—
30	C₂×#6157	—	4	16	no	—
31	D₄×C₅₃	—	4	6	no	—
32	C₂×C₄×C₅₃	C₂²×C₄	1	3	yes	—
33	C₂²⋊D₅₃	—	4	12	no	—
34	C₂×Dic₅₃	C₂²×C₄	1	3	yes	—
35	D₂₁₂	—	4	6	no	—
36	C₄×D₅₃	—	2	6	no	—
37	C₂²×C₅₃	—	2	12	no	—
38	C₂²×C₅₃	C₂³×C₄	1	1	yes	characteristic
39	C₂×D₅₃	—	2	12	no	—
40	C₂×D₅₃	—	4	4	no	—
41	C₂×D₅₃	—	4	12	no	—
42	C₂×D₅₃	H(5,𝔽₂)	1	6	yes	—
43	#6157	—	4	8	no	—
44	C₄×C₅₃	—	2	3	no	—
45	Dic₅₃	—	2	3	no	—
46	#2380	—	53	1	no	maximal, Sylow
47	C₂×C₅₃	—	4	4	no	—
48	C₂×C₅₃	#518	1	3	yes	—
49	D₅₃	—	4	4	no	—
50	D₅₃	#560	1	4	yes	—
51	C₂×#127	—	53	1	no	—
52	#386	—	53	12	no	—
53	#383	—	53	2	no	—
54	C₅₃	#2380	1	1	yes	characteristic, complemented, Sylow
55	C₂²×D₄	—	53	6	no	—
56	C₂³×C₄	—	53	4	no	—
57	#127	—	53	6	no	—
58	#127	—	53	1	no	—
59	#127	#6157	1	1	yes	characteristic, complemented
60	C₂×C₂²⋊C₄	—	53	12	no	—
61	C₂×C₄²	—	53	1	no	—
62	#95	—	53	12	no	—
63	C₂⁴	—	53	4	no	—
64	C₂×D₄	—	53	12	no	—
65	C₂×D₄	—	53	6	no	—
66	C₂×D₄	—	212	6	no	—
67	C₂×D₄	—	212	6	no	—
68	C₂×D₄	C₂×#6157	1	6	yes	complemented
69	C₂²×C₄	—	53	3	no	—
70	C₂²×C₄	—	53	8	no	—
71	C₂²×C₄	—	212	12	no	—
72	C₂²⋊C₄	—	212	12	no	—
73	C₄²	—	53	3	no	—
74	C₄²	C₂×#6157	1	1	yes	characteristic, complemented
75	C₂³	—	53	1	no	—
76	C₂³	—	53	4	no	—
77	C₂³	—	106	12	no	—
78	C₂³	—	106	12	no	—
79	C₂³	C₂²×#6157	1	4	yes	—
80	D₄	—	4	6	no	—
81	D₄	—	212	6	no	—
82	D₄	—	212	12	no	—
83	C₂×C₄	—	53	3	no	—
84	C₂×C₄	—	106	6	no	—
85	C₂×C₄	—	106	24	no	—
86	C₂×C₄	—	212	16	no	—
87	C₂×C₄	C₂²×#6157	1	3	yes	—
88	C₂²	—	2	12	no	—
89	C₂²	—	53	6	no	—
90	C₂²	—	106	12	no	—
91	C₂²	—	212	4	no	—
92	C₂²	—	212	12	no	—
93	C₂²	C₂³×#6157	1	1	yes	characteristic
94	C₄	—	2	3	no	—
95	C₄	—	106	3	no	—
96	C₄	—	212	8	no	—
97	C₂	—	4	4	no	—
98	C₂	—	53	4	no	—
99	C₂	—	212	4	no	—
100	C₂	#16811690	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16811571

C₂

yes

maximal, complemented

#16811568

C₂

yes

maximal, complemented

#127×D₅₃

C₂

yes

maximal, characteristic

C₂³×#6157

C₂²

yes

—

C₂×C₂²⋊#6157

C₂²

yes

—

(C₂×#6157)⋊C₄

C₂²

yes

—

C₂×D₄×D₅₃

C₂²

yes

—

#13133214

C₄

yes

characteristic, complemented

#13133213

C₄

yes

complemented

C₄²×D₅₃

C₂²

yes

characteristic

#127×C₅₃

C₄

yes

characteristic, complemented

C₂³×D₅₃

C₂³

yes

—

C₂²×#6157

—

C₂²×#6157

C₂³

yes

—

C₂×D₄×C₅₃

C₂×C₄

yes

complemented

C₂×C₂²⋊D₅₃

C₂×C₄

yes

complemented

D₄×D₅₃

—

D₄×D₅₃

—

C₂×D₂₁₂

C₂×C₄

yes

complemented

C₂×C₄×D₅₃

C₂³

yes

—

C₂²⋊#6157

—

C₄²×C₅₃

C₂×C₄

yes

characteristic, complemented

C₄×Dic₅₃

C₂×C₄

yes

complemented

C₂³×C₅₃

C₂²×C₄

yes

—

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂²×D₅₃

C₂²×C₄

yes

—

C₂²×D₅₃

C₂⁴

yes

characteristic

C₂×#6157

—

C₂×#6157

—

D₄×C₅₃

—

C₂×C₄×C₅₃

C₂²×C₄

yes

—

C₂²⋊D₅₃

—

C₂×Dic₅₃

C₂²×C₄

yes

—

D₂₁₂

—

C₄×D₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂³×C₄

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

H(5,𝔽₂)

yes

—

#6157

—

C₄×C₅₃

—

Dic₅₃

—

#2380

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

#518

yes

—

D₅₃

—

D₅₃

#560

yes

—

C₂×#127

—

#386

—

#383

—

C₅₃

#2380

yes

characteristic, complemented, Sylow

C₂²×D₄

—

C₂³×C₄

—

—

—

yes

characteristic, complemented

C₂×C₂²⋊C₄

—

C₂×C₄²

—

#95

—

C₂⁴

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

212

—

C₂×D₄

—

212

—

C₂×D₄

C₂×#6157

yes

complemented

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

212

—

C₂²⋊C₄

—

212

—

C₄²

—

C₄²

C₂×#6157

yes

characteristic, complemented

C₂³

—

C₂³

—

C₂³

—

106

—

C₂³

—

106

—

C₂³

C₂²×#6157

yes

—

D₄

—

D₄

—

212

—

D₄

—

212

—

C₂×C₄

—

C₂×C₄

—

106

—

C₂×C₄

—

106

—

C₂×C₄

—

212

—

C₂×C₄

C₂²×#6157

yes

—

C₂²

—

C₂²

—

C₂²

—

106

—

C₂²

—

212

—

C₂²

—

212

—

C₂²

C₂³×#6157

yes

characteristic

C₄

—

C₄

—

106

—

C₄

—

212

—

C₂

—

C₂

—

C₂

—

212

—

100

C₂

#16811690

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂³×#6157
commutator, socle	C₂²×C₅₃	C₂³×C₄
Fitting	#127×C₅₃	C₄
hypercenter	#127	#6157
hypocenter	C₅₃	#2380
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ #127

subgroup

G/H

center, Frattini

C₂²

C₂³×#6157

commutator, socle

C₂²×C₅₃

C₂³×C₄

Fitting

#127×C₅₃

C₄

C₅₃

G ▷ C₂²×C₅₃ ▷ C₅₃

C₁ ◁ C₂² ◁ #127

N	H ≅ G/N	type
1. #16811571	C₂	semidirect
2. #16811568	C₂	semidirect
8. #13133214	C₄	semidirect
9. #13133213	C₄	semidirect
11. #127×C₅₃	C₄	semidirect
15. C₂×D₄×C₅₃	C₂×C₄	semidirect
16. C₂×C₂²⋊D₅₃	C₂×C₄	semidirect
19. C₂×D₂₁₂	C₂×C₄	semidirect
22. C₄²×C₅₃	C₂×C₄	semidirect
23. C₄×Dic₅₃	C₂×C₄	semidirect
54. C₅₃	#2380	semidirect
59. #127	#6157	semidirect
68. C₂×D₄	C₂×#6157	semidirect
74. C₄²	C₂×#6157	semidirect

H ≅ G/N

type

1. #16811571

C₂

semidirect

2. #16811568

C₂

semidirect

8. #13133214

C₄

semidirect

9. #13133213

C₄

semidirect

11. #127×C₅₃

C₄

semidirect

15. C₂×D₄×C₅₃

C₂×C₄

semidirect

16. C₂×C₂²⋊D₅₃

C₂×C₄

semidirect

19. C₂×D₂₁₂

C₂×C₄

semidirect

22. C₄²×C₅₃

C₂×C₄

semidirect

23. C₄×Dic₅₃

C₂×C₄

semidirect

54. C₅₃

#2380

semidirect

59. #127

#6157

semidirect

68. C₂×D₄

C₂×#6157

semidirect

74. C₄²

C₂×#6157

semidirect

Subgroups — #20291259

Conjugacy classes of subgroups

Complements