Subgroups — C₂²×#13133211 #20287971

Order 6784.

1320 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#13133211	C₂	1	24	yes	maximal, direct product factor
2	C₂²×#11719919	C₂	1	3	yes	maximal, complemented
3	C₂²×#11719921	C₂	1	3	yes	maximal
4	C₂²×C₄²×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂⁴×D₅₃	C₂²	1	1	yes	characteristic
6	C₂³×Dic₅₃	C₂²	1	3	yes	—
7	#13133211	C₂²	1	64	yes	direct product factor
8	C₂×#11719919	C₂²	1	36	yes	complemented
9	C₂×#11719921	C₂²	1	36	yes	—
10	C₂³×C₄×C₅₃	C₂²	1	3	yes	—
11	C₂×C₄²×C₅₃	C₂²	1	12	yes	complemented
12	C₂⁴×C₅₃	C₂³	1	1	yes	characteristic
13	C₂³×D₅₃	—	2	12	no	—
14	C₂³×D₅₃	C₂³	1	6	yes	—
15	C₂²×C₄×C₅₃	—	2	12	no	—
16	C₂²×C₄×C₅₃	C₂³	1	18	yes	—
17	C₂²×Dic₅₃	—	2	12	no	—
18	C₂²×Dic₅₃	C₂³	1	18	yes	—
19	C₄²×C₅₃	C₂³	1	16	yes	complemented
20	#11719921	C₂³	1	48	yes	—
21	#11719919	C₂³	1	48	yes	complemented
22	C₂³×C₅₃	#41	1	12	yes	—
23	C₂³×C₅₃	C₂⁴	1	3	yes	—
24	C₂²×D₅₃	—	2	36	no	—
25	C₂²×D₅₃	—	4	16	no	—
26	C₂²×D₅₃	C₂⁴	1	4	yes	—
27	C₂×C₄×C₅₃	—	2	36	no	—
28	C₂×C₄×C₅₃	C₂⁴	1	12	yes	—
29	C₂×Dic₅₃	—	2	36	no	—
30	C₂×Dic₅₃	C₂⁴	1	12	yes	—
31	C₂²×C₅₃	#126	1	16	yes	complemented
32	C₂²×C₅₃	C₂×#41	1	18	yes	—
33	C₂²×C₅₃	C₂⁵	1	1	yes	characteristic
34	C₂×D₅₃	—	2	12	no	—
35	C₂×D₅₃	—	4	24	no	—
36	C₄×C₅₃	—	2	12	no	—
37	Dic₅₃	—	2	12	no	—
38	C₂²×#126	—	53	1	no	maximal, Sylow
39	C₂×C₅₃	C₂×#126	1	12	yes	complemented
40	C₂×C₅₃	C₂²×#41	1	3	yes	—
41	D₅₃	—	4	4	no	—
42	C₂×#126	—	53	24	no	—
43	C₂²×C₄⋊C₄	—	53	3	no	—
44	C₂²×C₂²⋊C₄	—	53	3	no	—
45	C₂²×C₄²	D₅₃	1	1	yes	characteristic, complemented
46	C₅₃	C₂²×#126	1	1	yes	characteristic, complemented, Sylow
47	C₂⁵	—	53	1	no	—
48	C₂³×C₄	—	53	3	no	—
49	C₂³×C₄	C₂×D₅₃	1	3	yes	—
50	#126	—	53	64	no	—
51	C₂×C₄⋊C₄	—	53	36	no	—
52	C₂×C₂²⋊C₄	—	53	36	no	—
53	C₂×C₄²	C₂×D₅₃	1	12	yes	complemented
54	C₂⁴	—	53	6	no	—
55	C₂⁴	—	106	12	no	—
56	C₂⁴	C₂²×D₅₃	1	1	yes	characteristic
57	C₂²×C₄	—	2	12	no	—
58	C₂²×C₄	—	53	18	no	—
59	C₂²×C₄	—	106	12	no	—
60	C₂²×C₄	C₂²×D₅₃	1	18	yes	—
61	C₄⋊C₄	—	53	48	no	—
62	C₂²⋊C₄	—	53	48	no	—
63	C₄²	C₂²×D₅₃	1	16	yes	complemented
64	C₂³	—	53	4	no	—
65	C₂³	—	106	36	no	—
66	C₂³	—	212	16	no	—
67	C₂³	#11719945	1	12	yes	—
68	C₂³	C₂³×D₅₃	1	3	yes	—
69	C₂×C₄	—	2	36	no	—
70	C₂×C₄	—	53	12	no	—
71	C₂×C₄	—	106	36	no	—
72	C₂×C₄	C₂³×D₅₃	1	12	yes	—
73	C₂²	—	106	12	no	—
74	C₂²	—	212	24	no	—
75	C₂²	#13133211	1	16	yes	direct product factor
76	C₂²	C₂×#11719945	1	18	yes	—
77	C₂²	C₂⁴×D₅₃	1	1	yes	characteristic
78	C₄	—	2	12	no	—
79	C₄	—	106	12	no	—
80	C₂	—	212	4	no	—
81	C₂	C₂×#13133211	1	12	yes	direct product factor
82	C₂	C₂²×#11719945	1	3	yes	—

	subgroup	G/H
center	C₂⁴	C₂²×D₅₃
commutator	C₂²×C₅₃	C₂⁵
Frattini	C₂²	C₂⁴×D₅₃
Fitting	C₂²×C₄²×C₅₃	C₂
socle	C₂⁴×C₅₃	C₂³
hypercenter	C₂²×C₄²	D₅₃
hypocenter	C₅₃	C₂²×#126
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄²

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂×#13133211	C₂	direct (N × H)
2. C₂²×#11719919	C₂	semidirect
4. C₂²×C₄²×C₅₃	C₂	semidirect
7. #13133211	C₂²	direct (N × H)
8. C₂×#11719919	C₂²	semidirect
11. C₂×C₄²×C₅₃	C₂²	semidirect
19. C₄²×C₅₃	C₂³	semidirect
21. #11719919	C₂³	semidirect
31. C₂²×C₅₃	#126	semidirect
39. C₂×C₅₃	C₂×#126	semidirect
45. C₂²×C₄²	D₅₃	semidirect
46. C₅₃	C₂²×#126	semidirect
53. C₂×C₄²	C₂×D₅₃	semidirect
63. C₄²	C₂²×D₅₃	semidirect
75. C₂²	#13133211	direct (N × H)
81. C₂	C₂×#13133211	direct (N × H)

Subgroups — C22×#13133211 #20287971

Conjugacy classes of subgroups

Complements

Subgroups — C₂²×#13133211 #20287971