Subgroups — C₂×#16810493 #20277116

Order 6784.

888 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊Dic₅₃	C₂	1	6	yes	maximal
2	#16810493	C₂	1	8	yes	maximal, direct product factor
3	C₂²×C₄²×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂³×Dic₅₃	C₂²	1	4	yes	—
5	C₂×C₄⋊Dic₅₃	—	2	24	no	—
6	C₂×C₄⋊Dic₅₃	C₂²	1	24	yes	—
7	C₂³×C₄×C₅₃	C₂²	1	3	yes	—
8	C₂×C₄²×C₅₃	C₄	1	8	yes	complemented
9	C₂×C₄²×C₅₃	C₂²	1	4	yes	—
10	C₂⁴×C₅₃	C₂³	1	1	yes	characteristic
11	C₂²×C₄×C₅₃	C₂×C₄	1	12	yes	—
12	C₂²×C₄×C₅₃	D₄	1	12	yes	—
13	C₂²×C₄×C₅₃	Q₈	1	12	yes	—
14	C₂²×C₄×C₅₃	C₂³	1	6	yes	—
15	C₂²×Dic₅₃	—	2	24	no	—
16	C₂²×Dic₅₃	C₂³	1	8	yes	—
17	C₄²×C₅₃	C₂×C₄	1	16	yes	complemented
18	C₄⋊Dic₅₃	—	2	48	no	—
19	C₂³×C₅₃	C₂²×C₄	1	2	yes	—
20	C₂³×C₅₃	C₂×D₄	1	6	yes	—
21	C₂³×C₅₃	C₂×Q₈	1	6	yes	—
22	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
23	C₂×C₄×C₅₃	C₄⋊C₄	1	48	yes	complemented
24	C₂×C₄×C₅₃	C₂²×C₄	1	12	yes	—
25	C₂×C₄×C₅₃	C₂×D₄	1	12	yes	—
26	C₂×C₄×C₅₃	C₂×Q₈	1	12	yes	—
27	C₂×Dic₅₃	—	2	24	no	—
28	C₂×Dic₅₃	—	4	16	no	—
29	C₂²×C₅₃	C₂×C₄⋊C₄	1	12	yes	—
30	C₂²×C₅₃	#127	1	4	yes	—
31	C₂²×C₅₃	C₄⋊Q₈	1	12	yes	—
32	C₂²×C₅₃	C₂³×C₄	1	1	yes	characteristic
33	C₂²×C₅₃	C₂²×D₄	1	3	yes	—
34	C₂²×C₅₃	C₂²×Q₈	1	3	yes	—
35	C₄×C₅₃	C₂×C₄⋊C₄	1	24	yes	complemented
36	Dic₅₃	—	4	8	no	—
37	C₂×#384	—	53	1	no	maximal, Sylow
38	C₂×C₅₃	#384	1	8	yes	complemented
39	C₂×C₅₃	C₂²×C₄⋊C₄	1	3	yes	—
40	C₂×C₅₃	C₂×#127	1	1	yes	characteristic
41	C₂×C₅₃	C₂×C₄⋊Q₈	1	3	yes	—
42	C₂²×C₄⋊C₄	—	53	6	no	—
43	C₂²×C₄²	D₅₃	1	1	yes	characteristic
44	#384	—	53	8	no	—
45	C₅₃	C₂×#384	1	1	yes	characteristic, complemented, Sylow
46	C₂³×C₄	—	53	4	no	—
47	C₂³×C₄	C₂×D₅₃	1	3	yes	—
48	C₂×C₄⋊C₄	—	53	24	no	—
49	C₂×C₄⋊C₄	—	106	24	no	—
50	C₂×C₄²	Dic₅₃	1	8	yes	complemented
51	C₂×C₄²	C₂×D₅₃	1	4	yes	—
52	C₂⁴	C₂²×D₅₃	1	1	yes	characteristic
53	C₂²×C₄	—	53	8	no	—
54	C₂²×C₄	—	106	24	no	—
55	C₂²×C₄	Dic₁₀₆	1	12	yes	—
56	C₂²×C₄	D₂₁₂	1	12	yes	—
57	C₂²×C₄	C₂×Dic₅₃	1	12	yes	—
58	C₂²×C₄	C₂²×D₅₃	1	6	yes	—
59	C₄⋊C₄	—	106	48	no	—
60	C₄²	C₂×Dic₅₃	1	16	yes	complemented
61	C₂³	C₂×Dic₁₀₆	1	6	yes	—
62	C₂³	C₂×D₂₁₂	1	6	yes	—
63	C₂³	C₂²×Dic₅₃	1	2	yes	—
64	C₂³	C₂³×D₅₃	1	1	yes	characteristic
65	C₂×C₄	—	106	24	no	—
66	C₂×C₄	—	212	16	no	—
67	C₂×C₄	C₄⋊Dic₅₃	1	48	yes	complemented
68	C₂×C₄	C₂×Dic₁₀₆	1	12	yes	—
69	C₂×C₄	C₂×D₂₁₂	1	12	yes	—
70	C₂×C₄	C₂²×Dic₅₃	1	12	yes	—
71	C₂²	C₂×C₄⋊Dic₅₃	1	12	yes	—
72	C₂²	#13133214	1	4	yes	—
73	C₂²	#13133217	1	12	yes	—
74	C₂²	C₂³×Dic₅₃	1	1	yes	characteristic
75	C₂²	C₂²×D₂₁₂	1	3	yes	—
76	C₂²	C₂²×Dic₁₀₆	1	3	yes	—
77	C₄	—	212	8	no	—
78	C₄	C₂×C₄⋊Dic₅₃	1	24	yes	complemented
79	C₂	#16810493	1	8	yes	direct product factor
80	C₂	C₂²×C₄⋊Dic₅₃	1	3	yes	—
81	C₂	C₂×#13133214	1	1	yes	characteristic
82	C₂	C₂×#13133217	1	3	yes	—

	subgroup	G/H
center	C₂⁴	C₂²×D₅₃
commutator	C₂²×C₅₃	C₂³×C₄
Frattini	C₂³	C₂³×D₅₃
Fitting	C₂²×C₄²×C₅₃	C₂
socle	C₂⁴×C₅₃	C₂³
hypercenter	C₂²×C₄²	D₅₃
hypocenter	C₅₃	C₂×#384
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
2. #16810493	C₂	direct (N × H)
8. C₂×C₄²×C₅₃	C₄	semidirect
17. C₄²×C₅₃	C₂×C₄	semidirect
23. C₂×C₄×C₅₃	C₄⋊C₄	semidirect
35. C₄×C₅₃	C₂×C₄⋊C₄	semidirect
38. C₂×C₅₃	#384	semidirect
45. C₅₃	C₂×#384	semidirect
50. C₂×C₄²	Dic₅₃	semidirect
60. C₄²	C₂×Dic₅₃	semidirect
67. C₂×C₄	C₄⋊Dic₅₃	semidirect
78. C₄	C₂×C₄⋊Dic₅₃	semidirect
79. C₂	#16810493	direct (N × H)

Subgroups — C2×#16810493 #20277116

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16810493 #20277116