Subgroups — #20274875

Order 6784.

280 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#13133151	C₂	1	4	yes	maximal
2	#16810493	C₂	1	2	yes	maximal
3	C₂×C₄×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊Dic₅₃	—	2	4	no	—
5	C₂×C₄⋊Dic₅₃	C₂²	1	4	yes	—
6	#13133151	—	2	8	no	—
7	C₂²×C₈×C₅₃	C₂²	1	2	yes	—
8	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
9	C₄×C₈×C₅₃	C₄	1	4	yes	complemented
10	C₂²×C₄×C₅₃	D₄	1	2	yes	—
11	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
12	C₂²×Dic₅₃	—	2	4	no	—
13	C₂×C₈×C₅₃	C₂×C₄	1	4	yes	—
14	C₂×C₈×C₅₃	D₄	1	4	yes	—
15	C₂×C₈×C₅₃	Q₈	1	4	yes	—
16	C₄²×C₅₃	C₂×C₄	1	2	yes	—
17	C₄²×C₅₃	Q₈	1	2	yes	—
18	C₄⋊Dic₅₃	—	2	8	no	—
19	C₄⋊Dic₅₃	—	4	8	no	—
20	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
21	C₂×C₄×C₅₃	C₄⋊C₄	1	4	yes	—
22	C₂×C₄×C₅₃	D₈	1	4	yes	—
23	C₂×C₄×C₅₃	Q₁₆	1	4	yes	—
24	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
25	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
26	C₂×C₄×C₅₃	C₂×Q₈	1	2	yes	—
27	C₂×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
28	C₂×Dic₅₃	—	4	4	no	—
29	C₂×Dic₅₃	—	4	8	no	—
30	C₈×C₅₃	C₄⋊C₄	1	8	yes	complemented
31	C₂²×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
32	C₂²×C₅₃	#127	1	1	yes	characteristic
33	C₂²×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
34	C₂²×C₅₃	C₂×D₈	1	2	yes	—
35	C₂²×C₅₃	C₂×Q₁₆	1	2	yes	—
36	C₄×C₅₃	#107	1	8	yes	complemented
37	C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
38	C₄×C₅₃	C₄⋊Q₈	1	2	yes	—
39	Dic₅₃	—	8	4	no	—
40	#1845	—	53	1	no	maximal, Sylow
41	C₂×C₅₃	#384	1	1	yes	characteristic
42	C₂×C₅₃	C₂×#107	1	2	yes	—
43	C₂×C₅₃	#493	1	1	yes	characteristic
44	C₂×C₅₃	#494	1	1	yes	characteristic
45	C₂×C₅₃	#500	1	2	yes	—
46	C₂×#107	—	53	4	no	—
47	C₂×C₄×C₈	D₅₃	1	1	yes	characteristic
48	#384	—	53	2	no	—
49	C₅₃	#1845	1	1	yes	characteristic, complemented, Sylow
50	C₂²×C₈	C₂×D₅₃	1	2	yes	—
51	C₂×C₄⋊C₄	—	53	4	no	—
52	C₂×C₄⋊C₄	—	106	4	no	—
53	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
54	#107	—	106	8	no	—
55	C₄×C₈	Dic₅₃	1	4	yes	complemented
56	C₂²×C₄	—	106	4	no	—
57	C₂²×C₄	D₂₁₂	1	2	yes	—
58	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
59	C₂×C₈	Dic₁₀₆	1	4	yes	—
60	C₂×C₈	D₂₁₂	1	4	yes	—
61	C₂×C₈	C₂×Dic₅₃	1	4	yes	—
62	C₄⋊C₄	—	106	8	no	—
63	C₄⋊C₄	—	212	8	no	—
64	C₄²	Dic₁₀₆	1	2	yes	—
65	C₄²	C₂×Dic₅₃	1	2	yes	—
66	C₂³	C₂×D₂₁₂	1	1	yes	characteristic
67	C₂×C₄	—	212	4	no	—
68	C₂×C₄	—	212	8	no	—
69	C₂×C₄	D₄₂₄	1	4	yes	—
70	C₂×C₄	Dic₂₁₂	1	4	yes	—
71	C₂×C₄	C₄⋊Dic₅₃	1	4	yes	—
72	C₂×C₄	C₂×Dic₁₀₆	1	2	yes	—
73	C₂×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
74	C₂×C₄	C₂×D₂₁₂	1	2	yes	—
75	C₂×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
76	C₈	C₄⋊Dic₅₃	1	8	yes	complemented
77	C₂²	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
78	C₂²	#13133214	1	1	yes	characteristic
79	C₂²	#13133217	1	1	yes	characteristic
80	C₂²	C₂×D₄₂₄	1	2	yes	—
81	C₂²	C₂×Dic₂₁₂	1	2	yes	—
82	C₄	—	424	4	no	—
83	C₄	#13133151	1	8	yes	complemented
84	C₄	C₂×C₄⋊Dic₅₃	1	2	yes	—
85	C₄	#13133217	1	2	yes	—
86	C₂	#16810493	1	1	yes	characteristic
87	C₂	C₂×#13133151	1	2	yes	—
88	C₂	#16811000	1	1	yes	characteristic
89	C₂	#16811003	1	1	yes	characteristic
90	C₂	#16811028	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₂×C₄×C₈×C₅₃	C₂
socle	C₂³×C₅₃	C₂×D₄
hypercenter	C₂×C₄×C₈	D₅₃
hypocenter	C₅₃	#1845
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
9. C₄×C₈×C₅₃	C₄	semidirect
30. C₈×C₅₃	C₄⋊C₄	semidirect
36. C₄×C₅₃	#107	semidirect
49. C₅₃	#1845	semidirect
55. C₄×C₈	Dic₅₃	semidirect
76. C₈	C₄⋊Dic₅₃	semidirect
83. C₄	#13133151	semidirect