Subgroups — #20272893

Order 6784.

82 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#13133123	C₂	1	1	yes	maximal, characteristic, complemented
2	#16810449	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×D₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
4	#13133151	—	2	1	no	—
5	#13133123	—	2	1	no	—
6	C₂×D₈×C₅₃	—	2	1	no	—
7	D₁₆×C₅₃	—	2	1	no	—
8	D₁₆×C₅₃	C₄	1	2	yes	complemented
9	C₂×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
10	C₂×D₄×C₅₃	—	4	1	no	—
11	D₈×C₅₃	—	2	2	no	—
12	D₈×C₅₃	—	4	1	no	—
13	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
14	C₄⋊Dic₅₃	—	4	1	no	—
15	C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
16	C₁₆×C₅₃	D₄	1	1	yes	characteristic
17	C₂³×C₅₃	—	8	1	no	—
18	D₄×C₅₃	—	4	2	no	—
19	D₄×C₅₃	—	8	1	no	—
20	C₂×C₄×C₅₃	D₈	1	1	yes	characteristic
21	C₂×Dic₅₃	—	8	1	no	—
22	C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
23	C₈×C₅₃	SD₁₆	1	1	yes	characteristic
24	C₂²×C₅₃	—	8	2	no	—
25	C₂²×C₅₃	—	16	1	no	—
26	C₂²×C₅₃	—	16	1	no	—
27	C₂²×C₅₃	D₁₆	1	1	yes	characteristic
28	C₄×C₅₃	#102	1	1	yes	characteristic
29	C₄×C₅₃	SD₃₂	1	1	yes	characteristic
30	Dic₅₃	—	16	1	no	—
31	#1415	—	53	1	no	maximal, Sylow
32	C₂×C₅₃	—	16	2	no	—
33	C₂×C₅₃	#357	1	1	yes	characteristic
34	C₂×C₅₃	D₃₂	1	1	yes	characteristic
35	C₂×C₅₃	SD₆₄	1	1	yes	characteristic
36	C₂×D₁₆	D₅₃	1	1	yes	characteristic
37	C₂×C₃₂	—	53	1	no	—
38	#366	—	53	1	no	—
39	C₅₃	#1415	1	1	yes	characteristic, complemented, Sylow
40	C₂×D₈	—	2	1	no	—
41	D₁₆	—	2	1	no	—
42	D₁₆	Dic₅₃	1	2	yes	complemented
43	C₂×C₁₆	C₂×D₅₃	1	1	yes	characteristic
44	#107	—	106	1	no	—
45	C₃₂	—	106	1	no	—
46	C₂×D₄	—	4	1	no	—
47	D₈	—	2	2	no	—
48	D₈	—	4	1	no	—
49	C₂×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
50	C₄⋊C₄	—	212	1	no	—
51	C₁₆	C₂×Dic₅₃	1	1	yes	characteristic
52	C₁₆	C₂²⋊D₅₃	1	1	yes	characteristic
53	C₂³	—	8	1	no	—
54	D₄	—	4	2	no	—
55	D₄	—	8	1	no	—
56	C₂×C₄	—	424	1	no	—
57	C₂×C₄	#11719922	1	1	yes	characteristic
58	C₈	#11719923	1	1	yes	characteristic
59	C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
60	C₂²	—	8	2	no	—
61	C₂²	—	16	1	no	—
62	C₂²	—	16	1	no	—
63	C₂²	#13133158	1	1	yes	characteristic
64	C₄	—	848	1	no	—
65	C₄	#13133138	1	1	yes	characteristic
66	C₄	#13133161	1	1	yes	characteristic
67	C₂	—	16	2	no	—
68	C₂	#16810424	1	1	yes	characteristic
69	C₂	#16810458	1	1	yes	characteristic
70	C₂	#16810461	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#13133158
commutator	C₁₆×C₅₃	C₂×C₄
Frattini	C₂×C₁₆	C₂×D₅₃
Fitting	C₂×D₁₆×C₅₃	C₂
socle	C₂²×C₅₃	D₁₆
hypercenter	C₂×D₁₆	D₅₃
hypocenter	C₅₃	#1415
LCS	G ▷ C₁₆×C₅₃ ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×D₁₆

Complements

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

N	H ≅ G/N	type
1. C₂×#13133123	C₂	semidirect
2. #16810449	C₂	semidirect
8. D₁₆×C₅₃	C₄	semidirect
39. C₅₃	#1415	semidirect
42. D₁₆	Dic₅₃	semidirect