Subgroups — #20272892

Order 6784.

82 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₈₄₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#13133123	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#366	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₄₂₄	—	2	1	no	—
5	D₈₄₈	—	2	1	no	—
6	D₈₄₈	C₄	1	2	yes	complemented
7	#13133123	—	2	1	no	—
8	C₂×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
9	#107×C₅₃	—	2	1	no	—
10	C₂×D₂₁₂	—	4	1	no	—
11	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
12	C₄⋊C₄×C₅₃	—	4	1	no	—
13	D₄₂₄	—	2	2	no	—
14	D₄₂₄	—	4	1	no	—
15	C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
16	C₁₆×C₅₃	D₄	1	1	yes	characteristic
17	C₂²×D₅₃	—	8	1	no	—
18	C₂×C₄×C₅₃	—	8	1	no	—
19	C₂×C₄×C₅₃	D₈	1	1	yes	characteristic
20	D₂₁₂	—	4	2	no	—
21	D₂₁₂	—	8	1	no	—
22	C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
23	C₈×C₅₃	SD₁₆	1	1	yes	characteristic
24	C₂²×C₅₃	D₁₆	1	1	yes	characteristic
25	C₂×D₅₃	—	8	2	no	—
26	C₂×D₅₃	—	16	1	no	—
27	C₂×D₅₃	—	16	1	no	—
28	C₄×C₅₃	—	16	1	no	—
29	C₄×C₅₃	#102	1	1	yes	characteristic
30	C₄×C₅₃	SD₃₂	1	1	yes	characteristic
31	#1415	—	53	1	no	maximal, Sylow
32	C₂×C₅₃	#357	1	1	yes	characteristic
33	C₂×C₅₃	D₃₂	1	1	yes	characteristic
34	C₂×C₅₃	SD₆₄	1	1	yes	characteristic
35	D₅₃	—	16	2	no	—
36	C₂×D₁₆	—	53	1	no	—
37	C₂×C₃₂	—	53	1	no	—
38	#366	D₅₃	1	1	yes	characteristic, complemented
39	C₅₃	#1415	1	1	yes	characteristic, complemented, Sylow
40	C₂×D₈	—	106	1	no	—
41	D₁₆	—	53	2	no	—
42	D₁₆	—	106	1	no	—
43	C₂×C₁₆	C₂×D₅₃	1	1	yes	characteristic
44	#107	—	2	1	no	—
45	C₃₂	—	106	1	no	—
46	C₂×D₄	—	212	1	no	—
47	D₈	—	106	2	no	—
48	D₈	—	212	1	no	—
49	C₂×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
50	C₄⋊C₄	—	4	1	no	—
51	C₁₆	C₄×D₅₃	1	1	yes	characteristic
52	C₁₆	D₂₁₂	1	1	yes	characteristic
53	C₂³	—	424	1	no	—
54	D₄	—	424	1	no	—
55	D₄	—	212	2	no	—
56	C₂×C₄	—	8	1	no	—
57	C₂×C₄	#11719922	1	1	yes	characteristic
58	C₈	#11719921	1	1	yes	characteristic
59	C₈	#11719924	1	1	yes	characteristic
60	C₂²	—	848	1	no	—
61	C₂²	—	848	1	no	—
62	C₂²	—	424	2	no	—
63	C₂²	#13133158	1	1	yes	characteristic
64	C₄	—	16	1	no	—
65	C₄	#13133137	1	1	yes	characteristic
66	C₄	#13133160	1	1	yes	characteristic
67	C₂	—	848	2	no	—
68	C₂	#16810423	1	1	yes	characteristic
69	C₂	#16810458	1	1	yes	characteristic
70	C₂	#16810460	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#13133158
commutator	C₁₆×C₅₃	C₂×C₄
Frattini	C₂×C₁₆	C₂×D₅₃
Fitting	C₅₃×#366	C₂
socle	C₂²×C₅₃	D₁₆
hypercenter	#366	D₅₃
hypocenter	C₅₃	#1415
LCS	G ▷ C₁₆×C₅₃ ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #366

Complements

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

N	H ≅ G/N	type
2. C₂×#13133123	C₂	semidirect
3. C₅₃×#366	C₂	semidirect
6. D₈₄₈	C₄	semidirect
38. #366	D₅₃	semidirect
39. C₅₃	#1415	semidirect