Subgroups — #20272787

Order 6784.

70 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16810376	C₂	1	2	yes	maximal
2	C₄×C₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₄⋊#88322	—	2	2	no	—
4	C₂×C₁₆×C₅₃	C₄	1	1	yes	characteristic
5	C₂×C₁₆×C₅₃	C₄	1	1	yes	characteristic
6	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
7	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
8	C₂×C₈×C₅₃	Q₈	1	1	yes	characteristic
9	C₄²×C₅₃	D₄	1	1	yes	characteristic
10	C₂×#88322	—	4	2	no	—
11	C₁₆×C₅₃	—	2	1	no	—
12	C₁₆×C₅₃	C₈	1	2	yes	complemented
13	C₂×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
14	C₂×C₄×C₅₃	D₈	1	1	yes	characteristic
15	C₂×C₄×C₅₃	Q₁₆	1	1	yes	characteristic
16	C₈×C₅₃	—	2	1	no	—
17	C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
18	C₈×C₅₃	M₁₆	1	1	yes	characteristic
19	#88322	—	8	2	no	—
20	C₂²×C₅₃	#107	1	1	yes	characteristic
21	C₄×C₅₃	—	2	1	no	—
22	C₄×C₅₃	C₄⋊C₈	1	1	yes	characteristic
23	C₄×C₅₃	#108	1	1	yes	characteristic
24	C₄×C₅₃	D₁₆	1	1	yes	characteristic
25	C₄×C₅₃	Q₃₂	1	1	yes	characteristic
26	#1371	—	53	1	no	maximal, Sylow
27	C₂×C₅₃	#335	1	1	yes	characteristic
28	C₂×C₅₃	#366	1	1	yes	characteristic
29	C₂×C₅₃	#368	1	1	yes	characteristic
30	C₄×C₁₆	D₅₃	1	1	yes	characteristic
31	#335	—	53	2	no	—
32	C₅₃	#1371	1	1	yes	characteristic, complemented, Sylow
33	C₂×C₁₆	Dic₅₃	1	1	yes	characteristic
34	C₂×C₁₆	Dic₅₃	1	1	yes	characteristic
35	C₄⋊C₈	—	106	2	no	—
36	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
37	C₂×C₈	—	212	2	no	—
38	C₂×C₈	Dic₁₀₆	1	1	yes	characteristic
39	C₂×C₈	C₂×Dic₅₃	1	1	yes	characteristic
40	C₄²	D₂₁₂	1	1	yes	characteristic
41	C₁₆	—	2	1	no	—
42	C₁₆	#88322	1	2	yes	complemented
43	C₂×C₄	D₄₂₄	1	1	yes	characteristic
44	C₂×C₄	Dic₂₁₂	1	1	yes	characteristic
45	C₂×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
46	C₈	—	2	1	no	—
47	C₈	—	424	2	no	—
48	C₈	C₂×#88322	1	1	yes	characteristic
49	C₈	#11719917	1	1	yes	characteristic
50	C₂²	#13133151	1	1	yes	characteristic
51	C₄	—	2	1	no	—
52	C₄	C₄⋊#88322	1	1	yes	characteristic
53	C₄	#13133153	1	1	yes	characteristic
54	C₄	D₈₄₈	1	1	yes	characteristic
55	C₄	Dic₄₂₄	1	1	yes	characteristic
56	C₂	#16810376	1	1	yes	characteristic
57	C₂	#16810449	1	1	yes	characteristic
58	C₂	#16810453	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₄₂₄
commutator	C₈×C₅₃	C₂×C₈
Frattini	C₄×C₈	C₂×D₅₃
Fitting	C₄×C₁₆×C₅₃	C₂
socle	C₂²×C₅₃	#107
hypercenter	C₄×C₁₆	D₅₃
hypocenter	C₅₃	#1371
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆

Complements

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

N	H ≅ G/N	type
12. C₁₆×C₅₃	C₈	semidirect
32. C₅₃	#1371	semidirect
42. C₁₆	#88322	semidirect