Subgroups — #20276997

Order 6784.

128 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16811028	C₂	1	2	yes	maximal
2	#16810451	C₂	1	4	yes	maximal
3	C₄×C₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
4	#13133217	—	2	2	no	—
5	#13133151	—	2	2	no	—
6	#13133151	C₂²	1	4	yes	—
7	C₂×C₁₆×C₅₃	C₂²	1	2	yes	—
8	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
9	C₂×Dic₁₀₆	—	4	2	no	—
10	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
11	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
12	C₄²×C₅₃	D₄	1	1	yes	characteristic
13	C₄⋊Dic₅₃	—	2	4	no	—
14	C₄⋊Dic₅₃	—	4	2	no	—
15	C₁₆×C₅₃	Q₈	1	4	yes	complemented
16	C₂×C₄×C₅₃	D₈	1	2	yes	—
17	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
18	C₂×Dic₅₃	—	4	4	no	—
19	Dic₁₀₆	—	8	4	no	—
20	C₈×C₅₃	Q₁₆	1	2	yes	—
21	C₈×C₅₃	C₂×Q₈	1	2	yes	—
22	C₂²×C₅₃	C₂×D₈	1	1	yes	characteristic
23	C₄×C₅₃	SD₃₂	1	4	yes	—
24	C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
25	C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
26	Dic₅₃	—	8	4	no	—
27	#2254	—	53	1	no	maximal, Sylow
28	C₂×C₅₃	#500	1	1	yes	characteristic
29	C₂×C₅₃	C₂×SD₃₂	1	2	yes	—
30	#500	—	53	2	no	—
31	#367	—	53	4	no	—
32	C₄×C₁₆	D₅₃	1	1	yes	characteristic
33	C₅₃	#2254	1	1	yes	characteristic, complemented, Sylow
34	C₄⋊Q₈	—	106	2	no	—
35	C₂×C₁₆	C₂×D₅₃	1	2	yes	—
36	#107	—	53	4	no	—
37	#107	—	106	2	no	—
38	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
39	C₂×Q₈	—	212	2	no	—
40	C₂×C₈	D₂₁₂	1	1	yes	characteristic
41	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
42	C₄⋊C₄	—	106	4	no	—
43	C₄⋊C₄	—	212	2	no	—
44	C₄²	D₂₁₂	1	1	yes	characteristic
45	C₁₆	Dic₁₀₆	1	4	yes	complemented
46	Q₈	—	424	4	no	—
47	C₂×C₄	—	212	4	no	—
48	C₂×C₄	D₄₂₄	1	2	yes	—
49	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
50	C₈	Dic₂₁₂	1	2	yes	—
51	C₈	C₂×Dic₁₀₆	1	2	yes	—
52	C₂²	C₂×D₄₂₄	1	1	yes	characteristic
53	C₄	—	424	4	no	—
54	C₄	#13133162	1	4	yes	—
55	C₄	#13133217	1	1	yes	characteristic
56	C₄	C₂×Dic₂₁₂	1	1	yes	characteristic
57	C₂	#16811028	1	1	yes	characteristic
58	C₂	C₂×#13133162	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₄₂₄
commutator	C₂×C₈×C₅₃	C₂³
Frattini	C₂×C₈	C₂²×D₅₃
Fitting	C₄×C₁₆×C₅₃	C₂
socle	C₂²×C₅₃	C₂×D₈
hypercenter	C₄×C₁₆	D₅₃
hypocenter	C₅₃	#2254
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆

Complements

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

N	H ≅ G/N	type
15. C₁₆×C₅₃	Q₈	semidirect
33. C₅₃	#2254	semidirect
45. C₁₆	Dic₁₀₆	semidirect