Subgroups — #20272779

Order 6784.

76 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊#11719908	C₂	1	2	yes	maximal
2	C₈²×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₂×#11719908	—	2	2	no	—
4	C₄×C₈×C₅₃	C₄	1	1	yes	characteristic
5	C₄×C₈×C₅₃	C₄	1	1	yes	characteristic
6	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
7	C₂×C₈×C₅₃	—	2	1	no	—
8	C₂×C₈×C₅₃	C₈	1	1	yes	characteristic
9	C₂×C₈×C₅₃	C₈	1	1	yes	characteristic
10	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
11	C₂×C₈×C₅₃	Q₈	1	1	yes	characteristic
12	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
13	#11719908	—	4	2	no	—
14	C₂×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
15	C₂×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
16	C₂×C₄×C₅₃	M₁₆	1	1	yes	characteristic
17	C₈×C₅₃	—	2	2	no	—
18	C₈×C₅₃	—	2	1	no	—
19	C₈×C₅₃	—	2	1	no	—
20	C₈×C₅₃	C₁₆	1	2	yes	complemented
21	C₈×C₅₃	SD₁₆	1	2	yes	—
22	C₂²×C₅₃	C₄⋊C₈	1	1	yes	characteristic
23	C₄×C₅₃	—	2	1	no	—
24	C₄×C₅₃	#106	1	1	yes	characteristic
25	C₄×C₅₃	#108	1	1	yes	characteristic
26	C₄×C₅₃	C₂×C₁₆	1	1	yes	characteristic
27	C₄×C₅₃	M₃₂	1	1	yes	characteristic
28	#1367	—	53	1	no	maximal, Sylow
29	C₂×C₅₃	#334	1	1	yes	characteristic
30	C₂×C₅₃	C₄⋊C₁₆	1	1	yes	characteristic
31	C₂×C₅₃	#364	1	1	yes	characteristic
32	C₄⋊C₁₆	—	53	2	no	—
33	C₈²	D₅₃	1	1	yes	characteristic
34	C₅₃	#1367	1	1	yes	characteristic, complemented, Sylow
35	C₂×C₁₆	—	106	2	no	—
36	C₄×C₈	Dic₅₃	1	1	yes	characteristic
37	C₄×C₈	Dic₅₃	1	1	yes	characteristic
38	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
39	C₂×C₈	—	2	1	no	—
40	C₂×C₈	#88322	1	1	yes	characteristic
41	C₂×C₈	#88322	1	1	yes	characteristic
42	C₂×C₈	Dic₁₀₆	1	1	yes	characteristic
43	C₂×C₈	D₂₁₂	1	1	yes	characteristic
44	C₄²	C₂×Dic₅₃	1	1	yes	characteristic
45	C₁₆	—	212	2	no	—
46	C₂×C₄	C₂×#88322	1	1	yes	characteristic
47	C₂×C₄	#11719917	1	1	yes	characteristic
48	C₂×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
49	C₈	—	2	2	no	—
50	C₈	—	2	1	no	—
51	C₈	—	2	1	no	—
52	C₈	#11719908	1	2	yes	complemented
53	C₈	#11719913	1	2	yes	—
54	C₂²	C₄⋊#88322	1	1	yes	characteristic
55	C₄	—	2	1	no	—
56	C₄	#13133149	1	1	yes	characteristic
57	C₄	#13133153	1	1	yes	characteristic
58	C₄	C₂×#11719908	1	1	yes	characteristic
59	C₄	#13133157	1	1	yes	characteristic
60	C₂	#16810374	1	1	yes	characteristic
61	C₂	C₄⋊#11719908	1	1	yes	characteristic
62	C₂	#16810444	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	D₂₁₂
commutator	C₄×C₅₃	C₂×C₁₆
Frattini	C₄×C₈	C₂×D₅₃
Fitting	C₈²×C₅₃	C₂
socle	C₂²×C₅₃	C₄⋊C₈
hypercenter	C₈²	D₅₃
hypocenter	C₅₃	#1367
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ C₈²

Complements

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

N	H ≅ G/N	type
20. C₈×C₅₃	C₁₆	semidirect
34. C₅₃	#1367	semidirect
52. C₈	#11719908	semidirect