Subgroups — #20272775

Order 6784.

74 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16811025	C₂	1	1	yes	maximal, characteristic
2	#16810444	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#364	C₂	1	1	yes	maximal, characteristic
4	#13133209	C₄	1	2	yes	—
5	#13133157	—	2	1	no	—
6	#13133151	—	2	1	no	—
7	#13133149	—	2	1	no	—
8	M₃₂×C₅₃	—	2	1	no	—
9	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
10	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
11	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
12	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
13	C₄⋊Dic₅₃	—	2	2	no	—
14	#11719919	—	4	2	no	—
15	C₁₆×C₅₃	—	4	1	no	—
16	#11719908	—	4	1	no	—
17	C₂×C₄×C₅₃	—	2	1	no	—
18	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
19	C₂×Dic₅₃	—	4	2	no	—
20	C₈×C₅₃	SD₁₆	1	1	yes	characteristic
21	C₈×C₅₃	SD₁₆	1	1	yes	characteristic
22	C₈×C₅₃	Q₁₆	1	1	yes	characteristic
23	C₈×C₅₃	Q₁₆	1	1	yes	characteristic
24	C₂²×C₅₃	#100	1	1	yes	characteristic
25	C₄×C₅₃	—	4	1	no	—
26	C₄×C₅₃	#103	1	1	yes	characteristic
27	C₄×C₅₃	#103	1	1	yes	characteristic
28	Dic₅₃	—	8	2	no	—
29	#1366	—	53	1	no	maximal, Sylow
30	C₂×C₅₃	—	2	1	no	—
31	C₂×C₅₃	#333	1	1	yes	characteristic
32	#499	—	53	1	no	—
33	#364	—	53	1	no	—
34	#364	D₅₃	1	1	yes	characteristic
35	C₅₃	#1366	1	1	yes	characteristic, complemented, Sylow
36	#125	—	53	2	no	—
37	M₃₂	—	2	1	no	—
38	M₃₂	—	106	1	no	—
39	#107	—	106	1	no	—
40	#106	—	106	1	no	—
41	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
42	C₂×C₈	D₂₁₂	1	1	yes	characteristic
43	C₂×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
44	C₄⋊C₄	—	106	2	no	—
45	C₄⋊C₄	—	212	2	no	—
46	C₄²	C₄×D₅₃	1	1	yes	characteristic
47	C₁₆	—	4	1	no	—
48	C₁₆	—	212	1	no	—
49	C₂×C₄	—	2	1	no	—
50	C₂×C₄	—	212	2	no	—
51	C₂×C₄	#11719921	1	1	yes	characteristic
52	C₈	#11719913	1	1	yes	characteristic
53	C₈	Dic₂₁₂	1	1	yes	characteristic
54	C₈	#11719923	1	1	yes	characteristic
55	C₈	#11719925	1	1	yes	characteristic
56	C₂²	#13133133	1	1	yes	characteristic
57	C₄	—	4	1	no	—
58	C₄	—	424	2	no	—
59	C₄	#13133140	1	1	yes	characteristic
60	C₄	#13133142	1	1	yes	characteristic
61	C₂	—	2	1	no	—
62	C₂	#16810371	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#16810371
commutator	C₄²×C₅₃	C₂×C₄
Frattini	C₄×C₈	C₂×D₅₃
Fitting	C₅₃×#364	C₂
socle	C₂×C₅₃	#333
hypercenter	#364	D₅₃
hypocenter	C₅₃	#1366
LCS	G ▷ C₄²×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #364

Complements

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

N	H ≅ G/N	type
35. C₅₃	#1366	semidirect