Subgroups — #20274471

Order 6784.

200 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16810581	C₂	1	3	yes	maximal
2	#16810380	C₂	1	3	yes	maximal
3	C₅₃×#338	C₂	1	1	yes	maximal, characteristic
4	C₂²×#88322	C₂²	1	1	yes	characteristic
5	#13133176	C₂²	1	3	yes	—
6	#13133127	C₄	1	6	yes	complemented
7	C₈×Dic₅₃	C₄	1	6	yes	complemented
8	C₂×M₁₆×C₅₃	C₂²	1	3	yes	—
9	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
10	M₁₆×C₅₃	—	2	6	no	—
11	C₂×C₈×C₅₃	C₂×C₄	1	6	yes	—
12	C₂²⋊Dic₅₃	—	2	3	no	—
13	C₄⋊Dic₅₃	—	2	3	no	—
14	C₄×Dic₅₃	C₂×C₄	1	6	yes	—
15	C₂×#88322	C₂×C₄	1	3	yes	—
16	C₂×#88322	C₂×C₄	1	3	yes	—
17	C₂³×C₅₃	#41	1	1	yes	characteristic
18	C₂×C₄×C₅₃	C₂²×C₄	1	3	yes	—
19	C₂×C₄×C₅₃	#41	1	3	yes	—
20	C₂×Dic₅₃	—	2	6	no	—
21	C₈×C₅₃	—	2	6	no	—
22	#88322	C₄²	1	1	yes	characteristic
23	#88322	C₄²	1	3	yes	—
24	C₂²×C₅₃	—	4	1	no	—
25	C₂²×C₅₃	#117	1	3	yes	—
26	C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
27	C₄×C₅₃	#117	1	3	yes	—
28	Dic₅₃	—	4	6	no	—
29	#1747	—	53	1	no	maximal, Sylow
30	C₂×C₅₃	—	2	3	no	—
31	C₂×C₅₃	#376	1	1	yes	characteristic
32	#405	—	53	3	no	—
33	#338	D₅₃	1	1	yes	characteristic
34	H(3,ℤ/4)	—	53	3	no	—
35	C₅₃	#1747	1	1	yes	characteristic, complemented, Sylow
36	C₂×M₁₆	C₂×D₅₃	1	3	yes	—
37	C₂²×C₈	—	53	1	no	—
38	#117	—	53	3	no	—
39	#97	—	53	6	no	—
40	C₄×C₈	—	53	6	no	—
41	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
42	M₁₆	—	2	6	no	—
43	C₂×C₈	—	53	3	no	—
44	C₂×C₈	—	53	3	no	—
45	C₂×C₈	C₄×D₅₃	1	6	yes	—
46	C₄⋊C₄	—	106	3	no	—
47	C₂²⋊C₄	—	106	3	no	—
48	C₄²	—	53	6	no	—
49	C₂³	#11719949	1	1	yes	characteristic
50	C₂×C₄	—	106	6	no	—
51	C₂×C₄	C₂×C₄×D₅₃	1	3	yes	—
52	C₂×C₄	#11719947	1	3	yes	—
53	C₈	—	2	6	no	—
54	C₈	—	53	1	no	—
55	C₈	—	53	3	no	—
56	C₂²	—	4	1	no	—
57	C₂²	#13133175	1	3	yes	—
58	C₄	—	212	6	no	—
59	C₄	C₄²×D₅₃	1	1	yes	characteristic
60	C₄	#13133173	1	3	yes	—
61	C₂	—	2	3	no	—
62	C₂	#16810468	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₄²×D₅₃
commutator	C₄×C₅₃	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₅₃×#338	C₂
socle	C₂×C₅₃	#376
hypercenter	#338	D₅₃
hypocenter	C₅₃	#1747
LCS	G ▷ C₄×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ #338

Complements

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

N	H ≅ G/N	type
6. #13133127	C₄	semidirect
7. C₈×Dic₅₃	C₄	semidirect
35. C₅₃	#1747	semidirect