Subgroups — #20272849

Order 6784.

68 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#13133123	C₂	1	2	yes	maximal
2	C₄×C₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
3	#13133123	C₄	1	4	yes	complemented
4	C₂×C₁₆×C₅₃	C₄	1	1	yes	characteristic
5	C₂×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
6	C₄×C₈×C₅₃	C₄	1	1	yes	characteristic
7	C₂×C₈×C₅₃	C₈	1	1	yes	characteristic
8	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
9	C₄²×C₅₃	C₈	1	1	yes	characteristic
10	C₁₆×C₅₃	C₈	1	2	yes	—
11	C₁₆×C₅₃	C₂×C₄	1	2	yes	—
12	C₂×C₄×C₅₃	C₁₆	1	2	yes	—
13	C₂×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
14	C₈×C₅₃	C₁₆	1	2	yes	—
15	C₈×C₅₃	C₄²	1	1	yes	characteristic
16	C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
17	C₂²×C₅₃	C₂×C₁₆	1	1	yes	characteristic
18	C₄×C₅₃	—	2	2	no	—
19	C₄×C₅₃	C₄×C₈	1	1	yes	characteristic
20	C₄×C₅₃	C₂×C₁₆	1	1	yes	characteristic
21	#1397	—	53	1	no	maximal, Sylow
22	C₂×C₅₃	C₄×C₁₆	1	1	yes	characteristic
23	C₂×C₅₃	M₆₄	1	2	yes	—
24	C₂×C₃₂	—	53	2	no	—
25	C₄×C₁₆	D₅₃	1	1	yes	characteristic
26	C₅₃	#1397	1	1	yes	characteristic, complemented, Sylow
27	C₂×C₁₆	Dic₅₃	1	1	yes	characteristic
28	C₂×C₁₆	C₂×D₅₃	1	1	yes	characteristic
29	C₄×C₈	Dic₅₃	1	1	yes	characteristic
30	C₃₂	—	53	4	no	—
31	C₂×C₈	#88322	1	1	yes	characteristic
32	C₂×C₈	C₂×Dic₅₃	1	1	yes	characteristic
33	C₄²	#88322	1	1	yes	characteristic
34	C₁₆	#88322	1	2	yes	—
35	C₁₆	C₄×D₅₃	1	2	yes	—
36	C₂×C₄	#11719908	1	2	yes	—
37	C₂×C₄	C₂×#88322	1	1	yes	characteristic
38	C₈	#11719908	1	2	yes	—
39	C₈	C₂×#88322	1	1	yes	characteristic
40	C₈	C₄×Dic₅₃	1	1	yes	characteristic
41	C₂²	C₂×#11719908	1	1	yes	characteristic
42	C₄	—	2	2	no	—
43	C₄	C₄×#88322	1	1	yes	characteristic
44	C₄	C₂×#11719908	1	1	yes	characteristic
45	C₂	C₄×#11719908	1	1	yes	characteristic
46	C₂	#16810457	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₁₆	C₂×D₅₃
commutator	C₂×C₅₃	C₄×C₁₆
Fitting	C₄×C₁₆×C₅₃	C₂
socle	C₂²×C₅₃	C₂×C₁₆
hypercenter	C₄×C₁₆	D₅₃
hypocenter	C₅₃	#1397
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₁₆ ◁ C₄×C₁₆

Complements

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

N	H ≅ G/N	type
3. #13133123	C₄	semidirect
26. C₅₃	#1397	semidirect