Subgroups — #20289821

Order 6784.

68 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₅₃⋊C₃₂	C₂	1	1	yes	maximal, characteristic
2	C₁₆×#6157	C₂	1	1	yes	maximal, characteristic
3	C₃₂×D₅₃	C₂	1	1	yes	maximal, characteristic
4	D₅₃⋊C₁₆	C₄	1	1	yes	characteristic
5	C₈×#6157	C₄	1	1	yes	characteristic
6	#13133267	C₄	1	2	yes	complemented
7	C₁₆×D₅₃	C₂²	1	1	yes	characteristic
8	#13133123	C₄	1	1	yes	characteristic, complemented
9	C₃₂×C₅₃	C₄	1	1	yes	characteristic, complemented
10	C₄×#6157	C₈	1	1	yes	characteristic
11	D₅₃⋊C₈	C₈	1	1	yes	characteristic
12	C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
13	#11719910	C₈	1	2	yes	—
14	C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
15	#11719908	C₂×C₄	1	1	yes	characteristic
16	C₂×#6157	C₁₆	1	2	yes	—
17	C₄×D₅₃	C₂×C₈	1	1	yes	characteristic
18	#88324	C₁₆	1	2	yes	—
19	C₈×C₅₃	C₄²	1	1	yes	characteristic
20	#88322	C₂×C₈	1	1	yes	characteristic
21	C₂×D₅₃	C₂×C₁₆	1	1	yes	characteristic
22	#6157	—	2	2	no	—
23	C₄×C₅₃	C₄×C₈	1	1	yes	characteristic
24	Dic₅₃	C₂×C₁₆	1	1	yes	characteristic
25	#1397	—	53	1	no	maximal, Sylow
26	C₂×C₅₃	C₄×C₁₆	1	1	yes	characteristic
27	D₅₃	M₆₄	1	2	yes	—
28	C₂×C₃₂	—	53	1	no	—
29	C₂×C₃₂	—	53	1	no	—
30	C₄×C₁₆	—	53	1	no	—
31	C₅₃	#1397	1	1	yes	characteristic, complemented, Sylow
32	C₂×C₁₆	—	53	1	no	—
33	C₂×C₁₆	—	53	1	no	—
34	C₄×C₈	—	53	1	no	—
35	C₃₂	—	53	2	no	—
36	C₃₂	—	53	1	no	—
37	C₃₂	#6157	1	1	yes	characteristic, complemented
38	C₂×C₈	—	53	1	no	—
39	C₂×C₈	—	53	1	no	—
40	C₄²	—	53	1	no	—
41	C₁₆	—	53	2	no	—
42	C₁₆	—	53	1	no	—
43	C₁₆	C₂×#6157	1	1	yes	characteristic
44	C₂×C₄	—	53	1	no	—
45	C₂×C₄	—	53	2	no	—
46	C₈	—	53	2	no	—
47	C₈	—	53	1	no	—
48	C₈	C₄×#6157	1	1	yes	characteristic
49	C₂²	—	53	1	no	—
50	C₄	—	53	1	no	—
51	C₄	—	106	2	no	—
52	C₄	C₈×#6157	1	1	yes	characteristic
53	C₂	—	53	2	no	—
54	C₂	C₁₆×#6157	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₁₆	C₂×#6157
commutator, socle	C₂×C₅₃	C₄×C₁₆
Fitting	C₃₂×C₅₃	C₄
hypercenter	C₃₂	#6157
hypocenter	C₅₃	#1397
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₁₆ ◁ C₃₂

Complements

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

N	H ≅ G/N	type
6. #13133267	C₄	semidirect
8. #13133123	C₄	semidirect
9. C₃₂×C₅₃	C₄	semidirect
31. C₅₃	#1397	semidirect
37. C₃₂	#6157	semidirect