Subgroups — #20272912

Order 6784.

58 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16810457	C₂	1	1	yes	maximal, characteristic
2	C₁₆×Dic₅₃	C₂	1	1	yes	maximal, characteristic
3	C₅₃×M₆₄	C₂	1	1	yes	maximal, characteristic
4	C₂×#11719908	C₄	1	1	yes	characteristic
5	C₈×Dic₅₃	C₄	1	1	yes	characteristic
6	#13133123	—	2	1	no	—
7	C₂×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
8	C₃₂×C₅₃	—	2	1	no	—
9	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
10	C₄×Dic₅₃	C₈	1	1	yes	characteristic
11	C₂×#88322	C₈	1	1	yes	characteristic
12	C₁₆×C₅₃	D₄	1	1	yes	characteristic
13	C₁₆×C₅₃	Q₈	1	1	yes	characteristic
14	#11719908	—	2	1	no	—
15	C₂×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
16	C₂×Dic₅₃	—	2	1	no	—
17	C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
18	C₈×C₅₃	M₁₆	1	1	yes	characteristic
19	#88322	C₁₆	1	2	yes	—
20	C₂²×C₅₃	M₃₂	1	1	yes	characteristic
21	C₄×C₅₃	C₄⋊C₈	1	1	yes	characteristic
22	C₄×C₅₃	C₂×C₁₆	1	1	yes	characteristic
23	Dic₅₃	—	2	2	no	—
24	#1422	—	53	1	no	maximal, Sylow
25	C₂×C₅₃	—	2	1	no	—
26	C₂×C₅₃	C₄⋊C₁₆	1	1	yes	characteristic
27	M₆₄	—	53	1	no	—
28	M₆₄	D₅₃	1	1	yes	characteristic
29	C₄×C₁₆	—	53	1	no	—
30	C₅₃	#1422	1	1	yes	characteristic, complemented, Sylow
31	C₂×C₁₆	—	53	1	no	—
32	C₂×C₁₆	C₂×D₅₃	1	1	yes	characteristic
33	C₄×C₈	—	53	1	no	—
34	C₃₂	—	2	1	no	—
35	C₃₂	—	106	1	no	—
36	C₂×C₈	—	53	1	no	—
37	C₂×C₈	C₄×D₅₃	1	1	yes	characteristic
38	C₄²	—	53	1	no	—
39	C₁₆	—	106	1	no	—
40	C₁₆	Dic₁₀₆	1	1	yes	characteristic
41	C₁₆	C₂²⋊D₅₃	1	1	yes	characteristic
42	C₂×C₄	—	106	1	no	—
43	C₂×C₄	C₈×D₅₃	1	1	yes	characteristic
44	C₈	—	53	2	no	—
45	C₈	#11719912	1	1	yes	characteristic
46	C₈	#11719919	1	1	yes	characteristic
47	C₂²	#13133156	1	1	yes	characteristic
48	C₄	—	106	2	no	—
49	C₄	#13133146	1	1	yes	characteristic
50	C₄	C₁₆×D₅₃	1	1	yes	characteristic
51	C₂	—	2	1	no	—
52	C₂	#16810441	1	1	yes	characteristic

	subgroup	G/H
center	C₁₆	C₂²⋊D₅₃
commutator	C₄×C₅₃	C₂×C₁₆
Frattini	C₂×C₁₆	C₂×D₅₃
Fitting	C₅₃×M₆₄	C₂
socle	C₂×C₅₃	C₄⋊C₁₆
hypercenter	M₆₄	D₅₃
hypocenter	C₅₃	#1422
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₁₆ ◁ C₂×C₁₆ ◁ M₆₄

Complements

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

N	H ≅ G/N	type
30. C₅₃	#1422	semidirect