Subgroups — #20272910

Order 6784.

62 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#13133123	C₂	1	1	yes	maximal, characteristic
2	C₁₆×Dic₅₃	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃₂×C₅₃	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	C₁₆	1	1	yes	characteristic
17	C₂×Dic₅₃	C₁₆	1	1	yes	characteristic
18	C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
19	C₈×C₅₃	M₁₆	1	1	yes	characteristic
20	#88322	—	2	1	no	—
21	C₂²×C₅₃	C₂×C₁₆	1	1	yes	characteristic
22	C₄×C₅₃	C₄⋊C₈	1	1	yes	characteristic
23	C₄×C₅₃	M₃₂	1	1	yes	characteristic
24	Dic₅₃	—	2	1	no	—
25	Dic₅₃	C₃₂	1	2	yes	complemented
26	C₄⋊C₃₂	—	53	1	no	maximal, Sylow
27	C₂×C₅₃	C₄⋊C₁₆	1	1	yes	characteristic
28	C₂×C₅₃	C₂×C₃₂	1	1	yes	characteristic
29	C₂×C₅₃	M₆₄	1	1	yes	characteristic
30	C₂×C₃₂	—	53	1	no	—
31	C₂×C₃₂	D₅₃	1	1	yes	characteristic
32	C₄×C₁₆	—	53	1	no	—
33	C₅₃	C₄⋊C₃₂	1	1	yes	characteristic, complemented, Sylow
34	C₂×C₁₆	—	53	1	no	—
35	C₂×C₁₆	C₂×D₅₃	1	1	yes	characteristic
36	C₄×C₈	—	53	1	no	—
37	C₃₂	—	2	1	no	—
38	C₃₂	—	106	1	no	—
39	C₂×C₈	—	53	1	no	—
40	C₂×C₈	C₄×D₅₃	1	1	yes	characteristic
41	C₄²	—	53	1	no	—
42	C₁₆	—	106	1	no	—
43	C₁₆	Dic₁₀₆	1	1	yes	characteristic
44	C₁₆	C₂²⋊D₅₃	1	1	yes	characteristic
45	C₂×C₄	—	53	1	no	—
46	C₂×C₄	—	53	1	no	—
47	C₂×C₄	C₈×D₅₃	1	1	yes	characteristic
48	C₈	—	106	1	no	—
49	C₈	#11719912	1	1	yes	characteristic
50	C₈	#11719919	1	1	yes	characteristic
51	C₂²	C₁₆×D₅₃	1	1	yes	characteristic
52	C₄	—	53	2	no	—
53	C₄	—	106	1	no	—
54	C₄	#13133146	1	1	yes	characteristic
55	C₄	#13133156	1	1	yes	characteristic
56	C₂	#16810441	1	1	yes	characteristic
57	C₂	C₃₂×D₅₃	1	1	yes	characteristic
58	C₂	#16810456	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₁₆	C₂×D₅₃
commutator	C₂×C₅₃	C₂×C₃₂
Fitting	C₂×C₃₂×C₅₃	C₂
socle	C₂²×C₅₃	C₂×C₁₆
hypercenter	C₂×C₃₂	D₅₃
hypocenter	C₅₃	C₄⋊C₃₂
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂×C₃₂

Complements

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

N	H ≅ G/N	type
25. Dic₅₃	C₃₂	semidirect
33. C₅₃	C₄⋊C₃₂	semidirect