Subgroups — #20272752

Order 6784.

112 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16810444	C₂	1	2	yes	maximal, complemented
2	C₅₃×#493	C₂	1	1	yes	maximal, characteristic
3	#13133157	—	2	2	no	—
4	C₂×D₈×C₅₃	—	2	2	no	—
5	#127×C₅₃	C₄	1	2	yes	—
6	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
7	C₂×D₄×C₅₃	—	2	2	no	—
8	C₂×D₄×C₅₃	—	4	2	no	—
9	D₈×C₅₃	—	4	2	no	—
10	D₈×C₅₃	—	4	2	no	—
11	C₂×C₈×C₅₃	D₄	1	2	yes	—
12	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
13	#11719908	—	4	2	no	—
14	C₂³×C₅₃	—	4	2	no	—
15	D₄×C₅₃	—	4	4	no	—
16	D₄×C₅₃	—	8	4	no	—
17	C₂×C₄×C₅₃	—	2	1	no	—
18	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
19	C₈×C₅₃	D₈	1	2	yes	—
20	C₈×C₅₃	SD₁₆	1	2	yes	—
21	C₂²×C₅₃	—	8	4	no	—
22	C₂²×C₅₃	—	8	2	no	—
23	C₂²×C₅₃	#100	1	1	yes	characteristic
24	C₄×C₅₃	—	2	2	no	—
25	C₄×C₅₃	#102	1	2	yes	—
26	#1357	—	53	1	no	maximal, Sylow
27	C₂×C₅₃	—	2	1	no	—
28	C₂×C₅₃	—	16	2	no	—
29	C₂×C₅₃	#331	1	1	yes	characteristic
30	#493	D₅₃	1	1	yes	characteristic
31	#364	—	53	2	no	—
32	C₅₃	#1357	1	1	yes	characteristic, complemented, Sylow
33	C₂×D₈	—	2	2	no	—
34	#127	Dic₅₃	1	2	yes	—
35	M₃₂	—	106	2	no	—
36	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
37	C₂×D₄	—	2	2	no	—
38	C₂×D₄	—	4	2	no	—
39	D₈	—	4	2	no	—
40	D₈	—	4	2	no	—
41	C₂×C₈	C₂²⋊D₅₃	1	2	yes	—
42	C₄²	C₂×Dic₅₃	1	1	yes	characteristic
43	C₁₆	—	212	2	no	—
44	C₂³	—	4	2	no	—
45	D₄	—	4	4	no	—
46	D₄	—	8	4	no	—
47	C₂×C₄	—	2	1	no	—
48	C₂×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
49	C₈	#11719922	1	2	yes	—
50	C₈	#11719923	1	2	yes	—
51	C₂²	—	8	4	no	—
52	C₂²	—	8	2	no	—
53	C₂²	#13133134	1	1	yes	characteristic
54	C₄	—	2	2	no	—
55	C₄	#13133138	1	2	yes	—
56	C₂	—	2	1	no	—
57	C₂	—	16	2	no	—
58	C₂	#16810367	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#16810367
commutator	C₄²×C₅₃	C₂×C₄
Frattini	C₄×C₈	C₂×D₅₃
Fitting	C₅₃×#493	C₂
socle	C₂×C₅₃	#331
hypercenter	#493	D₅₃
hypocenter	C₅₃	#1357
LCS	G ▷ C₄²×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #493

Complements

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

N	H ≅ G/N	type
1. #16810444	C₂	semidirect
32. C₅₃	#1357	semidirect