Subgroups — #20272878

Order 6784.

120 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16810416	C₂	1	2	yes	maximal, complemented
2	C₅₃×#493	C₂	1	1	yes	maximal, characteristic
3	#13133132	—	2	2	no	—
4	C₂×D₈×C₅₃	—	2	2	no	—
5	#127×C₅₃	C₄	1	1	yes	characteristic, complemented
6	#127×C₅₃	C₂²	1	1	yes	characteristic
7	C₄×C₈×C₅₃	C₄	1	1	yes	characteristic, complemented
8	C₂×D₄×C₅₃	—	2	1	no	—
9	C₂×D₄×C₅₃	—	4	1	no	—
10	C₂×D₄×C₅₃	—	4	1	no	—
11	C₂×D₄×C₅₃	D₄	1	2	yes	—
12	D₈×C₅₃	—	4	4	no	—
13	C₂×C₈×C₅₃	—	2	1	no	—
14	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
15	C₂²⋊Dic₅₃	—	4	2	no	—
16	C₂³×C₅₃	—	2	2	no	—
17	C₂³×C₅₃	—	4	1	no	—
18	D₄×C₅₃	—	4	2	no	—
19	D₄×C₅₃	—	4	2	no	—
20	D₄×C₅₃	—	8	2	no	—
21	D₄×C₅₃	—	8	2	no	—
22	C₂×C₄×C₅₃	—	2	1	no	—
23	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
24	C₂×Dic₅₃	—	8	2	no	—
25	C₈×C₅₃	—	2	2	no	—
26	C₂²×C₅₃	—	4	2	no	—
27	C₂²×C₅₃	—	8	2	no	—
28	C₂²×C₅₃	—	8	2	no	—
29	C₂²×C₅₃	—	8	1	no	—
30	C₂²×C₅₃	#99	1	1	yes	characteristic
31	C₄×C₅₃	—	2	2	no	—
32	C₄×C₅₃	—	2	1	no	—
33	Dic₅₃	—	16	2	no	—
34	#1408	—	53	1	no	maximal, Sylow
35	C₂×C₅₃	—	2	1	no	—
36	C₂×C₅₃	—	8	2	no	—
37	C₂×C₅₃	—	16	1	no	—
38	C₂×C₅₃	#353	1	1	yes	characteristic
39	#493	D₅₃	1	1	yes	characteristic
40	#353	—	53	2	no	—
41	C₅₃	#1408	1	1	yes	characteristic, complemented, Sylow
42	C₂×D₈	—	2	2	no	—
43	#127	Dic₅₃	1	1	yes	characteristic, complemented
44	#127	C₂×D₅₃	1	1	yes	characteristic
45	#99	—	106	2	no	—
46	C₄×C₈	Dic₅₃	1	1	yes	characteristic, complemented
47	C₂×D₄	—	2	1	no	—
48	C₂×D₄	—	4	1	no	—
49	C₂×D₄	—	4	1	no	—
50	C₂×D₄	C₂²⋊D₅₃	1	2	yes	—
51	D₈	—	4	4	no	—
52	C₂×C₈	—	2	1	no	—
53	C₂²⋊C₄	—	212	2	no	—
54	C₄²	C₂×Dic₅₃	1	1	yes	characteristic
55	C₂³	—	2	2	no	—
56	C₂³	—	4	1	no	—
57	D₄	—	4	2	no	—
58	D₄	—	4	2	no	—
59	D₄	—	8	2	no	—
60	D₄	—	8	2	no	—
61	C₂×C₄	—	2	1	no	—
62	C₂×C₄	—	424	2	no	—
63	C₂×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
64	C₈	—	2	2	no	—
65	C₂²	—	4	2	no	—
66	C₂²	—	8	2	no	—
67	C₂²	—	8	2	no	—
68	C₂²	—	8	1	no	—
69	C₂²	#13133132	1	1	yes	characteristic
70	C₄	—	2	2	no	—
71	C₄	—	2	1	no	—
72	C₄	—	848	2	no	—
73	C₂	—	2	1	no	—
74	C₂	—	8	2	no	—
75	C₂	—	16	1	no	—
76	C₂	#16810416	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#16810416
commutator	C₄²×C₅₃	C₂×C₄
Frattini	#127	C₂×D₅₃
Fitting	C₅₃×#493	C₂
socle	C₂×C₅₃	#353
hypercenter	#493	D₅₃
hypocenter	C₅₃	#1408
LCS	G ▷ C₄²×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ #127 ◁ #493

Complements

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

N	H ≅ G/N	type
1. #16810416	C₂	semidirect
5. #127×C₅₃	C₄	semidirect
7. C₄×C₈×C₅₃	C₄	semidirect
41. C₅₃	#1408	semidirect
43. #127	Dic₅₃	semidirect
46. C₄×C₈	Dic₅₃	semidirect