Subgroups — #20272886

Order 6784.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16810416	C₂	1	2	yes	maximal
2	C₅₃×#490	C₂	1	1	yes	maximal, characteristic
3	#13133132	—	2	2	no	—
4	#127×C₅₃	C₂²	1	1	yes	characteristic
5	#125×C₅₃	C₄	1	1	yes	characteristic, complemented
6	#102×C₅₃	—	2	2	no	—
7	#97×C₅₃	C₄	1	1	yes	characteristic, complemented
8	C₂×D₄×C₅₃	—	4	1	no	—
9	C₂×D₄×C₅₃	D₄	1	2	yes	—
10	C₂×C₈×C₅₃	—	2	1	no	—
11	C₄⋊C₄×C₅₃	—	2	1	no	—
12	C₄⋊C₄×C₅₃	—	4	1	no	—
13	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
14	C₂²⋊Dic₅₃	—	4	2	no	—
15	C₂³×C₅₃	—	2	2	no	—
16	D₄×C₅₃	—	4	2	no	—
17	D₄×C₅₃	—	8	2	no	—
18	C₂×C₄×C₅₃	—	2	1	no	—
19	C₂×C₄×C₅₃	—	4	1	no	—
20	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
21	C₂×Dic₅₃	—	8	2	no	—
22	C₈×C₅₃	—	4	1	no	—
23	C₂²×C₅₃	—	4	2	no	—
24	C₂²×C₅₃	—	8	2	no	—
25	C₂²×C₅₃	#99	1	1	yes	characteristic
26	C₄×C₅₃	—	2	1	no	—
27	C₄×C₅₃	—	4	1	no	—
28	C₄×C₅₃	—	8	1	no	—
29	Dic₅₃	—	16	2	no	—
30	#1412	—	53	1	no	maximal, Sylow
31	C₂×C₅₃	—	2	1	no	—
32	C₂×C₅₃	—	8	2	no	—
33	C₂×C₅₃	#354	1	1	yes	characteristic
34	#490	D₅₃	1	1	yes	characteristic
35	#353	—	53	2	no	—
36	C₅₃	#1412	1	1	yes	characteristic, complemented, Sylow
37	#127	C₂×D₅₃	1	1	yes	characteristic
38	#125	Dic₅₃	1	1	yes	characteristic, complemented
39	#102	—	2	2	no	—
40	#99	—	106	2	no	—
41	#97	Dic₅₃	1	1	yes	characteristic, complemented
42	C₂×D₄	—	4	1	no	—
43	C₂×D₄	C₂²⋊D₅₃	1	2	yes	—
44	C₂×C₈	—	2	1	no	—
45	C₄⋊C₄	—	2	1	no	—
46	C₄⋊C₄	—	4	1	no	—
47	C₂²⋊C₄	—	212	2	no	—
48	C₄²	C₂×Dic₅₃	1	1	yes	characteristic
49	C₂³	—	2	2	no	—
50	D₄	—	4	2	no	—
51	D₄	—	8	2	no	—
52	C₂×C₄	—	2	1	no	—
53	C₂×C₄	—	4	1	no	—
54	C₂×C₄	—	424	2	no	—
55	C₂×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
56	C₈	—	4	1	no	—
57	C₂²	—	4	2	no	—
58	C₂²	—	8	2	no	—
59	C₂²	#13133132	1	1	yes	characteristic
60	C₄	—	2	1	no	—
61	C₄	—	4	1	no	—
62	C₄	—	8	1	no	—
63	C₄	—	848	2	no	—
64	C₂	—	2	1	no	—
65	C₂	—	8	2	no	—
66	C₂	#16810418	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#16810418
commutator	C₄²×C₅₃	C₂×C₄
Frattini	#127	C₂×D₅₃
Fitting	C₅₃×#490	C₂
socle	C₂×C₅₃	#354
hypercenter	#490	D₅₃
hypocenter	C₅₃	#1412
LCS	G ▷ C₄²×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ #127 ◁ #490

Complements

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

N	H ≅ G/N	type
5. #125×C₅₃	C₄	semidirect
7. #97×C₅₃	C₄	semidirect
36. C₅₃	#1412	semidirect
38. #125	Dic₅₃	semidirect
41. #97	Dic₅₃	semidirect