Subgroups — #20272668

Order 6784.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16810405	C₂	1	2	yes	maximal
2	C₅₃×#424	C₂	1	1	yes	maximal, characteristic
3	#13133157	—	2	2	no	—
4	C₂×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
5	C₂²×C₈×C₅₃	C₄	1	1	yes	characteristic
6	#117×C₅₃	C₄	1	1	yes	characteristic
7	C₄⋊C₈×C₅₃	—	2	2	no	—
8	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
9	M₁₆×C₅₃	—	4	1	no	—
10	C₂×C₈×C₅₃	—	2	1	no	—
11	C₂×C₈×C₅₃	—	4	1	no	—
12	C₂×C₈×C₅₃	D₄	1	2	yes	—
13	C₄⋊C₄×C₅₃	—	2	1	no	—
14	C₂²⋊C₄×C₅₃	C₈	1	2	yes	—
15	C₄²×C₅₃	—	2	1	no	—
16	#11719908	—	4	2	no	—
17	C₂³×C₅₃	C₂×C₈	1	1	yes	characteristic
18	C₂×C₄×C₅₃	—	2	1	no	—
19	C₂×C₄×C₅₃	—	2	1	no	—
20	C₂×C₄×C₅₃	—	2	2	no	—
21	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
22	C₂×C₄×C₅₃	M₁₆	1	1	yes	characteristic
23	C₈×C₅₃	—	2	2	no	—
24	C₈×C₅₃	—	4	1	no	—
25	C₂²×C₅₃	—	2	1	no	—
26	C₂²×C₅₃	—	4	1	no	—
27	C₂²×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
28	C₄×C₅₃	—	2	1	no	—
29	C₄×C₅₃	—	4	2	no	—
30	C₄×C₅₃	#99	1	1	yes	characteristic
31	C₄×C₅₃	#101	1	1	yes	characteristic
32	#1328	—	53	1	no	maximal, Sylow
33	C₂×C₅₃	—	2	1	no	—
34	C₂×C₅₃	—	4	1	no	—
35	C₂×C₅₃	#324	1	1	yes	characteristic
36	#424	D₅₃	1	1	yes	characteristic
37	#349	—	53	2	no	—
38	C₅₃	#1328	1	1	yes	characteristic, complemented, Sylow
39	C₂×M₁₆	C₂×D₅₃	1	1	yes	characteristic
40	C₂²×C₈	Dic₅₃	1	1	yes	characteristic
41	#117	Dic₅₃	1	1	yes	characteristic
42	M₃₂	—	106	2	no	—
43	C₄⋊C₈	—	2	2	no	—
44	C₂²×C₄	C₂×Dic₅₃	1	1	yes	characteristic
45	M₁₆	—	4	1	no	—
46	C₂×C₈	—	2	1	no	—
47	C₂×C₈	—	4	1	no	—
48	C₂×C₈	C₂²⋊D₅₃	1	2	yes	—
49	C₄⋊C₄	—	2	1	no	—
50	C₂²⋊C₄	#88322	1	2	yes	—
51	C₄²	—	2	1	no	—
52	C₁₆	—	212	2	no	—
53	C₂³	C₂×#88322	1	1	yes	characteristic
54	C₂×C₄	—	2	1	no	—
55	C₂×C₄	—	2	1	no	—
56	C₂×C₄	—	2	2	no	—
57	C₂×C₄	#11719917	1	1	yes	characteristic
58	C₂×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
59	C₈	—	2	2	no	—
60	C₈	—	4	1	no	—
61	C₂²	—	2	1	no	—
62	C₂²	—	4	1	no	—
63	C₂²	C₂²⋊#88322	1	1	yes	characteristic
64	C₄	—	2	1	no	—
65	C₄	—	4	2	no	—
66	C₄	#13133132	1	1	yes	characteristic
67	C₄	#13133136	1	1	yes	characteristic
68	C₂	—	2	1	no	—
69	C₂	—	4	1	no	—
70	C₂	#16810349	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#13133132
commutator	C₂³×C₅₃	C₂×C₈
Frattini	C₂×M₁₆	C₂×D₅₃
Fitting	C₅₃×#424	C₂
socle	C₂×C₅₃	#324
hypercenter	#424	D₅₃
hypocenter	C₅₃	#1328
LCS	G ▷ C₂³×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×M₁₆ ◁ #424

Complements

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

N	H ≅ G/N	type
38. C₅₃	#1328	semidirect