Subgroups — #20272667

Order 6784.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16810651	C₂	1	1	yes	maximal, characteristic
2	#16810405	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#349	C₂	1	1	yes	maximal, characteristic
4	C₂²×#88322	C₄	1	1	yes	characteristic
5	#13133176	C₄	1	1	yes	characteristic
6	#13133157	—	2	1	no	—
7	#13133146	—	2	1	no	—
8	#13133146	—	2	1	no	—
9	C₂×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
10	M₃₂×C₅₃	—	2	1	no	—
11	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
12	M₁₆×C₅₃	—	4	1	no	—
13	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
14	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
15	C₂²⋊Dic₅₃	C₈	1	2	yes	—
16	C₄⋊Dic₅₃	—	2	1	no	—
17	C₄×Dic₅₃	—	2	1	no	—
18	C₂×#88322	—	2	1	no	—
19	C₂×#88322	—	4	1	no	—
20	C₁₆×C₅₃	—	4	1	no	—
21	#11719908	—	4	1	no	—
22	C₂³×C₅₃	C₂×C₈	1	1	yes	characteristic
23	C₂×C₄×C₅₃	—	2	1	no	—
24	C₂×C₄×C₅₃	—	2	1	no	—
25	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
26	C₂×C₄×C₅₃	M₁₆	1	1	yes	characteristic
27	C₂×Dic₅₃	—	2	2	no	—
28	C₈×C₅₃	—	2	1	no	—
29	C₈×C₅₃	—	2	1	no	—
30	#88322	—	4	1	no	—
31	C₂²×C₅₃	—	2	1	no	—
32	C₂²×C₅₃	—	4	1	no	—
33	C₂²×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
34	C₄×C₅₃	—	2	1	no	—
35	C₄×C₅₃	#99	1	1	yes	characteristic
36	C₄×C₅₃	#101	1	1	yes	characteristic
37	Dic₅₃	—	4	2	no	—
38	#1328	—	53	1	no	maximal, Sylow
39	C₂×C₅₃	—	2	1	no	—
40	C₂×C₅₃	—	4	1	no	—
41	C₂×C₅₃	#324	1	1	yes	characteristic
42	#424	—	53	1	no	—
43	#349	—	53	1	no	—
44	#349	D₅₃	1	1	yes	characteristic
45	C₅₃	#1328	1	1	yes	characteristic, complemented, Sylow
46	C₂×M₁₆	C₂×D₅₃	1	1	yes	characteristic
47	C₂²×C₈	—	53	1	no	—
48	#117	—	53	1	no	—
49	M₃₂	—	2	1	no	—
50	M₃₂	—	106	1	no	—
51	C₄⋊C₈	—	106	1	no	—
52	C₄⋊C₈	—	106	1	no	—
53	C₂²×C₄	C₄×D₅₃	1	1	yes	characteristic
54	M₁₆	—	4	1	no	—
55	C₂×C₈	—	106	1	no	—
56	C₂×C₈	—	212	1	no	—
57	C₂×C₈	D₂₁₂	1	1	yes	characteristic
58	C₂×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
59	C₄⋊C₄	—	106	1	no	—
60	C₂²⋊C₄	—	53	2	no	—
61	C₄²	—	106	1	no	—
62	C₁₆	—	4	1	no	—
63	C₁₆	—	212	1	no	—
64	C₂³	C₈×D₅₃	1	1	yes	characteristic
65	C₂×C₄	—	2	1	no	—
66	C₂×C₄	—	2	1	no	—
67	C₂×C₄	—	106	2	no	—
68	C₂×C₄	#11719912	1	1	yes	characteristic
69	C₂×C₄	#11719921	1	1	yes	characteristic
70	C₈	—	2	1	no	—
71	C₈	—	2	1	no	—
72	C₈	—	212	1	no	—
73	C₂²	—	2	1	no	—
74	C₂²	—	4	1	no	—
75	C₂²	#13133129	1	1	yes	characteristic
76	C₄	—	2	1	no	—
77	C₄	—	212	2	no	—
78	C₄	#13133131	1	1	yes	characteristic
79	C₄	#13133135	1	1	yes	characteristic
80	C₂	—	2	1	no	—
81	C₂	—	4	1	no	—
82	C₂	#16810348	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#13133131
commutator	C₂³×C₅₃	C₂×C₈
Frattini	C₂×M₁₆	C₂×D₅₃
Fitting	C₅₃×#349	C₂
socle	C₂×C₅₃	#324
hypercenter	#349	D₅₃
hypocenter	C₅₃	#1328
LCS	G ▷ C₂³×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×M₁₆ ◁ #349

Complements

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

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