Subgroups — #20272807

Order 6784.

152 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#13133151	C₂	1	2	yes	maximal
2	C₂²×C₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄⋊Dic₅₃	—	2	2	no	—
4	#13133151	—	2	2	no	—
5	#13133151	C₄	1	4	yes	complemented
6	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
7	C₂×C₁₆×C₅₃	—	2	2	no	—
8	C₂×C₁₆×C₅₃	C₄	1	2	yes	complemented
9	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
10	C₂²×Dic₅₃	—	4	2	no	—
11	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
12	C₂×C₈×C₅₃	C₂×C₄	1	2	yes	—
13	C₂×C₈×C₅₃	D₄	1	2	yes	—
14	C₂×C₈×C₅₃	Q₈	1	1	yes	characteristic
15	C₄⋊Dic₅₃	—	2	4	no	—
16	C₄⋊Dic₅₃	—	4	2	no	—
17	C₁₆×C₅₃	—	2	2	no	—
18	C₂³×C₅₃	D₈	1	1	yes	characteristic
19	C₂×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
20	C₂×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
21	C₂×C₄×C₅₃	SD₁₆	1	2	yes	—
22	C₂×C₄×C₅₃	Q₁₆	1	1	yes	characteristic
23	C₂×Dic₅₃	—	4	4	no	—
24	C₂×Dic₅₃	—	8	2	no	—
25	C₂×Dic₅₃	—	8	2	no	—
26	C₈×C₅₃	C₄²	1	1	yes	characteristic
27	C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
28	C₈×C₅₃	C₄⋊C₄	1	2	yes	—
29	C₂²×C₅₃	#102	1	2	yes	—
30	C₂²×C₅₃	#107	1	1	yes	characteristic
31	C₂²×C₅₃	D₁₆	1	1	yes	characteristic
32	C₂²×C₅₃	SD₃₂	1	2	yes	—
33	C₂²×C₅₃	Q₃₂	1	1	yes	characteristic
34	C₄×C₅₃	#95	1	1	yes	characteristic
35	C₄×C₅₃	#103	1	2	yes	—
36	C₄×C₅₃	#106	1	1	yes	characteristic
37	Dic₅₃	—	8	4	no	—
38	#1380	—	53	1	no	maximal, Sylow
39	C₂×C₅₃	#340	1	1	yes	characteristic
40	C₂×C₅₃	#357	1	2	yes	—
41	C₂×C₅₃	#358	1	2	yes	—
42	C₂×C₅₃	#366	1	1	yes	characteristic
43	C₂×C₅₃	#367	1	1	yes	characteristic
44	C₂²×C₁₆	D₅₃	1	1	yes	characteristic
45	C₂×#107	—	53	2	no	—
46	C₅₃	#1380	1	1	yes	characteristic, complemented, Sylow
47	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
48	C₂×C₄⋊C₄	—	106	2	no	—
49	C₂×C₁₆	—	2	2	no	—
50	C₂×C₁₆	Dic₅₃	1	2	yes	complemented
51	#107	—	53	4	no	—
52	#107	—	106	2	no	—
53	C₂²×C₄	—	212	2	no	—
54	C₂²×C₄	D₂₁₂	1	1	yes	characteristic
55	C₂×C₈	Dic₁₀₆	1	1	yes	characteristic
56	C₂×C₈	C₄×D₅₃	1	2	yes	—
57	C₂×C₈	C₂×Dic₅₃	1	1	yes	characteristic
58	C₂×C₈	C₂²⋊D₅₃	1	2	yes	—
59	C₄⋊C₄	—	106	4	no	—
60	C₄⋊C₄	—	212	2	no	—
61	C₁₆	—	2	2	no	—
62	C₂³	D₄₂₄	1	1	yes	characteristic
63	C₂×C₄	—	424	2	no	—
64	C₂×C₄	—	424	2	no	—
65	C₂×C₄	—	212	4	no	—
66	C₂×C₄	#11719913	1	2	yes	—
67	C₂×C₄	Dic₂₁₂	1	1	yes	characteristic
68	C₂×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
69	C₂×C₄	#11719921	1	2	yes	—
70	C₈	C₄×Dic₅₃	1	1	yes	characteristic
71	C₈	#11719919	1	2	yes	—
72	C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
73	C₂²	#13133139	1	2	yes	—
74	C₂²	#13133151	1	1	yes	characteristic
75	C₂²	D₈₄₈	1	1	yes	characteristic
76	C₂²	#13133162	1	2	yes	—
77	C₂²	Dic₄₂₄	1	1	yes	characteristic
78	C₄	—	424	4	no	—
79	C₄	#13133124	1	1	yes	characteristic
80	C₄	#13133142	1	2	yes	—
81	C₄	#13133149	1	1	yes	characteristic
82	C₂	#16810386	1	1	yes	characteristic
83	C₂	#16810425	1	2	yes	—
84	C₂	#16810428	1	2	yes	—
85	C₂	#16810449	1	1	yes	characteristic
86	C₂	#16810451	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₄₂₄
commutator	C₈×C₅₃	C₄²
Frattini	C₂²×C₈	C₂×D₅₃
Fitting	C₂²×C₁₆×C₅₃	C₂
socle	C₂³×C₅₃	D₈
hypercenter	C₂²×C₁₆	D₅₃
hypocenter	C₅₃	#1380
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆

Complements

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

N	H ≅ G/N	type
5. #13133151	C₄	semidirect
8. C₂×C₁₆×C₅₃	C₄	semidirect
46. C₅₃	#1380	semidirect
50. C₂×C₁₆	Dic₅₃	semidirect