Subgroups — #20272766

Order 6784.

80 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16811025	C₂	1	1	yes	maximal, characteristic
2	#16810374	C₂	1	1	yes	maximal, characteristic
3	C₅₃×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
4	#13133209	—	2	1	no	—
5	#13133151	C₄	1	1	yes	characteristic
6	#13133151	C₄	1	1	yes	characteristic
7	#13133149	—	2	1	no	—
8	C₄⋊#88322	—	2	1	no	—
9	C₂×C₁₆×C₅₃	—	2	1	no	—
10	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
11	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
12	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
13	C₄²×C₅₃	D₄	1	1	yes	characteristic
14	C₄⋊Dic₅₃	—	2	1	no	—
15	C₄⋊Dic₅₃	—	2	1	no	—
16	#11719919	—	4	1	no	—
17	#11719919	—	4	1	no	—
18	C₂×#88322	—	4	1	no	—
19	C₁₆×C₅₃	—	4	1	no	—
20	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
21	C₂×C₄×C₅₃	D₈	1	1	yes	characteristic
22	C₂×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
23	C₂×Dic₅₃	—	4	1	no	—
24	C₂×Dic₅₃	—	4	1	no	—
25	C₈×C₅₃	—	2	1	no	—
26	C₈×C₅₃	SD₁₆	1	1	yes	characteristic
27	C₈×C₅₃	Q₁₆	1	1	yes	characteristic
28	#88322	—	8	1	no	—
29	C₂²×C₅₃	#102	1	1	yes	characteristic
30	C₄×C₅₃	—	2	1	no	—
31	C₄×C₅₃	—	2	1	no	—
32	C₄×C₅₃	#101	1	1	yes	characteristic
33	C₄×C₅₃	#103	1	1	yes	characteristic
34	Dic₅₃	—	8	1	no	—
35	Dic₅₃	—	8	1	no	—
36	#1363	—	53	1	no	maximal, Sylow
37	C₂×C₅₃	#332	1	1	yes	characteristic
38	C₂×C₅₃	#359	1	1	yes	characteristic
39	C₂×C₅₃	#360	1	1	yes	characteristic
40	#499	—	53	1	no	—
41	C₄⋊C₁₆	D₅₃	1	1	yes	characteristic
42	#334	—	53	1	no	—
43	C₅₃	#1363	1	1	yes	characteristic, complemented, Sylow
44	#125	—	106	1	no	—
45	C₂×C₁₆	—	2	1	no	—
46	#107	—	53	1	no	—
47	#107	—	53	1	no	—
48	#106	—	106	1	no	—
49	C₄⋊C₈	—	106	1	no	—
50	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
51	C₂×C₈	—	212	1	no	—
52	C₂×C₈	C₄×D₅₃	1	1	yes	characteristic
53	C₂×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
54	C₄⋊C₄	—	106	1	no	—
55	C₄⋊C₄	—	106	1	no	—
56	C₄⋊C₄	—	212	1	no	—
57	C₄⋊C₄	—	212	1	no	—
58	C₄²	D₂₁₂	1	1	yes	characteristic
59	C₁₆	—	4	1	no	—
60	C₂×C₄	—	212	1	no	—
61	C₂×C₄	—	212	1	no	—
62	C₂×C₄	#11719913	1	1	yes	characteristic
63	C₂×C₄	D₄₂₄	1	1	yes	characteristic
64	C₂×C₄	#11719921	1	1	yes	characteristic
65	C₈	—	2	1	no	—
66	C₈	—	424	1	no	—
67	C₈	#11719923	1	1	yes	characteristic
68	C₈	#11719925	1	1	yes	characteristic
69	C₂²	#13133139	1	1	yes	characteristic
70	C₄	—	2	1	no	—
71	C₄	—	2	1	no	—
72	C₄	—	424	1	no	—
73	C₄	—	424	1	no	—
74	C₄	#13133135	1	1	yes	characteristic
75	C₄	#13133140	1	1	yes	characteristic
76	C₂	#16810368	1	1	yes	characteristic
77	C₂	#16810431	1	1	yes	characteristic
78	C₂	#16810434	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#13133139
commutator	C₂×C₈×C₅₃	C₂×C₄
Frattini	C₄×C₈	C₂×D₅₃
Fitting	C₅₃×C₄⋊C₁₆	C₂
socle	C₂²×C₅₃	#102
hypercenter	C₄⋊C₁₆	D₅₃
hypocenter	C₅₃	#1363
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
43. C₅₃	#1363	semidirect