Subgroups — #20289864

Order 6784.

82 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₅₃⋊C₃₂	C₂	1	1	yes	maximal, characteristic, complemented
2	#16811549	C₂	1	1	yes	maximal, characteristic, complemented
3	D₁₆×D₅₃	C₂	1	1	yes	maximal, characteristic
4	#13133286	—	2	1	no	—
5	#13133267	—	2	1	no	—
6	D₈×D₅₃	—	2	1	no	—
7	D₈₄₈	C₄	1	1	yes	characteristic, complemented
8	#13133158	—	2	1	no	—
9	C₁₆×D₅₃	C₂²	1	1	yes	characteristic
10	D₁₆×C₅₃	C₄	1	1	yes	characteristic, complemented
11	D₄×D₅₃	—	4	1	no	—
12	C₄⋊#6157	—	4	1	no	—
13	D₈×C₅₃	—	2	1	no	—
14	#11719922	—	4	1	no	—
15	D₄₂₄	—	2	1	no	—
16	C₈×D₅₃	D₄	1	1	yes	characteristic
17	C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
18	#11719908	D₄	1	1	yes	characteristic
19	C₂²×D₅₃	—	8	1	no	—
20	C₂×#6157	—	8	1	no	—
21	D₄×C₅₃	—	4	1	no	—
22	C₂²⋊D₅₃	—	8	1	no	—
23	D₂₁₂	—	4	1	no	—
24	C₄×D₅₃	D₈	1	1	yes	characteristic
25	C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
26	#88322	SD₁₆	1	1	yes	characteristic
27	C₂²×C₅₃	—	8	1	no	—
28	C₂×D₅₃	—	8	1	no	—
29	C₂×D₅₃	—	16	1	no	—
30	C₂×D₅₃	—	16	1	no	—
31	C₂×D₅₃	D₁₆	1	1	yes	characteristic
32	#6157	—	16	1	no	—
33	C₄×C₅₃	#102	1	1	yes	characteristic
34	Dic₅₃	SD₃₂	1	1	yes	characteristic
35	#1415	—	53	1	no	maximal, Sylow
36	C₂×C₅₃	—	16	1	no	—
37	C₂×C₅₃	#357	1	1	yes	characteristic
38	D₅₃	—	16	1	no	—
39	D₅₃	D₃₂	1	1	yes	characteristic
40	D₅₃	SD₆₄	1	1	yes	characteristic
41	C₂×D₁₆	—	53	1	no	—
42	C₂×C₃₂	—	53	1	no	—
43	#366	—	53	1	no	—
44	C₅₃	#1415	1	1	yes	characteristic, complemented, Sylow
45	C₂×D₈	—	106	1	no	—
46	D₁₆	—	53	1	no	—
47	D₁₆	—	106	1	no	—
48	D₁₆	#6157	1	1	yes	characteristic, complemented
49	C₂×C₁₆	—	53	1	no	—
50	#107	—	106	1	no	—
51	C₃₂	—	106	1	no	—
52	C₂×D₄	—	212	1	no	—
53	D₈	—	2	1	no	—
54	D₈	—	106	1	no	—
55	D₈	—	212	1	no	—
56	C₂×C₈	—	53	1	no	—
57	C₄⋊C₄	—	212	1	no	—
58	C₁₆	—	53	1	no	—
59	C₁₆	C₂×#6157	1	1	yes	characteristic
60	C₂³	—	424	1	no	—
61	D₄	—	4	1	no	—
62	D₄	—	424	1	no	—
63	D₄	—	212	1	no	—
64	C₂×C₄	—	53	1	no	—
65	C₂×C₄	—	424	1	no	—
66	C₈	—	53	1	no	—
67	C₈	C₂²⋊#6157	1	1	yes	characteristic
68	C₂²	—	8	1	no	—
69	C₂²	—	53	1	no	—
70	C₂²	—	848	1	no	—
71	C₂²	—	848	1	no	—
72	C₂²	—	424	1	no	—
73	C₄	—	53	1	no	—
74	C₄	—	848	1	no	—
75	C₄	#13133279	1	1	yes	characteristic
76	C₂	—	16	1	no	—
77	C₂	—	53	1	no	—
78	C₂	—	53	1	no	—
79	C₂	—	848	1	no	—
80	C₂	#16811535	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#16811535
commutator	C₁₆×C₅₃	C₂×C₄
Frattini	C₁₆	C₂×#6157
Fitting	D₁₆×C₅₃	C₄
socle	C₂×C₅₃	#357
hypercenter	D₁₆	#6157
hypocenter	C₅₃	#1415
LCS	G ▷ C₁₆×C₅₃ ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ D₁₆

Complements

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

N	H ≅ G/N	type
1. D₅₃⋊C₃₂	C₂	semidirect
2. #16811549	C₂	semidirect
7. D₈₄₈	C₄	semidirect
10. D₁₆×C₅₃	C₄	semidirect
44. C₅₃	#1415	semidirect
48. D₁₆	#6157	semidirect