Subgroups — #20289865

Order 6784.

74 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
2	#16811549	C₂	1	1	yes	maximal, characteristic
3	Q₃₂×D₅₃	C₂	1	1	yes	maximal, characteristic
4	#13133286	—	2	1	no	—
5	#13133267	—	2	1	no	—
6	Q₁₆×D₅₃	—	2	1	no	—
7	Dic₄₂₄	C₄	1	1	yes	characteristic, complemented
8	#13133163	—	2	1	no	—
9	C₁₆×D₅₃	C₂²	1	1	yes	characteristic
10	Q₃₂×C₅₃	C₄	1	1	yes	characteristic, complemented
11	Q₈×D₅₃	—	4	1	no	—
12	C₄⋊#6157	—	4	1	no	—
13	Q₁₆×C₅₃	—	2	1	no	—
14	#11719925	—	4	1	no	—
15	Dic₂₁₂	—	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	C₂×#6157	—	8	1	no	—
20	Q₈×C₅₃	—	4	1	no	—
21	C₄×D₅₃	—	8	1	no	—
22	C₄×D₅₃	D₈	1	1	yes	characteristic
23	Dic₁₀₆	—	4	1	no	—
24	Dic₁₀₆	—	8	1	no	—
25	C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
26	#88322	SD₁₆	1	1	yes	characteristic
27	C₂×D₅₃	D₁₆	1	1	yes	characteristic
28	#6157	—	16	1	no	—
29	C₄×C₅₃	—	8	1	no	—
30	C₄×C₅₃	#102	1	1	yes	characteristic
31	Dic₅₃	—	8	1	no	—
32	Dic₅₃	SD₃₂	1	1	yes	characteristic
33	#1416	—	53	1	no	maximal, Sylow
34	C₂×C₅₃	#357	1	1	yes	characteristic
35	D₅₃	SD₆₄	1	1	yes	characteristic
36	D₅₃	Q₆₄	1	1	yes	characteristic
37	C₂×Q₃₂	—	53	1	no	—
38	C₂×C₃₂	—	53	1	no	—
39	#366	—	53	1	no	—
40	C₅₃	#1416	1	1	yes	characteristic, complemented, Sylow
41	C₂×Q₁₆	—	106	1	no	—
42	Q₃₂	—	53	1	no	—
43	Q₃₂	—	106	1	no	—
44	Q₃₂	#6157	1	1	yes	characteristic, complemented
45	C₂×C₁₆	—	53	1	no	—
46	#107	—	106	1	no	—
47	C₃₂	—	106	1	no	—
48	C₂×Q₈	—	212	1	no	—
49	Q₁₆	—	2	1	no	—
50	Q₁₆	—	106	1	no	—
51	Q₁₆	—	212	1	no	—
52	C₂×C₈	—	53	1	no	—
53	C₄⋊C₄	—	212	1	no	—
54	C₁₆	—	53	1	no	—
55	C₁₆	C₂×#6157	1	1	yes	characteristic
56	Q₈	—	4	1	no	—
57	Q₈	—	424	1	no	—
58	Q₈	—	212	1	no	—
59	C₂×C₄	—	53	1	no	—
60	C₂×C₄	—	424	1	no	—
61	C₂×C₄	—	424	1	no	—
62	C₈	—	53	1	no	—
63	C₈	C₂²⋊#6157	1	1	yes	characteristic
64	C₂²	—	53	1	no	—
65	C₄	—	8	1	no	—
66	C₄	—	53	1	no	—
67	C₄	—	848	1	no	—
68	C₄	—	424	1	no	—
69	C₄	#13133279	1	1	yes	characteristic
70	C₂	—	53	1	no	—
71	C₂	—	53	1	no	—
72	C₂	#16811535	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#16811535
commutator	C₁₆×C₅₃	C₂×C₄
Frattini	C₁₆	C₂×#6157
Fitting	Q₃₂×C₅₃	C₄
socle	C₂×C₅₃	#357
hypercenter	Q₃₂	#6157
hypocenter	C₅₃	#1416
LCS	G ▷ C₁₆×C₅₃ ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ Q₃₂

Complements

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

N	H ≅ G/N	type
7. Dic₄₂₄	C₄	semidirect
10. Q₃₂×C₅₃	C₄	semidirect
40. C₅₃	#1416	semidirect
44. Q₃₂	#6157	semidirect