Subgroups — #20272909

Order 6784.

70 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₄₂₄	C₂	1	1	yes	maximal, characteristic
2	#16810453	C₂	1	1	yes	maximal, characteristic
3	C₅₃×M₆₄	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₂₁₂	—	2	1	no	—
5	Dic₄₂₄	—	2	1	no	—
6	Dic₄₂₄	C₄	1	2	yes	—
7	#13133153	—	2	1	no	—
8	C₂×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
9	C₃₂×C₅₃	—	2	1	no	—
10	C₂×Dic₁₀₆	—	4	1	no	—
11	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
12	#11719917	—	4	1	no	—
13	Dic₂₁₂	—	2	2	no	—
14	Dic₂₁₂	—	4	1	no	—
15	C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
16	C₁₆×C₅₃	D₄	1	1	yes	characteristic
17	C₂×C₄×C₅₃	D₈	1	1	yes	characteristic
18	C₂×Dic₅₃	—	8	1	no	—
19	Dic₁₀₆	—	4	2	no	—
20	Dic₁₀₆	—	8	1	no	—
21	C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
22	C₈×C₅₃	SD₁₆	1	1	yes	characteristic
23	#88322	—	8	1	no	—
24	C₂²×C₅₃	SD₃₂	1	1	yes	characteristic
25	C₄×C₅₃	#102	1	1	yes	characteristic
26	C₄×C₅₃	D₁₆	1	1	yes	characteristic
27	Dic₅₃	—	8	2	no	—
28	#1420	—	53	1	no	maximal, Sylow
29	C₂×C₅₃	—	2	1	no	—
30	C₂×C₅₃	#357	1	1	yes	characteristic
31	C₂×Q₃₂	—	53	1	no	—
32	M₆₄	D₅₃	1	1	yes	characteristic
33	#368	—	53	1	no	—
34	C₅₃	#1420	1	1	yes	characteristic, complemented, Sylow
35	C₂×Q₁₆	—	106	1	no	—
36	Q₃₂	—	53	2	no	—
37	Q₃₂	—	106	1	no	—
38	C₂×C₁₆	C₂×D₅₃	1	1	yes	characteristic
39	#108	—	106	1	no	—
40	C₃₂	—	2	1	no	—
41	C₂×Q₈	—	212	1	no	—
42	Q₁₆	—	106	2	no	—
43	Q₁₆	—	212	1	no	—
44	M₁₆	—	212	1	no	—
45	C₂×C₈	D₂₁₂	1	1	yes	characteristic
46	C₁₆	C₄×D₅₃	1	1	yes	characteristic
47	C₁₆	C₂²⋊D₅₃	1	1	yes	characteristic
48	Q₈	—	424	1	no	—
49	Q₈	—	212	2	no	—
50	C₂×C₄	—	424	1	no	—
51	C₂×C₄	D₄₂₄	1	1	yes	characteristic
52	C₈	—	424	1	no	—
53	C₈	#11719913	1	1	yes	characteristic
54	C₈	#11719921	1	1	yes	characteristic
55	C₂²	#13133162	1	1	yes	characteristic
56	C₄	—	424	2	no	—
57	C₄	#13133139	1	1	yes	characteristic
58	C₄	D₈₄₈	1	1	yes	characteristic
59	C₂	—	2	1	no	—
60	C₂	#16810425	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#16810425
commutator	C₁₆×C₅₃	C₂×C₄
Frattini	C₂×C₁₆	C₂×D₅₃
Fitting	C₅₃×M₆₄	C₂
socle	C₂×C₅₃	#357
hypercenter	M₆₄	D₅₃
hypocenter	C₅₃	#1420
LCS	G ▷ C₁₆×C₅₃ ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ M₆₄

Complements

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

N	H ≅ G/N	type
34. C₅₃	#1420	semidirect