Subgroups — #20272903

Order 6784.

74 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16811055	C₂	1	1	yes	maximal, characteristic
2	#16810451	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×M₆₄	C₂	1	1	yes	maximal, characteristic, complemented
4	#13133239	—	2	1	no	—
5	Dic₄₂₄	C₄	1	1	yes	characteristic, complemented
6	#13133162	—	2	1	no	—
7	D₈₄₈	C₄	1	1	yes	characteristic, complemented
8	#13133151	—	2	1	no	—
9	C₂×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
10	C₃₂×C₅₃	—	2	1	no	—
11	#11719945	—	4	1	no	—
12	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
13	C₄⋊Dic₅₃	—	4	1	no	—
14	Dic₂₁₂	—	2	1	no	—
15	D₄₂₄	—	2	1	no	—
16	#11719913	—	4	1	no	—
17	C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
18	C₁₆×C₅₃	D₄	1	1	yes	characteristic
19	C₂×C₄×C₅₃	D₈	1	1	yes	characteristic
20	C₂²⋊D₅₃	—	8	1	no	—
21	C₂×Dic₅₃	—	8	1	no	—
22	D₂₁₂	—	4	1	no	—
23	C₄×D₅₃	—	8	1	no	—
24	Dic₁₀₆	—	4	1	no	—
25	C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
26	C₈×C₅₃	SD₁₆	1	1	yes	characteristic
27	C₂²×C₅₃	D₁₆	1	1	yes	characteristic
28	C₂×D₅₃	—	8	1	no	—
29	C₄×C₅₃	#102	1	1	yes	characteristic
30	C₄×C₅₃	SD₃₂	1	1	yes	characteristic
31	Dic₅₃	—	8	1	no	—
32	Dic₅₃	—	16	1	no	—
33	#1418	—	53	1	no	maximal, Sylow
34	C₂×C₅₃	—	2	1	no	—
35	C₂×C₅₃	#357	1	1	yes	characteristic
36	D₅₃	—	16	1	no	—
37	#508	—	53	1	no	—
38	M₆₄	D₅₃	1	1	yes	characteristic, complemented
39	#367	—	53	1	no	—
40	C₅₃	#1418	1	1	yes	characteristic, complemented, Sylow
41	#135	—	106	1	no	—
42	Q₃₂	—	53	1	no	—
43	SD₃₂	—	106	1	no	—
44	D₁₆	—	53	1	no	—
45	C₂×C₁₆	C₂×D₅₃	1	1	yes	characteristic
46	#107	—	106	1	no	—
47	C₃₂	—	2	1	no	—
48	#41	—	212	1	no	—
49	Q₁₆	—	106	1	no	—
50	SD₁₆	—	212	1	no	—
51	D₈	—	106	1	no	—
52	C₂×C₈	D₂₁₂	1	1	yes	characteristic
53	C₄⋊C₄	—	212	1	no	—
54	C₁₆	C₄×D₅₃	1	1	yes	characteristic
55	C₁₆	C₂²⋊D₅₃	1	1	yes	characteristic
56	Q₈	—	212	1	no	—
57	D₄	—	424	1	no	—
58	D₄	—	212	1	no	—
59	C₂×C₄	—	424	1	no	—
60	C₂×C₄	—	424	1	no	—
61	C₂×C₄	D₄₂₄	1	1	yes	characteristic
62	C₈	#11719913	1	1	yes	characteristic
63	C₈	#11719921	1	1	yes	characteristic
64	C₂²	—	424	1	no	—
65	C₂²	D₈₄₈	1	1	yes	characteristic
66	C₄	—	848	1	no	—
67	C₄	—	424	1	no	—
68	C₄	#13133139	1	1	yes	characteristic
69	C₄	#13133162	1	1	yes	characteristic
70	C₂	—	2	1	no	—
71	C₂	—	848	1	no	—
72	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	#1418
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
2. #16810451	C₂	semidirect
3. C₅₃×M₆₄	C₂	semidirect
5. Dic₄₂₄	C₄	semidirect
7. D₈₄₈	C₄	semidirect
38. M₆₄	D₅₃	semidirect
40. C₅₃	#1418	semidirect