Subgroups — #20277024

Order 6784.

106 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16811055	C₂	1	2	yes	maximal, complemented
2	Dic₈₄₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#16810462	C₂	1	2	yes	maximal, complemented
4	D₁₆₉₆	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
6	#13133239	—	2	2	no	—
7	Dic₄₂₄	C₂²	1	2	yes	—
8	#13133162	—	2	2	no	—
9	D₈₄₈	C₂²	1	2	yes	—
10	C₂×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
11	C₃₂×C₅₃	C₂²	1	1	yes	characteristic
12	C₃₂×C₅₃	C₂²	1	1	yes	characteristic
13	#11719945	—	4	2	no	—
14	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
15	Dic₂₁₂	—	2	2	no	—
16	D₄₂₄	—	2	2	no	—
17	#11719913	—	4	2	no	—
18	C₁₆×C₅₃	D₄	1	1	yes	characteristic
19	C₁₆×C₅₃	C₂³	1	1	yes	characteristic
20	C₂×C₄×C₅₃	D₈	1	1	yes	characteristic
21	C₂²⋊D₅₃	—	8	2	no	—
22	D₂₁₂	—	4	2	no	—
23	C₄×D₅₃	—	8	2	no	—
24	Dic₁₀₆	—	4	2	no	—
25	C₈×C₅₃	D₈	1	1	yes	characteristic
26	C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
27	C₂²×C₅₃	D₁₆	1	1	yes	characteristic
28	C₂×D₅₃	—	8	2	no	—
29	C₄×C₅₃	D₁₆	1	1	yes	characteristic
30	C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
31	Dic₅₃	—	8	2	no	—
32	#2262	—	53	1	no	maximal, Sylow
33	C₂×C₅₃	—	2	1	no	—
34	C₂×C₅₃	C₂×D₁₆	1	1	yes	characteristic
35	D₅₃	—	16	2	no	—
36	#508	—	53	2	no	—
37	Q₆₄	—	53	1	no	—
38	SD₆₄	—	53	2	no	—
39	D₃₂	—	53	1	no	—
40	C₂×C₃₂	D₅₃	1	1	yes	characteristic, complemented
41	C₅₃	#2262	1	1	yes	characteristic, complemented, Sylow
42	#135	—	106	2	no	—
43	Q₃₂	—	53	2	no	—
44	SD₃₂	—	106	2	no	—
45	D₁₆	—	53	2	no	—
46	C₂×C₁₆	C₂×D₅₃	1	1	yes	characteristic
47	C₃₂	C₂×D₅₃	1	1	yes	characteristic
48	C₃₂	C₂×D₅₃	1	1	yes	characteristic
49	#41	—	212	2	no	—
50	Q₁₆	—	106	2	no	—
51	SD₁₆	—	212	2	no	—
52	D₈	—	106	2	no	—
53	C₂×C₈	D₂₁₂	1	1	yes	characteristic
54	C₁₆	D₂₁₂	1	1	yes	characteristic
55	C₁₆	C₂²×D₅₃	1	1	yes	characteristic
56	Q₈	—	212	2	no	—
57	D₄	—	424	2	no	—
58	D₄	—	212	2	no	—
59	C₂×C₄	—	424	2	no	—
60	C₂×C₄	D₄₂₄	1	1	yes	characteristic
61	C₈	D₄₂₄	1	1	yes	characteristic
62	C₈	C₂×D₂₁₂	1	1	yes	characteristic
63	C₂²	—	424	2	no	—
64	C₂²	D₈₄₈	1	1	yes	characteristic
65	C₄	—	424	2	no	—
66	C₄	D₈₄₈	1	1	yes	characteristic
67	C₄	C₂×D₄₂₄	1	1	yes	characteristic
68	C₂	—	2	1	no	—
69	C₂	—	848	2	no	—
70	C₂	C₂×D₈₄₈	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₈₄₈
commutator	C₁₆×C₅₃	C₂³
Frattini	C₁₆	C₂²×D₅₃
Fitting	C₂×C₃₂×C₅₃	C₂
socle	C₂×C₅₃	C₂×D₁₆
hypercenter	C₂×C₃₂	D₅₃
hypocenter	C₅₃	#2262
LCS	G ▷ C₁₆×C₅₃ ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×C₃₂

Complements

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

N	H ≅ G/N	type
1. #16811055	C₂	semidirect
2. Dic₈₄₈	C₂	semidirect
3. #16810462	C₂	semidirect
4. D₁₆₉₆	C₂	semidirect
5. C₂×C₃₂×C₅₃	C₂	semidirect
40. C₂×C₃₂	D₅₃	semidirect
41. C₅₃	#2262	semidirect