Subgroups — #20272856

Order 6784.

72 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16811039	C₂	1	1	yes	maximal, characteristic
2	#16810457	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
4	#13133223	C₄	1	1	yes	characteristic
5	#13133157	C₄	1	1	yes	characteristic
6	#13133156	—	2	1	no	—
7	C₁₆×D₅₃	—	2	1	no	—
8	#13133123	—	2	1	no	—
9	C₂×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
10	C₃₂×C₅₃	—	2	1	no	—
11	#11719945	C₈	1	1	yes	characteristic
12	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
13	#11719917	C₈	1	1	yes	characteristic
14	#11719912	—	2	1	no	—
15	C₈×D₅₃	—	2	1	no	—
16	C₁₆×C₅₃	D₄	1	1	yes	characteristic
17	C₁₆×C₅₃	D₄	1	1	yes	characteristic
18	#11719908	—	2	1	no	—
19	C₂×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
20	C₂²⋊D₅₃	—	2	1	no	—
21	D₂₁₂	C₁₆	1	1	yes	characteristic
22	C₄×D₅₃	—	2	1	no	—
23	Dic₁₀₆	C₁₆	1	1	yes	characteristic
24	C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
25	C₈×C₅₃	M₁₆	1	1	yes	characteristic
26	#88322	—	2	1	no	—
27	C₂²×C₅₃	M₃₂	1	1	yes	characteristic
28	C₂×D₅₃	—	2	1	no	—
29	C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
30	C₄×C₅₃	C₂×C₁₆	1	1	yes	characteristic
31	Dic₅₃	—	2	1	no	—
32	#1401	—	53	1	no	maximal, Sylow
33	C₂×C₅₃	—	2	1	no	—
34	C₂×C₅₃	C₂²⋊C₁₆	1	1	yes	characteristic
35	D₅₃	—	4	1	no	—
36	#504	—	53	1	no	—
37	M₆₄	—	53	1	no	—
38	C₂×C₃₂	D₅₃	1	1	yes	characteristic, complemented
39	C₅₃	#1401	1	1	yes	characteristic, complemented, Sylow
40	#131	—	53	1	no	—
41	M₃₂	—	53	1	no	—
42	M₃₂	—	106	1	no	—
43	C₂×C₁₆	—	106	1	no	—
44	C₂×C₁₆	C₂×D₅₃	1	1	yes	characteristic
45	C₃₂	—	2	1	no	—
46	C₃₂	—	106	1	no	—
47	#41	—	53	1	no	—
48	M₁₆	—	53	1	no	—
49	M₁₆	—	106	1	no	—
50	C₂×C₈	—	106	1	no	—
51	C₂×C₈	C₄×D₅₃	1	1	yes	characteristic
52	C₁₆	—	106	1	no	—
53	C₁₆	D₂₁₂	1	1	yes	characteristic
54	C₁₆	C₂²⋊D₅₃	1	1	yes	characteristic
55	Q₈	—	53	1	no	—
56	D₄	—	53	1	no	—
57	D₄	—	106	1	no	—
58	C₂×C₄	—	106	1	no	—
59	C₂×C₄	C₈×D₅₃	1	1	yes	characteristic
60	C₈	—	106	1	no	—
61	C₈	#11719912	1	1	yes	characteristic
62	C₈	#11719921	1	1	yes	characteristic
63	C₂²	—	106	1	no	—
64	C₂²	#13133156	1	1	yes	characteristic
65	C₄	—	106	1	no	—
66	C₄	#13133129	1	1	yes	characteristic
67	C₄	C₁₆×D₅₃	1	1	yes	characteristic
68	C₂	—	2	1	no	—
69	C₂	—	212	1	no	—
70	C₂	#16810402	1	1	yes	characteristic

	subgroup	G/H
center	C₁₆	D₂₁₂
commutator	C₄×C₅₃	C₂×C₁₆
Frattini	C₂×C₁₆	C₂×D₅₃
Fitting	C₂×C₃₂×C₅₃	C₂
socle	C₂×C₅₃	C₂²⋊C₁₆
hypercenter	C₂×C₃₂	D₅₃
hypocenter	C₅₃	#1401
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₁₆ ◁ C₂×C₁₆ ◁ C₂×C₃₂

Complements

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

N	H ≅ G/N	type
2. #16810457	C₂	semidirect
3. C₂×C₃₂×C₅₃	C₂	semidirect
38. C₂×C₃₂	D₅₃	semidirect
39. C₅₃	#1401	semidirect