Subgroups — #20272857

Order 6784.

72 conjugacy classes of subgroups.

Conjugacy classes of subgroups

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

	subgroup	G/H
center	C₁₆	C₂²⋊D₅₃
commutator	C₄×C₅₃	C₂×C₁₆
Frattini	C₂×C₁₆	C₂×D₅₃
Fitting	C₅₃×#504	C₂
socle	C₂×C₅₃	C₂²⋊C₁₆
hypercenter	#504	D₅₃
hypocenter	C₅₃	#1401
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₁₆ ◁ C₂×C₁₆ ◁ #504

Complements

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

N	H ≅ G/N	type
1. #16810457	C₂	semidirect
2. C₂×#13133123	C₂	semidirect
39. C₅₃	#1401	semidirect