Subgroups — #20272882

Order 6784.

104 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16810422	C₂	1	2	yes	maximal, complemented
2	C₅₃×#492	C₂	1	1	yes	maximal, characteristic
3	#13133136	—	2	2	no	—
4	C₂×SD₁₆×C₅₃	—	2	2	no	—
5	C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
6	#127×C₅₃	C₄	1	1	yes	characteristic
7	C₄×C₈×C₅₃	C₄	1	1	yes	characteristic
8	C₂×Q₈×C₅₃	D₄	1	2	yes	—
9	C₂×D₄×C₅₃	—	2	1	no	—
10	C₂×D₄×C₅₃	—	4	1	no	—
11	SD₁₆×C₅₃	—	4	4	no	—
12	C₂×C₈×C₅₃	—	2	1	no	—
13	C₄⋊C₄×C₅₃	—	4	1	no	—
14	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
15	#11719917	—	4	2	no	—
16	C₂³×C₅₃	—	4	1	no	—
17	Q₈×C₅₃	—	4	2	no	—
18	D₄×C₅₃	—	4	2	no	—
19	D₄×C₅₃	—	8	2	no	—
20	C₂×C₄×C₅₃	—	2	1	no	—
21	C₂×C₄×C₅₃	—	2	2	no	—
22	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
23	C₈×C₅₃	—	2	2	no	—
24	#88322	—	8	2	no	—
25	C₂²×C₅₃	—	8	2	no	—
26	C₂²×C₅₃	—	8	1	no	—
27	C₂²×C₅₃	#99	1	1	yes	characteristic
28	C₄×C₅₃	—	2	2	no	—
29	C₄×C₅₃	—	2	1	no	—
30	C₄×C₅₃	—	4	2	no	—
31	#1410	—	53	1	no	maximal, Sylow
32	C₂×C₅₃	—	2	1	no	—
33	C₂×C₅₃	—	16	1	no	—
34	C₂×C₅₃	#353	1	1	yes	characteristic
35	#492	D₅₃	1	1	yes	characteristic
36	#356	—	53	2	no	—
37	C₅₃	#1410	1	1	yes	characteristic, complemented, Sylow
38	C₂×SD₁₆	—	2	2	no	—
39	C₄⋊Q₈	C₂×D₅₃	1	1	yes	characteristic
40	#127	Dic₅₃	1	1	yes	characteristic
41	#101	—	106	2	no	—
42	C₄×C₈	Dic₅₃	1	1	yes	characteristic
43	C₂×Q₈	C₂²⋊D₅₃	1	2	yes	—
44	C₂×D₄	—	2	1	no	—
45	C₂×D₄	—	4	1	no	—
46	SD₁₆	—	4	4	no	—
47	M₁₆	—	212	2	no	—
48	C₂×C₈	—	2	1	no	—
49	C₄⋊C₄	—	4	1	no	—
50	C₄²	C₂×Dic₅₃	1	1	yes	characteristic
51	C₂³	—	4	1	no	—
52	Q₈	—	4	2	no	—
53	D₄	—	4	2	no	—
54	D₄	—	8	2	no	—
55	C₂×C₄	—	2	1	no	—
56	C₂×C₄	—	2	2	no	—
57	C₂×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
58	C₈	—	2	2	no	—
59	C₈	—	424	2	no	—
60	C₂²	—	8	2	no	—
61	C₂²	—	8	1	no	—
62	C₂²	#13133132	1	1	yes	characteristic
63	C₄	—	2	2	no	—
64	C₄	—	2	1	no	—
65	C₄	—	4	2	no	—
66	C₂	—	2	1	no	—
67	C₂	—	16	1	no	—
68	C₂	#16810416	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#16810416
commutator	C₄²×C₅₃	C₂×C₄
Frattini	C₄⋊Q₈	C₂×D₅₃
Fitting	C₅₃×#492	C₂
socle	C₂×C₅₃	#353
hypercenter	#492	D₅₃
hypocenter	C₅₃	#1410
LCS	G ▷ C₄²×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ C₄⋊Q₈ ◁ #492

Complements

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

N	H ≅ G/N	type
1. #16810422	C₂	semidirect
37. C₅₃	#1410	semidirect