Subgroups — #20274325

Order 6784.

168 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16811028	C₂	1	2	yes	maximal
2	#16810974	C₂	1	4	yes	maximal
3	C₈²×C₅₃	C₂	1	1	yes	maximal, characteristic
4	#13133217	C₂²	1	4	yes	—
5	#13133151	—	2	4	no	—
6	#13133142	—	2	8	no	—
7	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
8	C₄×C₈×C₅₃	C₂²	1	2	yes	—
9	C₂×Dic₁₀₆	—	2	4	no	—
10	C₂×C₈×C₅₃	D₄	1	4	yes	—
11	C₂×C₈×C₅₃	D₄	1	2	yes	—
12	C₄²×C₅₃	C₂³	1	1	yes	characteristic
13	C₄⋊Dic₅₃	—	2	4	no	—
14	C₄⋊Dic₅₃	—	2	4	no	—
15	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
16	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
17	C₂×Dic₅₃	—	4	4	no	—
18	Dic₁₀₆	—	4	8	no	—
19	C₈×C₅₃	—	2	4	no	—
20	C₈×C₅₃	Q₁₆	1	4	yes	—
21	C₂²×C₅₃	#127	1	1	yes	characteristic
22	C₄×C₅₃	C₂×Q₁₆	1	2	yes	—
23	C₄×C₅₃	#135	1	4	yes	—
24	Dic₅₃	—	8	4	no	—
25	#1710	—	53	1	no	maximal, Sylow
26	C₂×C₅₃	#494	1	1	yes	characteristic
27	C₂×C₅₃	#495	1	2	yes	—
28	#500	—	53	2	no	—
29	#487	—	53	4	no	—
30	C₈²	D₅₃	1	1	yes	characteristic
31	C₅₃	#1710	1	1	yes	characteristic, complemented, Sylow
32	C₄⋊Q₈	—	53	4	no	—
33	#107	—	106	4	no	—
34	#103	—	106	8	no	—
35	C₄×C₈	C₂×D₅₃	1	2	yes	—
36	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
37	C₂×Q₈	—	106	4	no	—
38	C₂×C₈	D₂₁₂	1	4	yes	—
39	C₂×C₈	D₂₁₂	1	2	yes	—
40	C₄⋊C₄	—	106	4	no	—
41	C₄⋊C₄	—	106	4	no	—
42	C₄²	C₂²×D₅₃	1	1	yes	characteristic
43	Q₈	—	212	8	no	—
44	C₂×C₄	—	212	4	no	—
45	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
46	C₂×C₄	C₂×D₂₁₂	1	2	yes	—
47	C₈	—	2	4	no	—
48	C₈	Dic₂₁₂	1	4	yes	—
49	C₂²	#13133214	1	1	yes	characteristic
50	C₄	—	424	4	no	—
51	C₄	C₂×Dic₂₁₂	1	2	yes	—
52	C₄	#13133239	1	4	yes	—
53	C₂	#16811003	1	1	yes	characteristic
54	C₂	#16811007	1	2	yes	—

	subgroup	G/H
center	C₂²	#13133214
commutator	C₄²×C₅₃	C₂³
Frattini	C₄²	C₂²×D₅₃
Fitting	C₈²×C₅₃	C₂
socle	C₂²×C₅₃	#127
hypercenter	C₈²	D₅₃
hypocenter	C₅₃	#1710
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈²

Complements

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

N	H ≅ G/N	type
31. C₅₃	#1710	semidirect