Subgroups — #20274327

Order 6784.

178 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16810974	C₂	1	3	yes	maximal, complemented
2	#16810970	C₂	1	3	yes	maximal
3	C₈²×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
4	#13133217	C₂²	1	3	yes	—
5	#13133214	C₂²	1	1	yes	characteristic
6	#13133142	—	2	6	no	—
7	#13133139	—	2	6	no	—
8	C₄×C₈×C₅₃	C₂²	1	3	yes	—
9	C₂×D₂₁₂	—	2	3	no	—
10	C₂×Dic₁₀₆	—	2	3	no	—
11	C₂×C₈×C₅₃	D₄	1	6	yes	—
12	C₄²×C₅₃	C₂³	1	1	yes	characteristic
13	C₄⋊Dic₅₃	—	2	3	no	—
14	C₄⋊Dic₅₃	—	2	3	no	—
15	C₂²×D₅₃	—	4	1	no	—
16	C₂×C₄×C₅₃	C₂×D₄	1	3	yes	—
17	C₂×Dic₅₃	—	4	3	no	—
18	D₂₁₂	—	4	6	no	—
19	Dic₁₀₆	—	4	6	no	—
20	C₈×C₅₃	—	2	6	no	—
21	C₂²×C₅₃	#127	1	1	yes	characteristic
22	C₂×D₅₃	—	8	3	no	—
23	C₄×C₅₃	#135	1	6	yes	—
24	Dic₅₃	—	8	3	no	—
25	#1711	—	53	1	no	maximal, Sylow
26	C₂×C₅₃	#495	1	3	yes	—
27	D₅₃	—	16	1	no	—
28	#487	—	53	3	no	—
29	#486	—	53	3	no	—
30	C₈²	D₅₃	1	1	yes	characteristic, complemented
31	C₅₃	#1711	1	1	yes	characteristic, complemented, Sylow
32	C₄⋊Q₈	—	53	3	no	—
33	#127	—	53	1	no	—
34	#103	—	106	6	no	—
35	#102	—	106	6	no	—
36	C₄×C₈	C₂×D₅₃	1	3	yes	—
37	C₂×Q₈	—	106	3	no	—
38	C₂×D₄	—	106	3	no	—
39	C₂×C₈	D₂₁₂	1	6	yes	—
40	C₄⋊C₄	—	106	3	no	—
41	C₄⋊C₄	—	106	3	no	—
42	C₄²	C₂²×D₅₃	1	1	yes	characteristic
43	C₂³	—	212	1	no	—
44	Q₈	—	212	6	no	—
45	D₄	—	212	6	no	—
46	C₂×C₄	—	212	3	no	—
47	C₂×C₄	C₂×D₂₁₂	1	3	yes	—
48	C₈	—	2	6	no	—
49	C₂²	—	424	3	no	—
50	C₂²	#13133214	1	1	yes	characteristic
51	C₄	—	424	3	no	—
52	C₄	#13133239	1	6	yes	—
53	C₂	—	848	1	no	—
54	C₂	#16811007	1	3	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	#1711
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
1. #16810974	C₂	semidirect
3. C₈²×C₅₃	C₂	semidirect
30. C₈²	D₅₃	semidirect
31. C₅₃	#1711	semidirect