Subgroups — #20272900

Order 6784.

72 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16811055	C₂	1	1	yes	maximal, characteristic
2	#16810453	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
4	#13133239	—	2	1	no	—
5	Dic₄₂₄	C₄	1	1	yes	characteristic
6	#13133162	—	2	1	no	—
7	D₈₄₈	C₄	1	1	yes	characteristic
8	#13133153	—	2	1	no	—
9	C₂×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
10	C₃₂×C₅₃	—	2	1	no	—
11	#11719945	—	4	1	no	—
12	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
13	#11719917	—	4	1	no	—
14	Dic₂₁₂	—	2	1	no	—
15	D₄₂₄	—	2	1	no	—
16	#11719913	—	4	1	no	—
17	C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
18	C₁₆×C₅₃	D₄	1	1	yes	characteristic
19	C₂×C₄×C₅₃	D₈	1	1	yes	characteristic
20	C₂²⋊D₅₃	—	8	1	no	—
21	D₂₁₂	—	4	1	no	—
22	C₄×D₅₃	—	8	1	no	—
23	Dic₁₀₆	—	4	1	no	—
24	C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
25	C₈×C₅₃	SD₁₆	1	1	yes	characteristic
26	#88322	—	8	1	no	—
27	C₂²×C₅₃	SD₃₂	1	1	yes	characteristic
28	C₂×D₅₃	—	8	1	no	—
29	C₄×C₅₃	#102	1	1	yes	characteristic
30	C₄×C₅₃	D₁₆	1	1	yes	characteristic
31	Dic₅₃	—	8	1	no	—
32	#1417	—	53	1	no	maximal, Sylow
33	C₂×C₅₃	—	2	1	no	—
34	C₂×C₅₃	#357	1	1	yes	characteristic
35	D₅₃	—	16	1	no	—
36	#508	—	53	1	no	—
37	C₂×C₃₂	D₅₃	1	1	yes	characteristic, complemented
38	#368	—	53	1	no	—
39	C₅₃	#1417	1	1	yes	characteristic, complemented, Sylow
40	#135	—	106	1	no	—
41	Q₃₂	—	53	1	no	—
42	SD₃₂	—	106	1	no	—
43	D₁₆	—	53	1	no	—
44	C₂×C₁₆	C₂×D₅₃	1	1	yes	characteristic
45	#108	—	106	1	no	—
46	C₃₂	—	2	1	no	—
47	#41	—	212	1	no	—
48	Q₁₆	—	106	1	no	—
49	SD₁₆	—	212	1	no	—
50	D₈	—	106	1	no	—
51	M₁₆	—	212	1	no	—
52	C₂×C₈	D₂₁₂	1	1	yes	characteristic
53	C₁₆	C₄×D₅₃	1	1	yes	characteristic
54	C₁₆	C₂²⋊D₅₃	1	1	yes	characteristic
55	Q₈	—	212	1	no	—
56	D₄	—	424	1	no	—
57	D₄	—	212	1	no	—
58	C₂×C₄	—	424	1	no	—
59	C₂×C₄	D₄₂₄	1	1	yes	characteristic
60	C₈	—	424	1	no	—
61	C₈	#11719913	1	1	yes	characteristic
62	C₈	#11719921	1	1	yes	characteristic
63	C₂²	—	424	1	no	—
64	C₂²	#13133162	1	1	yes	characteristic
65	C₄	—	424	1	no	—
66	C₄	#13133139	1	1	yes	characteristic
67	C₄	D₈₄₈	1	1	yes	characteristic
68	C₂	—	2	1	no	—
69	C₂	—	848	1	no	—
70	C₂	#16810425	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₈₄₈
commutator	C₁₆×C₅₃	C₂×C₄
Frattini	C₂×C₁₆	C₂×D₅₃
Fitting	C₂×C₃₂×C₅₃	C₂
socle	C₂×C₅₃	#357
hypercenter	C₂×C₃₂	D₅₃
hypocenter	C₅₃	#1417
LCS	G ▷ C₁₆×C₅₃ ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×C₃₂

Complements

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

N	H ≅ G/N	type
2. #16810453	C₂	semidirect
3. C₂×C₃₂×C₅₃	C₂	semidirect
37. C₂×C₃₂	D₅₃	semidirect
39. C₅₃	#1417	semidirect