Subgroups — #20272906

Order 6784.

78 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₈₄₈	C₂	1	1	yes	maximal, characteristic
2	#16810453	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×M₆₄	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₄₂₄	—	2	1	no	—
5	D₈₄₈	—	2	1	no	—
6	D₈₄₈	C₄	1	2	yes	—
7	#13133153	—	2	1	no	—
8	C₂×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
9	C₃₂×C₅₃	—	2	1	no	—
10	C₂×D₂₁₂	—	4	1	no	—
11	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
12	#11719917	—	4	1	no	—
13	D₄₂₄	—	2	2	no	—
14	D₄₂₄	—	4	1	no	—
15	C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
16	C₁₆×C₅₃	D₄	1	1	yes	characteristic
17	C₂²×D₅₃	—	8	1	no	—
18	C₂×C₄×C₅₃	D₈	1	1	yes	characteristic
19	D₂₁₂	—	4	2	no	—
20	D₂₁₂	—	8	1	no	—
21	C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
22	C₈×C₅₃	SD₁₆	1	1	yes	characteristic
23	#88322	—	8	1	no	—
24	C₂²×C₅₃	SD₃₂	1	1	yes	characteristic
25	C₂×D₅₃	—	8	2	no	—
26	C₂×D₅₃	—	16	2	no	—
27	C₄×C₅₃	#102	1	1	yes	characteristic
28	C₄×C₅₃	D₁₆	1	1	yes	characteristic
29	#1419	—	53	1	no	maximal, Sylow
30	C₂×C₅₃	—	2	1	no	—
31	C₂×C₅₃	#357	1	1	yes	characteristic
32	D₅₃	—	16	2	no	—
33	C₂×D₁₆	—	53	1	no	—
34	M₆₄	D₅₃	1	1	yes	characteristic, complemented
35	#368	—	53	1	no	—
36	C₅₃	#1419	1	1	yes	characteristic, complemented, Sylow
37	C₂×D₈	—	106	1	no	—
38	D₁₆	—	53	2	no	—
39	D₁₆	—	106	1	no	—
40	C₂×C₁₆	C₂×D₅₃	1	1	yes	characteristic
41	#108	—	106	1	no	—
42	C₃₂	—	2	1	no	—
43	C₂×D₄	—	212	1	no	—
44	D₈	—	106	2	no	—
45	D₈	—	212	1	no	—
46	M₁₆	—	212	1	no	—
47	C₂×C₈	D₂₁₂	1	1	yes	characteristic
48	C₁₆	C₄×D₅₃	1	1	yes	characteristic
49	C₁₆	C₂²⋊D₅₃	1	1	yes	characteristic
50	C₂³	—	424	1	no	—
51	D₄	—	424	1	no	—
52	D₄	—	212	2	no	—
53	C₂×C₄	D₄₂₄	1	1	yes	characteristic
54	C₈	—	424	1	no	—
55	C₈	#11719913	1	1	yes	characteristic
56	C₈	#11719921	1	1	yes	characteristic
57	C₂²	—	848	2	no	—
58	C₂²	—	424	2	no	—
59	C₂²	#13133162	1	1	yes	characteristic
60	C₄	#13133139	1	1	yes	characteristic
61	C₄	D₈₄₈	1	1	yes	characteristic
62	C₂	—	2	1	no	—
63	C₂	—	848	2	no	—
64	C₂	#16810425	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#16810425
commutator	C₁₆×C₅₃	C₂×C₄
Frattini	C₂×C₁₆	C₂×D₅₃
Fitting	C₅₃×M₆₄	C₂
socle	C₂×C₅₃	#357
hypercenter	M₆₄	D₅₃
hypocenter	C₅₃	#1419
LCS	G ▷ C₁₆×C₅₃ ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ M₆₄

Complements

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

N	H ≅ G/N	type
2. #16810453	C₂	semidirect
3. C₅₃×M₆₄	C₂	semidirect
34. M₆₄	D₅₃	semidirect
36. C₅₃	#1419	semidirect