Subgroups — #23781417

Order 10432.

54 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊(C₁₆₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
2	C₄×C₁₆₃⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₄⋊C₈×C₁₆₃	C₂	1	1	yes	maximal, characteristic
4	C₂×C₁₆₃⋊C₈	—	2	1	no	—
5	C₂×C₁₆₃⋊C₈	C₄	1	1	yes	characteristic
6	C₂×C₁₆₃⋊C₈	C₄	1	1	yes	characteristic
7	C₂×C₈×C₁₆₃	—	2	1	no	—
8	C₄²×C₁₆₃	C₂²	1	1	yes	characteristic
9	C₁₆₃⋊C₈	—	2	1	no	—
10	C₁₆₃⋊C₈	—	4	1	no	—
11	C₁₆₃⋊C₈	C₈	1	2	yes	complemented
12	C₂×C₄×C₁₆₃	C₂×C₄	1	1	yes	characteristic
13	C₂×C₄×C₁₆₃	D₄	1	1	yes	characteristic
14	C₂×C₄×C₁₆₃	Q₈	1	1	yes	characteristic
15	C₈×C₁₆₃	—	4	1	no	—
16	C₂²×C₁₆₃	C₄⋊C₄	1	1	yes	characteristic
17	C₄×C₁₆₃	—	2	1	no	—
18	C₄×C₁₆₃	C₂×C₈	1	1	yes	characteristic
19	C₄×C₁₆₃	M₁₆	1	1	yes	characteristic
20	C₄×C₁₆₃	D₈	1	1	yes	characteristic
21	C₄×C₁₆₃	Q₁₆	1	1	yes	characteristic
22	C₂×C₁₆₃	C₄⋊C₈	1	1	yes	characteristic
23	C₂×C₁₆₃	#107	1	1	yes	characteristic
24	C₂×C₁₆₃	#108	1	1	yes	characteristic
25	C₁₆₃	#335	1	1	yes	characteristic, complemented, Sylow
26	#335	—	163	1	no	maximal, Sylow
27	C₄⋊C₈	—	163	1	no	—
28	C₄⋊C₈	D₁₆₃	1	1	yes	characteristic
29	C₄×C₈	—	163	1	no	—
30	C₂×C₈	—	2	1	no	—
31	C₂×C₈	—	326	1	no	—
32	C₂×C₈	—	163	1	no	—
33	C₂×C₈	—	163	1	no	—
34	C₄²	C₂×D₁₆₃	1	1	yes	characteristic
35	C₂×C₄	C₄×D₁₆₃	1	1	yes	characteristic
36	C₂×C₄	C₂²⋊D₁₆₃	1	1	yes	characteristic
37	C₂×C₄	Dic₃₂₆	1	1	yes	characteristic
38	C₈	—	4	1	no	—
39	C₈	—	326	1	no	—
40	C₈	—	652	1	no	—
41	C₈	—	163	2	no	—
42	C₂²	Dic₁₆₃⋊C₄	1	1	yes	characteristic
43	C₄	—	2	1	no	—
44	C₄	C₈×D₁₆₃	1	1	yes	characteristic
45	C₄	#14579682	1	1	yes	characteristic
46	C₄	#14579684	1	1	yes	characteristic
47	C₄	#14579689	1	1	yes	characteristic
48	C₂	Dic₁₆₃⋊C₈	1	1	yes	characteristic
49	C₂	#19104668	1	1	yes	characteristic
50	C₂	#19104670	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²⋊D₁₆₃
commutator	C₄×C₁₆₃	C₂×C₈
Frattini	C₄²	C₂×D₁₆₃
Fitting	C₄⋊C₈×C₁₆₃	C₂
socle, prime radical	C₂²×C₁₆₃	C₄⋊C₄
cosocle	C₄²×C₁₆₃	C₂²
hypercenter	C₄⋊C₈	D₁₆₃
hypocenter	C₁₆₃	#335
LCS	G ▷ C₄×C₁₆₃ ▷ C₂×C₁₆₃ ▷ C₁₆₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
11. C₁₆₃⋊C₈	C₈	semidirect
25. C₁₆₃	#335	semidirect