Subgroups — #35212737

Order 20864.

56 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₆₃⋊C₃₂	C₂	1	1	yes	maximal, characteristic
2	#23781491	C₂	1	1	yes	maximal, characteristic
3	#366×C₁₆₃	C₂	1	1	yes	maximal, characteristic
4	#19104669	—	2	1	no	—
5	C₁₆₃⋊C₃₂	C₄	1	2	yes	complemented
6	C₂×C₁₆×C₁₆₃	C₂²	1	1	yes	characteristic
7	#107×C₁₆₃	—	2	1	no	—
8	C₄⋊Dic₁₆₃	—	4	1	no	—
9	C₂×C₈×C₁₆₃	D₄	1	1	yes	characteristic
10	C₄⋊C₄×C₁₆₃	—	4	1	no	—
11	C₁₆×C₁₆₃	C₂×C₄	1	1	yes	characteristic
12	C₁₆×C₁₆₃	Q₈	1	1	yes	characteristic
13	C₂×Dic₁₆₃	—	8	1	no	—
14	C₂×C₄×C₁₆₃	—	8	1	no	—
15	C₂×C₄×C₁₆₃	D₈	1	1	yes	characteristic
16	C₈×C₁₆₃	C₄⋊C₄	1	1	yes	characteristic
17	C₈×C₁₆₃	Q₁₆	1	1	yes	characteristic
18	C₂²×C₁₆₃	D₁₆	1	1	yes	characteristic
19	C₄×C₁₆₃	—	16	1	no	—
20	C₄×C₁₆₃	#107	1	1	yes	characteristic
21	C₄×C₁₆₃	Q₃₂	1	1	yes	characteristic
22	Dic₁₆₃	—	16	1	no	—
23	C₂×C₁₆₃	#366	1	1	yes	characteristic
24	C₂×C₁₆₃	SD₆₄	1	1	yes	characteristic
25	C₂×C₁₆₃	SD₆₄	1	1	yes	characteristic
26	C₁₆₃	#1424	1	1	yes	characteristic, complemented, Sylow
27	#1424	—	163	1	no	maximal, Sylow
28	C₂×C₃₂	—	163	1	no	—
29	#366	—	163	1	no	—
30	#366	D₁₆₃	1	1	yes	characteristic
31	C₂×C₁₆	C₂×D₁₆₃	1	1	yes	characteristic
32	#107	—	2	1	no	—
33	#107	—	326	1	no	—
34	C₃₂	—	163	2	no	—
35	C₂×C₈	C₂²⋊D₁₆₃	1	1	yes	characteristic
36	C₄⋊C₄	—	4	1	no	—
37	C₄⋊C₄	—	652	1	no	—
38	C₁₆	C₄×D₁₆₃	1	1	yes	characteristic
39	C₁₆	Dic₃₂₆	1	1	yes	characteristic
40	C₂×C₄	—	8	1	no	—
41	C₂×C₄	—	1304	1	no	—
42	C₂×C₄	#14579684	1	1	yes	characteristic
43	C₈	Dic₁₆₃⋊C₄	1	1	yes	characteristic
44	C₈	#14579689	1	1	yes	characteristic
45	C₂²	#19104676	1	1	yes	characteristic
46	C₄	—	16	1	no	—
47	C₄	—	2608	1	no	—
48	C₄	#19104668	1	1	yes	characteristic
49	C₄	#19104681	1	1	yes	characteristic
50	C₂	#23781490	1	1	yes	characteristic
51	C₂	#23781502	1	1	yes	characteristic
52	C₂	#23781503	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#19104676
commutator	C₁₆×C₁₆₃	C₂×C₄
Frattini	C₂×C₁₆	C₂×D₁₆₃
Fitting	#366×C₁₆₃	C₂
socle, prime radical	C₂²×C₁₆₃	D₁₆
cosocle	C₂×C₁₆×C₁₆₃	C₂²
hypercenter	#366	D₁₆₃
hypocenter	C₁₆₃	#1424
LCS	G ▷ C₁₆×C₁₆₃ ▷ C₈×C₁₆₃ ▷ C₄×C₁₆₃ ▷ C₂×C₁₆₃ ▷ C₁₆₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #366

Complements

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

N	H ≅ G/N	type
5. C₁₆₃⋊C₃₂	C₄	semidirect
26. C₁₆₃	#1424	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₁₆₃⋊C₃₂	2
#23781491	2
#366×C₁₆₃	2
#1424	163