Subgroups — #35212736

Order 20864.

56 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23781491	C₂	1	2	yes	maximal
2	C₂×C₃₂×C₁₆₃	C₂	1	1	yes	maximal, characteristic
3	#19104669	—	2	2	no	—
4	C₂×C₁₆×C₁₆₃	C₂²	1	1	yes	characteristic
5	C₃₂×C₁₆₃	C₄	1	2	yes	complemented
6	C₄⋊Dic₁₆₃	—	4	2	no	—
7	C₂×C₈×C₁₆₃	D₄	1	1	yes	characteristic
8	C₁₆×C₁₆₃	C₂×C₄	1	1	yes	characteristic
9	C₁₆×C₁₆₃	Q₈	1	1	yes	characteristic
10	C₂×Dic₁₆₃	—	8	2	no	—
11	C₂×C₄×C₁₆₃	D₈	1	1	yes	characteristic
12	C₈×C₁₆₃	C₄⋊C₄	1	1	yes	characteristic
13	C₈×C₁₆₃	Q₁₆	1	1	yes	characteristic
14	C₂²×C₁₆₃	D₁₆	1	1	yes	characteristic
15	C₄×C₁₆₃	#107	1	1	yes	characteristic
16	C₄×C₁₆₃	Q₃₂	1	1	yes	characteristic
17	Dic₁₆₃	—	16	2	no	—
18	C₂×C₁₆₃	#366	1	1	yes	characteristic
19	C₂×C₁₆₃	D₃₂	1	1	yes	characteristic
20	C₂×C₁₆₃	Q₆₄	1	1	yes	characteristic
21	C₁₆₃	#1423	1	1	yes	characteristic, complemented, Sylow
22	#1423	—	163	1	no	maximal, Sylow
23	C₂×C₃₂	D₁₆₃	1	1	yes	characteristic
24	#366	—	163	2	no	—
25	C₂×C₁₆	C₂×D₁₆₃	1	1	yes	characteristic
26	#107	—	326	2	no	—
27	C₃₂	Dic₁₆₃	1	2	yes	complemented
28	C₂×C₈	D₆₅₂	1	1	yes	characteristic
29	C₄⋊C₄	—	652	2	no	—
30	C₁₆	C₂×Dic₁₆₃	1	1	yes	characteristic
31	C₁₆	Dic₃₂₆	1	1	yes	characteristic
32	C₂×C₄	—	1304	2	no	—
33	C₂×C₄	D₁₃₀₄	1	1	yes	characteristic
34	C₈	C₄⋊Dic₁₆₃	1	1	yes	characteristic
35	C₈	Dic₆₅₂	1	1	yes	characteristic
36	C₂²	D₂₆₀₈	1	1	yes	characteristic
37	C₄	—	2608	2	no	—
38	C₄	#19104669	1	1	yes	characteristic
39	C₄	Dic₁₃₀₄	1	1	yes	characteristic
40	C₂	#23781491	1	1	yes	characteristic
41	C₂	D₅₂₁₆	1	1	yes	characteristic
42	C₂	Dic₂₆₀₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₂₆₀₈
commutator	C₁₆×C₁₆₃	C₂×C₄
Frattini	C₂×C₁₆	C₂×D₁₆₃
Fitting	C₂×C₃₂×C₁₆₃	C₂
socle, prime radical	C₂²×C₁₆₃	D₁₆
cosocle	C₂×C₁₆×C₁₆₃	C₂²
hypercenter	C₂×C₃₂	D₁₆₃
hypocenter	C₁₆₃	#1423
LCS	G ▷ C₁₆×C₁₆₃ ▷ C₈×C₁₆₃ ▷ C₄×C₁₆₃ ▷ C₂×C₁₆₃ ▷ C₁₆₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×C₃₂

Complements

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

N	H ≅ G/N	type
5. C₃₂×C₁₆₃	C₄	semidirect
21. C₁₆₃	#1423	semidirect
27. C₃₂	Dic₁₆₃	semidirect

Maximal subgroups

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

subgroup	index
#23781491	2
C₂×C₃₂×C₁₆₃	2
#1423	163