Subgroups — #35212734

Order 20864.

58 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23781499	C₂	1	2	yes	maximal
2	C₄×C₁₆×C₁₆₃	C₂	1	1	yes	maximal, characteristic
3	C₁₆₃⋊C₃₂	—	2	2	no	—
4	C₂×C₁₆×C₁₆₃	C₄	1	1	yes	characteristic
5	C₂×C₁₆×C₁₆₃	C₂²	1	1	yes	characteristic
6	C₄×C₈×C₁₆₃	C₄	1	1	yes	characteristic
7	C₂×C₈×C₁₆₃	C₈	1	1	yes	characteristic
8	C₂×C₈×C₁₆₃	C₂×C₄	1	1	yes	characteristic
9	C₄²×C₁₆₃	C₈	1	1	yes	characteristic
10	C₁₆×C₁₆₃	—	2	1	no	—
11	C₁₆×C₁₆₃	D₄	1	1	yes	characteristic
12	C₁₆×C₁₆₃	Q₈	1	1	yes	characteristic
13	C₂×C₄×C₁₆₃	—	2	1	no	—
14	C₂×C₄×C₁₆₃	C₂×C₈	1	1	yes	characteristic
15	C₈×C₁₆₃	C₁₆	1	2	yes	—
16	C₈×C₁₆₃	C₄⋊C₄	1	1	yes	characteristic
17	C₈×C₁₆₃	M₁₆	1	1	yes	characteristic
18	C₂²×C₁₆₃	M₃₂	1	1	yes	characteristic
19	C₄×C₁₆₃	—	2	2	no	—
20	C₄×C₁₆₃	C₄⋊C₈	1	1	yes	characteristic
21	C₄×C₁₆₃	C₂×C₁₆	1	1	yes	characteristic
22	C₂×C₁₆₃	—	2	1	no	—
23	C₂×C₁₆₃	C₄⋊C₁₆	1	1	yes	characteristic
24	C₁₆₃	#1422	1	1	yes	characteristic, complemented, Sylow
25	#1422	—	163	1	no	maximal, Sylow
26	M₆₄	—	163	2	no	—
27	C₄×C₁₆	D₁₆₃	1	1	yes	characteristic
28	C₂×C₁₆	Dic₁₆₃	1	1	yes	characteristic
29	C₂×C₁₆	C₂×D₁₆₃	1	1	yes	characteristic
30	C₄×C₈	Dic₁₆₃	1	1	yes	characteristic
31	C₃₂	—	326	2	no	—
32	C₂×C₈	C₁₆₃⋊C₈	1	1	yes	characteristic
33	C₂×C₈	C₂×Dic₁₆₃	1	1	yes	characteristic
34	C₄²	C₁₆₃⋊C₈	1	1	yes	characteristic
35	C₁₆	—	2	1	no	—
36	C₁₆	D₆₅₂	1	1	yes	characteristic
37	C₁₆	Dic₃₂₆	1	1	yes	characteristic
38	C₂×C₄	—	2	1	no	—
39	C₂×C₄	C₂×C₁₆₃⋊C₈	1	1	yes	characteristic
40	C₈	C₁₆₃⋊C₁₆	1	2	yes	—
41	C₈	C₄⋊Dic₁₆₃	1	1	yes	characteristic
42	C₈	#14579683	1	1	yes	characteristic
43	C₂²	#19104675	1	1	yes	characteristic
44	C₄	—	2	2	no	—
45	C₄	C₄⋊(C₁₆₃⋊C₈)	1	1	yes	characteristic
46	C₄	C₂×C₁₆₃⋊C₁₆	1	1	yes	characteristic
47	C₂	—	2	1	no	—
48	C₂	C₄⋊(C₁₆₃⋊C₁₆)	1	1	yes	characteristic

	subgroup	G/H
center	C₁₆	D₆₅₂
commutator	C₄×C₁₆₃	C₂×C₁₆
Frattini	C₂×C₁₆	C₂×D₁₆₃
Fitting	C₄×C₁₆×C₁₆₃	C₂
socle	C₂×C₁₆₃	C₄⋊C₁₆
prime radical	C₂²×C₁₆₃	M₃₂
cosocle	C₂×C₁₆×C₁₆₃	C₂²
hypercenter	C₄×C₁₆	D₁₆₃
hypocenter	C₁₆₃	#1422
LCS	G ▷ C₄×C₁₆₃ ▷ C₂×C₁₆₃ ▷ C₁₆₃
UCS	C₁ ◁ C₁₆ ◁ C₂×C₁₆ ◁ C₄×C₁₆

Complements

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

N	H ≅ G/N	type
24. C₁₆₃	#1422	semidirect

Maximal subgroups

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

subgroup	index
#23781499	2
C₄×C₁₆×C₁₆₃	2
#1422	163