Subgroups — #35212608

Order 20864.

70 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23781418	C₂	1	2	yes	maximal
2	C₄×C₁₆×C₁₆₃	C₂	1	1	yes	maximal, characteristic
3	C₄⋊(C₁₆₃⋊C₈)	—	2	2	no	—
4	C₂×C₁₆×C₁₆₃	C₄	1	1	yes	characteristic
5	C₂×C₁₆×C₁₆₃	C₄	1	1	yes	characteristic
6	C₄×C₈×C₁₆₃	C₂²	1	1	yes	characteristic
7	C₂×C₁₆₃⋊C₈	—	4	2	no	—
8	C₂×C₈×C₁₆₃	C₂×C₄	1	1	yes	characteristic
9	C₂×C₈×C₁₆₃	Q₈	1	1	yes	characteristic
10	C₄²×C₁₆₃	D₄	1	1	yes	characteristic
11	C₁₆×C₁₆₃	—	2	1	no	—
12	C₁₆×C₁₆₃	C₈	1	2	yes	complemented
13	C₁₆₃⋊C₈	—	8	2	no	—
14	C₂×C₄×C₁₆₃	C₄⋊C₄	1	1	yes	characteristic
15	C₂×C₄×C₁₆₃	D₈	1	1	yes	characteristic
16	C₂×C₄×C₁₆₃	Q₁₆	1	1	yes	characteristic
17	C₈×C₁₆₃	—	2	1	no	—
18	C₈×C₁₆₃	C₂×C₈	1	1	yes	characteristic
19	C₈×C₁₆₃	M₁₆	1	1	yes	characteristic
20	C₂²×C₁₆₃	#107	1	1	yes	characteristic
21	C₄×C₁₆₃	—	2	1	no	—
22	C₄×C₁₆₃	C₄⋊C₈	1	1	yes	characteristic
23	C₄×C₁₆₃	#108	1	1	yes	characteristic
24	C₄×C₁₆₃	D₁₆	1	1	yes	characteristic
25	C₄×C₁₆₃	Q₃₂	1	1	yes	characteristic
26	C₂×C₁₆₃	#335	1	1	yes	characteristic
27	C₂×C₁₆₃	#366	1	1	yes	characteristic
28	C₂×C₁₆₃	#368	1	1	yes	characteristic
29	C₁₆₃	#1371	1	1	yes	characteristic, complemented, Sylow
30	#1371	—	163	1	no	maximal, Sylow
31	C₄×C₁₆	D₁₆₃	1	1	yes	characteristic
32	#335	—	163	2	no	—
33	C₂×C₁₆	Dic₁₆₃	1	1	yes	characteristic
34	C₂×C₁₆	Dic₁₆₃	1	1	yes	characteristic
35	C₄⋊C₈	—	326	2	no	—
36	C₄×C₈	C₂×D₁₆₃	1	1	yes	characteristic
37	C₂×C₈	—	652	2	no	—
38	C₂×C₈	C₂×Dic₁₆₃	1	1	yes	characteristic
39	C₂×C₈	Dic₃₂₆	1	1	yes	characteristic
40	C₄²	D₆₅₂	1	1	yes	characteristic
41	C₁₆	—	2	1	no	—
42	C₁₆	C₁₆₃⋊C₈	1	2	yes	complemented
43	C₂×C₄	C₄⋊Dic₁₆₃	1	1	yes	characteristic
44	C₂×C₄	D₁₃₀₄	1	1	yes	characteristic
45	C₂×C₄	Dic₆₅₂	1	1	yes	characteristic
46	C₈	—	2	1	no	—
47	C₈	—	1304	2	no	—
48	C₈	C₂×C₁₆₃⋊C₈	1	1	yes	characteristic
49	C₈	#14579683	1	1	yes	characteristic
50	C₂²	#19104669	1	1	yes	characteristic
51	C₄	—	2	1	no	—
52	C₄	C₄⋊(C₁₆₃⋊C₈)	1	1	yes	characteristic
53	C₄	#19104671	1	1	yes	characteristic
54	C₄	D₂₆₀₈	1	1	yes	characteristic
55	C₄	Dic₁₃₀₄	1	1	yes	characteristic
56	C₂	#23781418	1	1	yes	characteristic
57	C₂	#23781491	1	1	yes	characteristic
58	C₂	#23781495	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₁₃₀₄
commutator	C₈×C₁₆₃	C₂×C₈
Frattini	C₄×C₈	C₂×D₁₆₃
Fitting	C₄×C₁₆×C₁₆₃	C₂
socle, prime radical	C₂²×C₁₆₃	#107
cosocle	C₄×C₈×C₁₆₃	C₂²
hypercenter	C₄×C₁₆	D₁₆₃
hypocenter	C₁₆₃	#1371
LCS	G ▷ C₈×C₁₆₃ ▷ C₄×C₁₆₃ ▷ C₂×C₁₆₃ ▷ C₁₆₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆

Complements

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

N	H ≅ G/N	type
12. C₁₆×C₁₆₃	C₈	semidirect
29. C₁₆₃	#1371	semidirect
42. C₁₆	C₁₆₃⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
#23781418	2
C₄×C₁₆×C₁₆₃	2
#1371	163