Subgroups — #35212716

Order 20864.

74 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₁₃₀₄	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₆₃⋊C₃₂	C₂	1	1	yes	maximal, characteristic
3	#366×C₁₆₃	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₆₅₂	—	2	1	no	—
5	Dic₁₃₀₄	—	2	1	no	—
6	Dic₁₃₀₄	C₄	1	2	yes	complemented
7	C₁₆₃⋊C₃₂	—	2	1	no	—
8	C₂×C₁₆×C₁₆₃	C₂²	1	1	yes	characteristic
9	#107×C₁₆₃	—	2	1	no	—
10	C₂×Dic₃₂₆	—	4	1	no	—
11	Dic₆₅₂	—	2	2	no	—
12	Dic₆₅₂	—	4	1	no	—
13	C₂×C₈×C₁₆₃	D₄	1	1	yes	characteristic
14	C₄⋊C₄×C₁₆₃	—	4	1	no	—
15	C₁₆×C₁₆₃	C₂×C₄	1	1	yes	characteristic
16	C₁₆×C₁₆₃	D₄	1	1	yes	characteristic
17	Dic₃₂₆	—	4	2	no	—
18	Dic₃₂₆	—	8	1	no	—
19	C₂×Dic₁₆₃	—	8	1	no	—
20	C₂×C₄×C₁₆₃	—	8	1	no	—
21	C₂×C₄×C₁₆₃	D₈	1	1	yes	characteristic
22	C₈×C₁₆₃	C₂²⋊C₄	1	1	yes	characteristic
23	C₈×C₁₆₃	SD₁₆	1	1	yes	characteristic
24	C₂²×C₁₆₃	D₁₆	1	1	yes	characteristic
25	C₄×C₁₆₃	—	16	1	no	—
26	C₄×C₁₆₃	#102	1	1	yes	characteristic
27	C₄×C₁₆₃	SD₃₂	1	1	yes	characteristic
28	Dic₁₆₃	—	8	2	no	—
29	C₂×C₁₆₃	#357	1	1	yes	characteristic
30	C₂×C₁₆₃	SD₆₄	1	1	yes	characteristic
31	C₂×C₁₆₃	Q₆₄	1	1	yes	characteristic
32	C₁₆₃	#1416	1	1	yes	characteristic, complemented, Sylow
33	#1416	—	163	1	no	maximal, Sylow
34	C₂×Q₃₂	—	163	1	no	—
35	C₂×C₃₂	—	163	1	no	—
36	#366	D₁₆₃	1	1	yes	characteristic
37	C₂×Q₁₆	—	326	1	no	—
38	Q₃₂	—	326	1	no	—
39	Q₃₂	—	163	2	no	—
40	C₂×C₁₆	C₂×D₁₆₃	1	1	yes	characteristic
41	#107	—	2	1	no	—
42	C₃₂	—	326	1	no	—
43	C₂×Q₈	—	652	1	no	—
44	Q₁₆	—	326	2	no	—
45	Q₁₆	—	652	1	no	—
46	C₂×C₈	C₂²⋊D₁₆₃	1	1	yes	characteristic
47	C₄⋊C₄	—	4	1	no	—
48	C₁₆	C₄×D₁₆₃	1	1	yes	characteristic
49	C₁₆	D₆₅₂	1	1	yes	characteristic
50	Q₈	—	1304	1	no	—
51	Q₈	—	652	2	no	—
52	C₂×C₄	—	8	1	no	—
53	C₂×C₄	—	1304	1	no	—
54	C₂×C₄	#14579684	1	1	yes	characteristic
55	C₈	(C₂×D₁₆₃)⋊C₄	1	1	yes	characteristic
56	C₈	#14579686	1	1	yes	characteristic
57	C₂²	#19104676	1	1	yes	characteristic
58	C₄	—	16	1	no	—
59	C₄	—	1304	2	no	—
60	C₄	#19104655	1	1	yes	characteristic
61	C₄	#19104678	1	1	yes	characteristic
62	C₂	#23781465	1	1	yes	characteristic
63	C₂	#23781503	1	1	yes	characteristic
64	C₂	#23781505	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	#1416
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
6. Dic₁₃₀₄	C₄	semidirect
32. C₁₆₃	#1416	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₁₃₀₄	2
C₂×C₁₆₃⋊C₃₂	2
#366×C₁₆₃	2
#1416	163