Subgroups — #35212678

Order 20864.

72 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23781499	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₁₆₃⋊C₃₂	C₂	1	1	yes	maximal, characteristic, complemented
3	#504×C₁₆₃	C₂	1	1	yes	maximal, characteristic
4	C₁₆₃⋊C₃₂	—	2	1	no	—
5	C₁₆₃⋊C₃₂	—	2	1	no	—
6	#131×C₁₆₃	C₄	1	1	yes	characteristic
7	M₃₂×C₁₆₃	—	2	1	no	—
8	M₃₂×C₁₆₃	C₄	1	1	yes	characteristic
9	C₂×C₁₆×C₁₆₃	—	2	1	no	—
10	C₂×C₁₆×C₁₆₃	C₂²	1	1	yes	characteristic
11	#41×C₁₆₃	C₈	1	1	yes	characteristic
12	M₁₆×C₁₆₃	—	2	1	no	—
13	M₁₆×C₁₆₃	C₈	1	1	yes	characteristic
14	C₂×C₈×C₁₆₃	—	2	1	no	—
15	C₂×C₈×C₁₆₃	C₂×C₄	1	1	yes	characteristic
16	C₁₆×C₁₆₃	—	2	1	no	—
17	C₁₆×C₁₆₃	D₄	1	1	yes	characteristic
18	C₁₆×C₁₆₃	D₄	1	1	yes	characteristic
19	Q₈×C₁₆₃	C₁₆	1	1	yes	characteristic
20	D₄×C₁₆₃	—	2	1	no	—
21	D₄×C₁₆₃	C₁₆	1	1	yes	characteristic
22	C₂×C₄×C₁₆₃	—	2	1	no	—
23	C₂×C₄×C₁₆₃	C₂×C₈	1	1	yes	characteristic
24	C₈×C₁₆₃	—	2	1	no	—
25	C₈×C₁₆₃	C₂²⋊C₄	1	1	yes	characteristic
26	C₈×C₁₆₃	M₁₆	1	1	yes	characteristic
27	C₂²×C₁₆₃	—	2	1	no	—
28	C₂²×C₁₆₃	M₃₂	1	1	yes	characteristic
29	C₄×C₁₆₃	—	2	1	no	—
30	C₄×C₁₆₃	C₂²⋊C₈	1	1	yes	characteristic
31	C₄×C₁₆₃	C₂×C₁₆	1	1	yes	characteristic
32	C₂×C₁₆₃	—	2	1	no	—
33	C₂×C₁₆₃	—	4	1	no	—
34	C₂×C₁₆₃	C₂²⋊C₁₆	1	1	yes	characteristic
35	C₁₆₃	#1401	1	1	yes	characteristic, complemented, Sylow
36	#1401	—	163	1	no	maximal, Sylow
37	#504	D₁₆₃	1	1	yes	characteristic
38	M₆₄	—	163	1	no	—
39	C₂×C₃₂	—	163	1	no	—
40	#131	Dic₁₆₃	1	1	yes	characteristic
41	M₃₂	—	2	1	no	—
42	M₃₂	Dic₁₆₃	1	1	yes	characteristic
43	C₂×C₁₆	—	2	1	no	—
44	C₂×C₁₆	C₂×D₁₆₃	1	1	yes	characteristic
45	C₃₂	—	326	1	no	—
46	C₃₂	—	326	1	no	—
47	#41	C₁₆₃⋊C₈	1	1	yes	characteristic
48	M₁₆	—	2	1	no	—
49	M₁₆	C₁₆₃⋊C₈	1	1	yes	characteristic
50	C₂×C₈	—	2	1	no	—
51	C₂×C₈	C₂×Dic₁₆₃	1	1	yes	characteristic
52	C₁₆	—	2	1	no	—
53	C₁₆	C₂²⋊D₁₆₃	1	1	yes	characteristic
54	C₁₆	C₂²⋊D₁₆₃	1	1	yes	characteristic
55	Q₈	C₁₆₃⋊C₁₆	1	1	yes	characteristic
56	D₄	—	2	1	no	—
57	D₄	C₁₆₃⋊C₁₆	1	1	yes	characteristic
58	C₂×C₄	—	2	1	no	—
59	C₂×C₄	C₂×C₁₆₃⋊C₈	1	1	yes	characteristic
60	C₈	—	2	1	no	—
61	C₈	C₂²⋊Dic₁₆₃	1	1	yes	characteristic
62	C₈	#14579683	1	1	yes	characteristic
63	C₂²	—	2	1	no	—
64	C₂²	#19104675	1	1	yes	characteristic
65	C₄	—	2	1	no	—
66	C₄	C₂²⋊(C₁₆₃⋊C₈)	1	1	yes	characteristic
67	C₄	C₂×C₁₆₃⋊C₁₆	1	1	yes	characteristic
68	C₂	—	2	1	no	—
69	C₂	—	4	1	no	—
70	C₂	C₂²⋊(C₁₆₃⋊C₁₆)	1	1	yes	characteristic

	subgroup	G/H
center	C₁₆	C₂²⋊D₁₆₃
commutator	C₄×C₁₆₃	C₂×C₁₆
Frattini	C₂×C₁₆	C₂×D₁₆₃
Fitting	#504×C₁₆₃	C₂
socle	C₂×C₁₆₃	C₂²⋊C₁₆
prime radical	#41×C₁₆₃	C₈
cosocle	C₂×C₁₆×C₁₆₃	C₂²
hypercenter	#504	D₁₆₃
hypocenter	C₁₆₃	#1401
LCS	G ▷ C₄×C₁₆₃ ▷ C₂×C₁₆₃ ▷ C₁₆₃
UCS	C₁ ◁ C₁₆ ◁ C₂×C₁₆ ◁ #504

Complements

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

N	H ≅ G/N	type
1. #23781499	C₂	semidirect
2. C₂×C₁₆₃⋊C₃₂	C₂	semidirect
35. C₁₆₃	#1401	semidirect

Maximal subgroups

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

subgroup	index
#23781499	2
C₂×C₁₆₃⋊C₃₂	2
#504×C₁₆₃	2
#1401	163