Subgroups — #35212679

Order 20864.

72 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23782081	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₆₃⋊C₃₂	C₂	1	1	yes	maximal, characteristic, complemented
3	M₆₄×C₁₆₃	C₂	1	1	yes	maximal, characteristic, complemented
4	#19104741	C₄	1	1	yes	characteristic
5	#19104675	C₄	1	1	yes	characteristic
6	#19104674	—	2	1	no	—
7	C₁₆×D₁₆₃	—	2	1	no	—
8	C₁₆₃⋊C₃₂	—	2	1	no	—
9	C₂×C₁₆×C₁₆₃	C₂²	1	1	yes	characteristic
10	C₃₂×C₁₆₃	—	2	1	no	—
11	#14579699	C₈	1	1	yes	characteristic
12	#14579683	C₈	1	1	yes	characteristic
13	#14579682	—	2	1	no	—
14	C₈×D₁₆₃	—	2	1	no	—
15	C₁₆₃⋊C₁₆	—	2	1	no	—
16	C₂×C₈×C₁₆₃	C₂×C₄	1	1	yes	characteristic
17	C₁₆×C₁₆₃	D₄	1	1	yes	characteristic
18	C₁₆×C₁₆₃	D₄	1	1	yes	characteristic
19	Dic₃₂₆	C₁₆	1	1	yes	characteristic
20	D₆₅₂	C₁₆	1	1	yes	characteristic
21	C₂²⋊D₁₆₃	—	2	1	no	—
22	C₄×D₁₆₃	—	2	1	no	—
23	C₁₆₃⋊C₈	—	2	1	no	—
24	C₂×C₄×C₁₆₃	C₂×C₈	1	1	yes	characteristic
25	C₈×C₁₆₃	C₂²⋊C₄	1	1	yes	characteristic
26	C₈×C₁₆₃	M₁₆	1	1	yes	characteristic
27	C₂²×C₁₆₃	M₃₂	1	1	yes	characteristic
28	C₂×D₁₆₃	—	2	1	no	—
29	C₄×C₁₆₃	C₂²⋊C₈	1	1	yes	characteristic
30	C₄×C₁₆₃	C₂×C₁₆	1	1	yes	characteristic
31	Dic₁₆₃	—	2	1	no	—
32	C₂×C₁₆₃	—	2	1	no	—
33	C₂×C₁₆₃	C₂²⋊C₁₆	1	1	yes	characteristic
34	D₁₆₃	—	4	1	no	—
35	C₁₆₃	#1401	1	1	yes	characteristic, complemented, Sylow
36	#1401	—	163	1	no	maximal, Sylow
37	#504	—	163	1	no	—
38	M₆₄	D₁₆₃	1	1	yes	characteristic, complemented
39	C₂×C₃₂	—	163	1	no	—
40	#131	—	163	1	no	—
41	M₃₂	—	326	1	no	—
42	M₃₂	—	163	1	no	—
43	C₂×C₁₆	—	326	1	no	—
44	C₂×C₁₆	C₂×D₁₆₃	1	1	yes	characteristic
45	C₃₂	—	2	1	no	—
46	C₃₂	—	326	1	no	—
47	#41	—	163	1	no	—
48	M₁₆	—	326	1	no	—
49	M₁₆	—	163	1	no	—
50	C₂×C₈	—	326	1	no	—
51	C₂×C₈	C₄×D₁₆₃	1	1	yes	characteristic
52	C₁₆	—	326	1	no	—
53	C₁₆	C₂²⋊D₁₆₃	1	1	yes	characteristic
54	C₁₆	D₆₅₂	1	1	yes	characteristic
55	Q₈	—	163	1	no	—
56	D₄	—	326	1	no	—
57	D₄	—	163	1	no	—
58	C₂×C₄	—	326	1	no	—
59	C₂×C₄	C₈×D₁₆₃	1	1	yes	characteristic
60	C₈	—	326	1	no	—
61	C₈	(C₂×D₁₆₃)⋊C₄	1	1	yes	characteristic
62	C₈	#14579682	1	1	yes	characteristic
63	C₂²	—	326	1	no	—
64	C₂²	#19104674	1	1	yes	characteristic
65	C₄	—	326	1	no	—
66	C₄	(C₂×D₁₆₃)⋊C₈	1	1	yes	characteristic
67	C₄	C₁₆×D₁₆₃	1	1	yes	characteristic
68	C₂	—	2	1	no	—
69	C₂	—	652	1	no	—
70	C₂	(C₂×D₁₆₃)⋊C₁₆	1	1	yes	characteristic

	subgroup	G/H
center	C₁₆	C₂²⋊D₁₆₃
commutator	C₄×C₁₆₃	C₂×C₁₆
Frattini	C₂×C₁₆	C₂×D₁₆₃
Fitting	M₆₄×C₁₆₃	C₂
socle	C₂×C₁₆₃	C₂²⋊C₁₆
prime radical	#14579699	C₈
cosocle	C₂×C₁₆×C₁₆₃	C₂²
hypercenter	M₆₄	D₁₆₃
hypocenter	C₁₆₃	#1401
LCS	G ▷ C₄×C₁₆₃ ▷ C₂×C₁₆₃ ▷ C₁₆₃
UCS	C₁ ◁ C₁₆ ◁ C₂×C₁₆ ◁ M₆₄

Complements

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

N	H ≅ G/N	type
2. C₂×C₁₆₃⋊C₃₂	C₂	semidirect
3. M₆₄×C₁₆₃	C₂	semidirect
35. C₁₆₃	#1401	semidirect
38. M₆₄	D₁₆₃	semidirect

Maximal subgroups

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

subgroup	index
#23782081	2
C₂×C₁₆₃⋊C₃₂	2
M₆₄×C₁₆₃	2
#1401	163