Subgroups — #35212720

Order 20864.

72 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₆₃⋊C₃₂	C₂	1	1	yes	maximal, characteristic, complemented
2	#23781495	C₂	1	1	yes	maximal, characteristic, complemented
3	#508×C₁₆₃	C₂	1	1	yes	maximal, characteristic
4	#19104671	—	2	1	no	—
5	C₁₆₃⋊C₃₂	—	2	1	no	—
6	#135×C₁₆₃	—	2	1	no	—
7	Q₃₂×C₁₆₃	C₄	1	1	yes	characteristic
8	SD₃₂×C₁₆₃	—	2	1	no	—
9	D₁₆×C₁₆₃	C₄	1	1	yes	characteristic
10	C₂×C₁₆×C₁₆₃	C₂²	1	1	yes	characteristic
11	#14579683	—	4	1	no	—
12	#41×C₁₆₃	—	4	1	no	—
13	Q₁₆×C₁₆₃	—	2	1	no	—
14	SD₁₆×C₁₆₃	—	4	1	no	—
15	D₈×C₁₆₃	—	2	1	no	—
16	C₂×C₈×C₁₆₃	D₄	1	1	yes	characteristic
17	C₁₆×C₁₆₃	C₂×C₄	1	1	yes	characteristic
18	C₁₆×C₁₆₃	D₄	1	1	yes	characteristic
19	C₁₆₃⋊C₈	—	8	1	no	—
20	Q₈×C₁₆₃	—	4	1	no	—
21	D₄×C₁₆₃	—	4	1	no	—
22	D₄×C₁₆₃	—	8	1	no	—
23	C₂×C₄×C₁₆₃	—	8	1	no	—
24	C₂×C₄×C₁₆₃	D₈	1	1	yes	characteristic
25	C₈×C₁₆₃	C₂²⋊C₄	1	1	yes	characteristic
26	C₈×C₁₆₃	SD₁₆	1	1	yes	characteristic
27	C₂²×C₁₆₃	—	8	1	no	—
28	C₂²×C₁₆₃	SD₃₂	1	1	yes	characteristic
29	C₄×C₁₆₃	—	8	1	no	—
30	C₄×C₁₆₃	#102	1	1	yes	characteristic
31	C₄×C₁₆₃	D₁₆	1	1	yes	characteristic
32	C₂×C₁₆₃	—	2	1	no	—
33	C₂×C₁₆₃	—	16	1	no	—
34	C₂×C₁₆₃	#357	1	1	yes	characteristic
35	C₁₆₃	#1417	1	1	yes	characteristic, complemented, Sylow
36	#1417	—	163	1	no	maximal, Sylow
37	#508	D₁₆₃	1	1	yes	characteristic
38	C₂×C₃₂	—	163	1	no	—
39	#368	—	163	1	no	—
40	#135	—	2	1	no	—
41	Q₃₂	Dic₁₆₃	1	1	yes	characteristic
42	SD₃₂	—	2	1	no	—
43	D₁₆	Dic₁₆₃	1	1	yes	characteristic
44	C₂×C₁₆	C₂×D₁₆₃	1	1	yes	characteristic
45	#108	—	326	1	no	—
46	C₃₂	—	326	1	no	—
47	#41	—	4	1	no	—
48	Q₁₆	—	2	1	no	—
49	SD₁₆	—	4	1	no	—
50	D₈	—	2	1	no	—
51	M₁₆	—	652	1	no	—
52	C₂×C₈	C₂²⋊D₁₆₃	1	1	yes	characteristic
53	C₁₆	C₂×Dic₁₆₃	1	1	yes	characteristic
54	C₁₆	C₂²⋊D₁₆₃	1	1	yes	characteristic
55	Q₈	—	4	1	no	—
56	D₄	—	4	1	no	—
57	D₄	—	8	1	no	—
58	C₂×C₄	—	8	1	no	—
59	C₂×C₄	#14579684	1	1	yes	characteristic
60	C₈	—	1304	1	no	—
61	C₈	C₂²⋊Dic₁₆₃	1	1	yes	characteristic
62	C₈	#14579687	1	1	yes	characteristic
63	C₂²	—	8	1	no	—
64	C₂²	#19104679	1	1	yes	characteristic
65	C₄	—	8	1	no	—
66	C₄	#19104656	1	1	yes	characteristic
67	C₄	#19104676	1	1	yes	characteristic
68	C₂	—	2	1	no	—
69	C₂	—	16	1	no	—
70	C₂	#23781466	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#19104676
commutator	C₁₆×C₁₆₃	C₂×C₄
Frattini	C₂×C₁₆	C₂×D₁₆₃
Fitting, prime radical	#508×C₁₆₃	C₂
socle	C₂×C₁₆₃	#357
cosocle	C₂×C₁₆×C₁₆₃	C₂²
hypercenter	#508	D₁₆₃
hypocenter	C₁₆₃	#1417
LCS	G ▷ C₁₆×C₁₆₃ ▷ C₈×C₁₆₃ ▷ C₄×C₁₆₃ ▷ C₂×C₁₆₃ ▷ C₁₆₃
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #508

Complements

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

N	H ≅ G/N	type
1. C₂×C₁₆₃⋊C₃₂	C₂	semidirect
2. #23781495	C₂	semidirect
35. C₁₆₃	#1417	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₁₆₃⋊C₃₂	2
#23781495	2
#508×C₁₆₃	2
#1417	163