Subgroups — #35212726

Order 20864.

78 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	#23781495	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×D₁₆×C₁₆₃	C₂	1	1	yes	maximal, characteristic
4	#19104671	—	2	1	no	—
5	C₁₆₃⋊C₃₂	—	2	1	no	—
6	C₂×D₈×C₁₆₃	—	2	1	no	—
7	D₁₆×C₁₆₃	—	2	1	no	—
8	D₁₆×C₁₆₃	C₄	1	2	yes	—
9	C₂×C₁₆×C₁₆₃	C₂²	1	1	yes	characteristic
10	#14579683	—	4	1	no	—
11	C₂×D₄×C₁₆₃	—	4	1	no	—
12	D₈×C₁₆₃	—	2	2	no	—
13	D₈×C₁₆₃	—	4	1	no	—
14	C₂×C₈×C₁₆₃	D₄	1	1	yes	characteristic
15	C₁₆×C₁₆₃	C₂×C₄	1	1	yes	characteristic
16	C₁₆×C₁₆₃	D₄	1	1	yes	characteristic
17	C₁₆₃⋊C₈	—	8	1	no	—
18	C₂³×C₁₆₃	—	8	1	no	—
19	D₄×C₁₆₃	—	4	2	no	—
20	D₄×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	—	8	2	no	—
25	C₂²×C₁₆₃	—	16	2	no	—
26	C₂²×C₁₆₃	SD₃₂	1	1	yes	characteristic
27	C₄×C₁₆₃	#102	1	1	yes	characteristic
28	C₄×C₁₆₃	D₁₆	1	1	yes	characteristic
29	C₂×C₁₆₃	—	2	1	no	—
30	C₂×C₁₆₃	—	16	2	no	—
31	C₂×C₁₆₃	#357	1	1	yes	characteristic
32	C₁₆₃	#1419	1	1	yes	characteristic, complemented, Sylow
33	#1419	—	163	1	no	maximal, Sylow
34	C₂×D₁₆	D₁₆₃	1	1	yes	characteristic
35	M₆₄	—	163	1	no	—
36	#368	—	163	1	no	—
37	C₂×D₈	—	2	1	no	—
38	D₁₆	—	2	1	no	—
39	D₁₆	Dic₁₆₃	1	2	yes	—
40	C₂×C₁₆	C₂×D₁₆₃	1	1	yes	characteristic
41	#108	—	326	1	no	—
42	C₃₂	—	326	1	no	—
43	C₂×D₄	—	4	1	no	—
44	D₈	—	2	2	no	—
45	D₈	—	4	1	no	—
46	M₁₆	—	652	1	no	—
47	C₂×C₈	C₂²⋊D₁₆₃	1	1	yes	characteristic
48	C₁₆	C₂×Dic₁₆₃	1	1	yes	characteristic
49	C₁₆	C₂²⋊D₁₆₃	1	1	yes	characteristic
50	C₂³	—	8	1	no	—
51	D₄	—	4	2	no	—
52	D₄	—	8	1	no	—
53	C₂×C₄	#14579684	1	1	yes	characteristic
54	C₈	—	1304	1	no	—
55	C₈	C₂²⋊Dic₁₆₃	1	1	yes	characteristic
56	C₈	#14579687	1	1	yes	characteristic
57	C₂²	—	8	2	no	—
58	C₂²	—	16	2	no	—
59	C₂²	#19104679	1	1	yes	characteristic
60	C₄	#19104656	1	1	yes	characteristic
61	C₄	#19104676	1	1	yes	characteristic
62	C₂	—	2	1	no	—
63	C₂	—	16	2	no	—
64	C₂	#23781466	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#23781466
commutator	C₁₆×C₁₆₃	C₂×C₄
Frattini	C₂×C₁₆	C₂×D₁₆₃
Fitting, prime radical	C₂×D₁₆×C₁₆₃	C₂
socle	C₂×C₁₆₃	#357
cosocle	C₂×C₁₆×C₁₆₃	C₂²
hypercenter	C₂×D₁₆	D₁₆₃
hypocenter	C₁₆₃	#1419
LCS	G ▷ C₁₆×C₁₆₃ ▷ C₈×C₁₆₃ ▷ C₄×C₁₆₃ ▷ C₂×C₁₆₃ ▷ C₁₆₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×D₁₆

Complements

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

N	H ≅ G/N	type
1. #23781499	C₂	semidirect
2. #23781495	C₂	semidirect
32. C₁₆₃	#1419	semidirect

Maximal subgroups

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

subgroup	index
#23781499	2
#23781495	2
C₂×D₁₆×C₁₆₃	2
#1419	163