Subgroups — #35212598

Order 20864.

74 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23782067	C₂	1	1	yes	maximal, characteristic
2	#23781486	C₂	1	1	yes	maximal, characteristic
3	#364×C₁₆₃	C₂	1	1	yes	maximal, characteristic
4	#19104727	C₄	1	2	yes	—
5	#19104675	—	2	1	no	—
6	#19104669	—	2	1	no	—
7	#19104667	—	2	1	no	—
8	M₃₂×C₁₆₃	—	2	1	no	—
9	C₄×C₈×C₁₆₃	C₂²	1	1	yes	characteristic
10	C₄⋊Dic₁₆₃	—	2	2	no	—
11	Dic₁₆₃⋊C₄	—	4	2	no	—
12	C₁₆₃⋊C₁₆	—	4	1	no	—
13	C₂×C₈×C₁₆₃	D₄	1	1	yes	characteristic
14	C₂×C₈×C₁₆₃	D₄	1	1	yes	characteristic
15	C₄²×C₁₆₃	C₂×C₄	1	1	yes	characteristic
16	C₁₆×C₁₆₃	—	4	1	no	—
17	C₂×Dic₁₆₃	—	4	2	no	—
18	C₂×C₄×C₁₆₃	—	2	1	no	—
19	C₂×C₄×C₁₆₃	C₂²⋊C₄	1	1	yes	characteristic
20	C₈×C₁₆₃	SD₁₆	1	1	yes	characteristic
21	C₈×C₁₆₃	SD₁₆	1	1	yes	characteristic
22	C₈×C₁₆₃	Q₁₆	1	1	yes	characteristic
23	C₈×C₁₆₃	Q₁₆	1	1	yes	characteristic
24	C₂²×C₁₆₃	#100	1	1	yes	characteristic
25	C₄×C₁₆₃	—	4	1	no	—
26	C₄×C₁₆₃	#103	1	1	yes	characteristic
27	C₄×C₁₆₃	#103	1	1	yes	characteristic
28	Dic₁₆₃	—	8	2	no	—
29	C₂×C₁₆₃	—	2	1	no	—
30	C₂×C₁₆₃	#333	1	1	yes	characteristic
31	C₁₆₃	#1366	1	1	yes	characteristic, complemented, Sylow
32	#1366	—	163	1	no	maximal, Sylow
33	#499	—	163	1	no	—
34	#364	—	163	1	no	—
35	#364	D₁₆₃	1	1	yes	characteristic
36	#125	—	163	2	no	—
37	M₃₂	—	2	1	no	—
38	M₃₂	—	326	1	no	—
39	#107	—	326	1	no	—
40	#106	—	326	1	no	—
41	C₄×C₈	C₂×D₁₆₃	1	1	yes	characteristic
42	C₂×C₈	C₂²⋊D₁₆₃	1	1	yes	characteristic
43	C₂×C₈	D₆₅₂	1	1	yes	characteristic
44	C₄⋊C₄	—	326	2	no	—
45	C₄⋊C₄	—	652	2	no	—
46	C₄²	C₄×D₁₆₃	1	1	yes	characteristic
47	C₁₆	—	4	1	no	—
48	C₁₆	—	652	1	no	—
49	C₂×C₄	—	2	1	no	—
50	C₂×C₄	—	652	2	no	—
51	C₂×C₄	(C₂×D₁₆₃)⋊C₄	1	1	yes	characteristic
52	C₈	#14579687	1	1	yes	characteristic
53	C₈	#14579688	1	1	yes	characteristic
54	C₈	#14579689	1	1	yes	characteristic
55	C₈	Dic₆₅₂	1	1	yes	characteristic
56	C₂²	#19104651	1	1	yes	characteristic
57	C₄	—	4	1	no	—
58	C₄	—	1304	2	no	—
59	C₄	#19104658	1	1	yes	characteristic
60	C₄	#19104660	1	1	yes	characteristic
61	C₂	—	2	1	no	—
62	C₂	#23781413	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#23781413
commutator	C₄²×C₁₆₃	C₂×C₄
Frattini	C₄×C₈	C₂×D₁₆₃
Fitting	#364×C₁₆₃	C₂
socle	C₂×C₁₆₃	#333
prime radical	C₂²×C₁₆₃	#100
cosocle	C₄×C₈×C₁₆₃	C₂²
hypercenter	#364	D₁₆₃
hypocenter	C₁₆₃	#1366
LCS	G ▷ C₄²×C₁₆₃ ▷ C₂×C₄×C₁₆₃ ▷ C₂×C₁₆₃ ▷ C₁₆₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #364

Complements

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

N	H ≅ G/N	type
31. C₁₆₃	#1366	semidirect

Maximal subgroups

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

subgroup	index
#23782067	2
#23781486	2
#364×C₁₆₃	2
#1366	163