Subgroups — #35212573

Order 20864.

112 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23781486	C₂	1	2	yes	maximal, complemented
2	#493×C₁₆₃	C₂	1	1	yes	maximal, characteristic
3	#19104675	—	2	2	no	—
4	C₂×D₈×C₁₆₃	—	2	2	no	—
5	#127×C₁₆₃	C₄	1	2	yes	—
6	C₄×C₈×C₁₆₃	C₂²	1	1	yes	characteristic
7	C₁₆₃⋊C₁₆	—	4	2	no	—
8	C₂×D₄×C₁₆₃	—	2	2	no	—
9	C₂×D₄×C₁₆₃	—	4	2	no	—
10	D₈×C₁₆₃	—	4	2	no	—
11	D₈×C₁₆₃	—	4	2	no	—
12	C₂×C₈×C₁₆₃	D₄	1	2	yes	—
13	C₄²×C₁₆₃	C₂×C₄	1	1	yes	characteristic
14	C₂³×C₁₆₃	—	4	2	no	—
15	D₄×C₁₆₃	—	4	4	no	—
16	D₄×C₁₆₃	—	8	4	no	—
17	C₂×C₄×C₁₆₃	—	2	1	no	—
18	C₂×C₄×C₁₆₃	C₂²⋊C₄	1	1	yes	characteristic
19	C₈×C₁₆₃	D₈	1	2	yes	—
20	C₈×C₁₆₃	SD₁₆	1	2	yes	—
21	C₂²×C₁₆₃	—	8	4	no	—
22	C₂²×C₁₆₃	—	8	2	no	—
23	C₂²×C₁₆₃	#100	1	1	yes	characteristic
24	C₄×C₁₆₃	—	2	2	no	—
25	C₄×C₁₆₃	#102	1	2	yes	—
26	C₂×C₁₆₃	—	2	1	no	—
27	C₂×C₁₆₃	—	16	2	no	—
28	C₂×C₁₆₃	#331	1	1	yes	characteristic
29	C₁₆₃	#1357	1	1	yes	characteristic, complemented, Sylow
30	#1357	—	163	1	no	maximal, Sylow
31	#493	D₁₆₃	1	1	yes	characteristic
32	#364	—	163	2	no	—
33	C₂×D₈	—	2	2	no	—
34	#127	Dic₁₆₃	1	2	yes	—
35	M₃₂	—	326	2	no	—
36	C₄×C₈	C₂×D₁₆₃	1	1	yes	characteristic
37	C₂×D₄	—	2	2	no	—
38	C₂×D₄	—	4	2	no	—
39	D₈	—	4	2	no	—
40	D₈	—	4	2	no	—
41	C₂×C₈	C₂²⋊D₁₆₃	1	2	yes	—
42	C₄²	C₂×Dic₁₆₃	1	1	yes	characteristic
43	C₁₆	—	652	2	no	—
44	C₂³	—	4	2	no	—
45	D₄	—	4	4	no	—
46	D₄	—	8	4	no	—
47	C₂×C₄	—	2	1	no	—
48	C₂×C₄	C₂²⋊Dic₁₆₃	1	1	yes	characteristic
49	C₈	#14579684	1	2	yes	—
50	C₈	#14579687	1	2	yes	—
51	C₂²	—	8	4	no	—
52	C₂²	—	8	2	no	—
53	C₂²	#19104652	1	1	yes	characteristic
54	C₄	—	2	2	no	—
55	C₄	#19104656	1	2	yes	—
56	C₂	—	2	1	no	—
57	C₂	—	16	2	no	—
58	C₂	#23781409	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#23781409
commutator	C₄²×C₁₆₃	C₂×C₄
Frattini	C₄×C₈	C₂×D₁₆₃
Fitting, prime radical	#493×C₁₆₃	C₂
socle	C₂×C₁₆₃	#331
cosocle	C₄×C₈×C₁₆₃	C₂²
hypercenter	#493	D₁₆₃
hypocenter	C₁₆₃	#1357
LCS	G ▷ C₄²×C₁₆₃ ▷ C₂×C₄×C₁₆₃ ▷ C₂×C₁₆₃ ▷ C₁₆₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #493

Complements

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

N	H ≅ G/N	type
1. #23781486	C₂	semidirect
29. C₁₆₃	#1357	semidirect

Maximal subgroups

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

subgroup	index
#23781486	2
#493×C₁₆₃	2
#1357	163