Subgroups — #35212575

Order 20864.

92 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23781486	C₂	1	2	yes	maximal
2	#494×C₁₆₃	C₂	1	1	yes	maximal, characteristic
3	#19104675	—	2	2	no	—
4	C₂×Q₁₆×C₁₆₃	—	2	2	no	—
5	C₄⋊Q₈×C₁₆₃	C₄	1	2	yes	—
6	C₄×C₈×C₁₆₃	C₂²	1	1	yes	characteristic
7	C₁₆₃⋊C₁₆	—	4	2	no	—
8	C₂×Q₈×C₁₆₃	—	2	2	no	—
9	Q₁₆×C₁₆₃	—	4	2	no	—
10	Q₁₆×C₁₆₃	—	4	2	no	—
11	C₂×C₈×C₁₆₃	D₄	1	2	yes	—
12	C₄⋊C₄×C₁₆₃	—	4	2	no	—
13	C₄²×C₁₆₃	C₂×C₄	1	1	yes	characteristic
14	Q₈×C₁₆₃	—	4	4	no	—
15	C₂×C₄×C₁₆₃	—	2	1	no	—
16	C₂×C₄×C₁₆₃	—	4	2	no	—
17	C₂×C₄×C₁₆₃	C₂²⋊C₄	1	1	yes	characteristic
18	C₈×C₁₆₃	D₈	1	2	yes	—
19	C₈×C₁₆₃	SD₁₆	1	2	yes	—
20	C₂²×C₁₆₃	#100	1	1	yes	characteristic
21	C₄×C₁₆₃	—	2	2	no	—
22	C₄×C₁₆₃	—	8	2	no	—
23	C₄×C₁₆₃	#102	1	2	yes	—
24	C₂×C₁₆₃	—	2	1	no	—
25	C₂×C₁₆₃	#331	1	1	yes	characteristic
26	C₁₆₃	#1358	1	1	yes	characteristic, complemented, Sylow
27	#1358	—	163	1	no	maximal, Sylow
28	#494	D₁₆₃	1	1	yes	characteristic
29	#364	—	163	2	no	—
30	C₂×Q₁₆	—	2	2	no	—
31	C₄⋊Q₈	Dic₁₆₃	1	2	yes	—
32	M₃₂	—	326	2	no	—
33	C₄×C₈	C₂×D₁₆₃	1	1	yes	characteristic
34	C₂×Q₈	—	2	2	no	—
35	Q₁₆	—	4	2	no	—
36	Q₁₆	—	4	2	no	—
37	C₂×C₈	C₂²⋊D₁₆₃	1	2	yes	—
38	C₄⋊C₄	—	4	2	no	—
39	C₄²	C₂×Dic₁₆₃	1	1	yes	characteristic
40	C₁₆	—	652	2	no	—
41	Q₈	—	4	4	no	—
42	C₂×C₄	—	2	1	no	—
43	C₂×C₄	—	4	2	no	—
44	C₂×C₄	C₂²⋊Dic₁₆₃	1	1	yes	characteristic
45	C₈	#14579684	1	2	yes	—
46	C₈	#14579687	1	2	yes	—
47	C₂²	#19104652	1	1	yes	characteristic
48	C₄	—	2	2	no	—
49	C₄	—	8	2	no	—
50	C₄	#19104656	1	2	yes	—
51	C₂	—	2	1	no	—
52	C₂	#23781409	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#23781409
commutator	C₄²×C₁₆₃	C₂×C₄
Frattini	C₄×C₈	C₂×D₁₆₃
Fitting	#494×C₁₆₃	C₂
socle	C₂×C₁₆₃	#331
prime radical	C₂²×C₁₆₃	#100
cosocle	C₄×C₈×C₁₆₃	C₂²
hypercenter	#494	D₁₆₃
hypocenter	C₁₆₃	#1358
LCS	G ▷ C₄²×C₁₆₃ ▷ C₂×C₄×C₁₆₃ ▷ C₂×C₁₆₃ ▷ C₁₆₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #494

Complements

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

N	H ≅ G/N	type
26. C₁₆₃	#1358	semidirect

Maximal subgroups

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

subgroup	index
#23781486	2
#494×C₁₆₃	2
#1358	163