Subgroups — #23781408

Order 10432.

84 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19104732	C₂	1	1	yes	maximal, characteristic
2	C₄⋊(C₁₆₃⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄⋊C₈×C₁₆₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₆₅₂	—	2	1	no	—
5	C₂×D₆₅₂	—	2	1	no	—
6	C₂×D₆₅₂	C₄	1	2	yes	—
7	C₂×C₁₆₃⋊C₈	—	2	1	no	—
8	C₂×C₈×C₁₆₃	—	2	1	no	—
9	C₄²×C₁₆₃	C₂²	1	1	yes	characteristic
10	C₂²×D₁₆₃	—	2	2	no	—
11	D₆₅₂	—	4	2	no	—
12	D₆₅₂	—	4	1	no	—
13	D₆₅₂	—	4	1	no	—
14	D₆₅₂	—	4	1	no	—
15	D₆₅₂	—	4	1	no	—
16	C₁₆₃⋊C₈	—	4	1	no	—
17	C₂×C₄×C₁₆₃	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×C₁₆₃	D₄	1	1	yes	characteristic
19	C₂×C₄×C₁₆₃	D₄	1	1	yes	characteristic
20	C₈×C₁₆₃	—	4	1	no	—
21	C₂²×C₁₆₃	C₂²⋊C₄	1	1	yes	characteristic
22	C₂×D₁₆₃	—	4	2	no	—
23	C₂×D₁₆₃	—	4	2	no	—
24	C₂×D₁₆₃	—	4	2	no	—
25	C₄×C₁₆₃	—	2	1	no	—
26	C₄×C₁₆₃	D₈	1	1	yes	characteristic
27	C₄×C₁₆₃	D₈	1	1	yes	characteristic
28	C₄×C₁₆₃	SD₁₆	1	1	yes	characteristic
29	C₄×C₁₆₃	SD₁₆	1	1	yes	characteristic
30	C₂×C₁₆₃	#100	1	1	yes	characteristic
31	C₂×C₁₆₃	#102	1	1	yes	characteristic
32	C₂×C₁₆₃	#102	1	1	yes	characteristic
33	D₁₆₃	—	8	2	no	—
34	C₁₆₃	#331	1	1	yes	characteristic, complemented, Sylow
35	#331	—	163	1	no	maximal, Sylow
36	#127	—	163	1	no	—
37	C₄⋊C₈	—	163	1	no	—
38	C₄⋊C₈	D₁₆₃	1	1	yes	characteristic, complemented
39	C₂×D₄	—	326	1	no	—
40	C₂×D₄	—	326	1	no	—
41	C₂×D₄	—	163	2	no	—
42	C₂×C₈	—	2	1	no	—
43	C₂×C₈	—	326	1	no	—
44	C₄²	C₂×D₁₆₃	1	1	yes	characteristic
45	C₂³	—	326	2	no	—
46	D₄	—	652	2	no	—
47	D₄	—	652	1	no	—
48	D₄	—	652	1	no	—
49	D₄	—	652	1	no	—
50	D₄	—	652	1	no	—
51	C₂×C₄	C₄×D₁₆₃	1	1	yes	characteristic
52	C₂×C₄	C₂²⋊D₁₆₃	1	1	yes	characteristic
53	C₂×C₄	D₆₅₂	1	1	yes	characteristic
54	C₈	—	4	1	no	—
55	C₈	—	652	1	no	—
56	C₂²	—	652	2	no	—
57	C₂²	—	652	2	no	—
58	C₂²	—	652	2	no	—
59	C₂²	(C₂×D₁₆₃)⋊C₄	1	1	yes	characteristic
60	C₄	—	2	1	no	—
61	C₄	#14579684	1	1	yes	characteristic
62	C₄	D₁₃₀₄	1	1	yes	characteristic
63	C₄	#14579686	1	1	yes	characteristic
64	C₄	#14579688	1	1	yes	characteristic
65	C₂	—	1304	2	no	—
66	C₂	#19104651	1	1	yes	characteristic
67	C₂	#19104655	1	1	yes	characteristic
68	C₂	#19104657	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	(C₂×D₁₆₃)⋊C₄
commutator	C₂×C₄×C₁₆₃	C₂×C₄
Frattini	C₄²	C₂×D₁₆₃
Fitting	C₄⋊C₈×C₁₆₃	C₂
socle	C₂²×C₁₆₃	C₂²⋊C₄
prime radical	#19104732	C₂
cosocle	C₄²×C₁₆₃	C₂²
hypercenter	C₄⋊C₈	D₁₆₃
hypocenter	C₁₆₃	#331
LCS	G ▷ C₂×C₄×C₁₆₃ ▷ C₂²×C₁₆₃ ▷ C₁₆₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
2. C₄⋊(C₁₆₃⋊C₈)	C₂	semidirect
3. C₄⋊C₈×C₁₆₃	C₂	semidirect
34. C₁₆₃	#331	semidirect
38. C₄⋊C₈	D₁₆₃	semidirect