Subgroups — #35216846

Order 20864.

106 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23782098	C₂	1	2	yes	maximal, complemented
2	#23781502	C₂	1	2	yes	maximal, complemented
3	D₅₂₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₂×D₁₆₃	C₂	1	1	yes	maximal, characteristic, complemented
5	Q₆₄×C₁₆₃	C₂	1	1	yes	maximal, characteristic, complemented
6	#19104758	—	2	2	no	—
7	#19104678	—	2	2	no	—
8	D₂₆₀₈	C₂²	1	2	yes	—
9	C₁₆×D₁₆₃	C₂²	1	1	yes	characteristic
10	C₁₆₃⋊C₃₂	C₂²	1	1	yes	characteristic
11	Q₃₂×C₁₆₃	C₂²	1	2	yes	—
12	C₃₂×C₁₆₃	C₂²	1	1	yes	characteristic
13	#14579700	—	4	2	no	—
14	#14579686	—	4	2	no	—
15	D₁₃₀₄	—	2	2	no	—
16	C₈×D₁₆₃	D₄	1	1	yes	characteristic
17	C₁₆₃⋊C₁₆	D₄	1	1	yes	characteristic
18	Q₁₆×C₁₆₃	—	2	2	no	—
19	C₁₆×C₁₆₃	C₂³	1	1	yes	characteristic
20	D₆₅₂	—	4	2	no	—
21	D₆₅₂	—	8	2	no	—
22	C₄×D₁₆₃	—	8	2	no	—
23	C₄×D₁₆₃	D₈	1	1	yes	characteristic
24	C₁₆₃⋊C₈	D₈	1	1	yes	characteristic
25	Q₈×C₁₆₃	—	4	2	no	—
26	C₈×C₁₆₃	C₂×D₄	1	1	yes	characteristic
27	C₂×D₁₆₃	—	8	2	no	—
28	C₂×D₁₆₃	D₁₆	1	1	yes	characteristic
29	C₄×C₁₆₃	—	8	2	no	—
30	C₄×C₁₆₃	C₂×D₈	1	1	yes	characteristic
31	Dic₁₆₃	D₁₆	1	1	yes	characteristic
32	C₂×C₁₆₃	C₂×D₁₆	1	1	yes	characteristic
33	D₁₆₃	—	2	1	no	—
34	D₁₆₃	—	16	2	no	—
35	C₁₆₃	#2262	1	1	yes	characteristic, complemented, Sylow
36	#2262	—	163	1	no	maximal, Sylow
37	#508	—	163	2	no	—
38	Q₆₄	D₁₆₃	1	1	yes	characteristic, complemented
39	SD₆₄	—	163	2	no	—
40	D₃₂	—	163	1	no	—
41	C₂×C₃₂	—	163	1	no	—
42	#135	—	326	2	no	—
43	Q₃₂	C₂×D₁₆₃	1	2	yes	—
44	SD₃₂	—	326	2	no	—
45	D₁₆	—	163	2	no	—
46	C₂×C₁₆	—	163	1	no	—
47	C₃₂	—	163	1	no	—
48	C₃₂	C₂×D₁₆₃	1	1	yes	characteristic
49	#41	—	652	2	no	—
50	Q₁₆	—	2	2	no	—
51	SD₁₆	—	652	2	no	—
52	D₈	—	326	2	no	—
53	C₂×C₈	—	163	1	no	—
54	C₁₆	—	163	1	no	—
55	C₁₆	C₂²×D₁₆₃	1	1	yes	characteristic
56	Q₈	—	4	2	no	—
57	D₄	—	1304	2	no	—
58	D₄	—	652	2	no	—
59	C₂×C₄	—	1304	2	no	—
60	C₂×C₄	—	163	1	no	—
61	C₈	—	163	1	no	—
62	C₈	D₄×D₁₆₃	1	1	yes	characteristic
63	C₂²	—	1304	2	no	—
64	C₂²	—	163	1	no	—
65	C₄	—	8	2	no	—
66	C₄	—	163	1	no	—
67	C₄	D₈×D₁₆₃	1	1	yes	characteristic
68	C₂	—	2608	2	no	—
69	C₂	—	326	1	no	—
70	C₂	D₁₆×D₁₆₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₁₆×D₁₆₃
commutator, cosocle	C₁₆×C₁₆₃	C₂³
Frattini	C₁₆	C₂²×D₁₆₃
Fitting	Q₆₄×C₁₆₃	C₂
socle	C₂×C₁₆₃	C₂×D₁₆
hypercenter	Q₆₄	D₁₆₃
hypocenter	C₁₆₃	#2262
LCS	G ▷ C₁₆×C₁₆₃ ▷ C₈×C₁₆₃ ▷ C₄×C₁₆₃ ▷ C₂×C₁₆₃ ▷ C₁₆₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ Q₆₄

Complements

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

N	H ≅ G/N	type
1. #23782098	C₂	semidirect
2. #23781502	C₂	semidirect
3. D₅₂₁₆	C₂	semidirect
4. C₃₂×D₁₆₃	C₂	semidirect
5. Q₆₄×C₁₆₃	C₂	semidirect
35. C₁₆₃	#2262	semidirect
38. Q₆₄	D₁₆₃	semidirect

Maximal subgroups

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

subgroup	index
#23782098	2
#23781502	2
D₅₂₁₆	2
C₃₂×D₁₆₃	2
Q₆₄×C₁₆₃	2
#2262	163