Subgroups — D₈×Dic₅₀₉ #45655321

Order 32576.

134 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₄×Dic₅₀₉	C₂	1	2	yes	maximal, complemented
2	#30654149	C₂	1	1	yes	maximal, characteristic, complemented
3	#30654136	C₂	1	2	yes	maximal, complemented
4	C₈×Dic₅₀₉	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×D₈×C₅₀₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₅₀₉	—	2	2	no	—
7	C₂×#16861325	C₂²	1	1	yes	characteristic
8	C₄⋊Dic₅₀₉	C₂²	1	2	yes	—
9	C₂²⋊Dic₅₀₉	—	2	2	no	—
10	C₄×Dic₅₀₉	C₂²	1	1	yes	characteristic
11	C₂×D₄×C₅₀₉	C₂²	1	2	yes	—
12	D₈×C₅₀₉	C₄	1	4	yes	complemented
13	C₂×C₈×C₅₀₉	C₂²	1	1	yes	characteristic
14	C₂×Dic₅₀₉	—	2	2	no	—
15	C₂×Dic₅₀₉	—	4	4	no	—
16	C₂×Dic₅₀₉	D₄	1	1	yes	characteristic
17	C₂×Dic₅₀₉	D₄	1	1	yes	characteristic
18	#16861325	—	2	1	no	—
19	C₂³×C₅₀₉	—	2	2	no	—
20	D₄×C₅₀₉	—	2	2	no	—
21	D₄×C₅₀₉	C₂×C₄	1	4	yes	complemented
22	C₂×C₄×C₅₀₉	C₂³	1	1	yes	characteristic
23	C₈×C₅₀₉	C₂×C₄	1	2	yes	complemented
24	C₂²×C₅₀₉	—	2	4	no	—
25	C₂²×C₅₀₉	—	4	2	no	—
26	C₂²×C₅₀₉	—	4	2	no	—
27	C₂²×C₅₀₉	C₂×D₄	1	1	yes	characteristic
28	C₄×C₅₀₉	C₂²×C₄	1	1	yes	characteristic
29	C₄×C₅₀₉	#41	1	1	yes	characteristic
30	Dic₅₀₉	—	2	1	no	—
31	Dic₅₀₉	—	4	2	no	—
32	Dic₅₀₉	D₈	1	2	yes	direct product factor
33	C₂×C₅₀₉	—	4	4	no	—
34	C₂×C₅₀₉	C₄×D₄	1	1	yes	characteristic
35	C₂×C₅₀₉	C₂×D₈	1	1	yes	characteristic
36	C₂×C₅₀₉	#135	1	1	yes	characteristic
37	C₅₀₉	C₄×D₈	1	1	yes	characteristic, complemented, Sylow
38	C₄×D₈	—	509	1	no	maximal, Sylow
39	C₂×D₈	D₅₀₉	1	1	yes	characteristic
40	C₄×D₄	—	509	2	no	—
41	#107	—	509	1	no	—
42	#102	—	509	2	no	—
43	C₄×C₈	—	509	1	no	—
44	C₂×D₄	C₂×D₅₀₉	1	2	yes	—
45	C₂²×C₄	—	1018	2	no	—
46	D₈	Dic₅₀₉	1	4	yes	direct product factor
47	C₂×C₈	—	509	1	no	—
48	C₂×C₈	C₂×D₅₀₉	1	1	yes	characteristic
49	C₄⋊C₄	—	509	2	no	—
50	C₂²⋊C₄	—	1018	2	no	—
51	C₄²	—	509	1	no	—
52	C₂³	—	2	2	no	—
53	D₄	—	2	2	no	—
54	D₄	C₂×Dic₅₀₉	1	4	yes	complemented
55	C₂×C₄	—	2036	4	no	—
56	C₂×C₄	—	1018	2	no	—
57	C₂×C₄	—	509	1	no	—
58	C₂×C₄	—	509	1	no	—
59	C₂×C₄	C₂²×D₅₀₉	1	1	yes	characteristic
60	C₈	—	1018	1	no	—
61	C₈	C₂×Dic₅₀₉	1	2	yes	complemented
62	C₂²	—	2	4	no	—
63	C₂²	—	4	2	no	—
64	C₂²	—	4	2	no	—
65	C₂²	D₄×D₅₀₉	1	1	yes	characteristic
66	C₄	—	2036	2	no	—
67	C₄	—	1018	1	no	—
68	C₄	—	509	2	no	—
69	C₄	C₂²×Dic₅₀₉	1	1	yes	characteristic
70	C₄	#22550860	1	1	yes	characteristic
71	C₂	—	4	4	no	—
72	C₂	D₄×Dic₅₀₉	1	1	yes	characteristic
73	C₂	D₈×D₅₀₉	1	1	yes	characteristic
74	C₂	#30654234	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₅₀₉
commutator	C₄×C₅₀₉	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₅₀₉
Fitting, prime radical	C₂×D₈×C₅₀₉	C₂
socle	C₂²×C₅₀₉	C₂×D₄
cosocle	C₂×C₄×C₅₀₉	C₂³
hypercenter	C₂×D₈	D₅₀₉
hypocenter	C₅₀₉	C₄×D₈
LCS	G ▷ C₄×C₅₀₉ ▷ C₂×C₅₀₉ ▷ C₅₀₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×D₈

Complements

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

N	H ≅ G/N	type
1. D₄×Dic₅₀₉	C₂	semidirect
2. #30654149	C₂	semidirect
3. #30654136	C₂	semidirect
4. C₈×Dic₅₀₉	C₂	semidirect
12. D₈×C₅₀₉	C₄	semidirect
21. D₄×C₅₀₉	C₂×C₄	semidirect
23. C₈×C₅₀₉	C₂×C₄	semidirect
32. Dic₅₀₉	D₈	direct (N × H)
37. C₅₀₉	C₄×D₈	semidirect
46. D₈	Dic₅₀₉	direct (N × H)
54. D₄	C₂×Dic₅₀₉	semidirect
61. C₈	C₂×Dic₅₀₉	semidirect

Maximal subgroups

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

subgroup	index
D₄×Dic₅₀₉	2
#30654149	2
#30654136	2
C₈×Dic₅₀₉	2
C₂×D₈×C₅₀₉	2
C₄×D₈	509

Subgroups — D8×Dic509 #45655321

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₈×Dic₅₀₉ #45655321