Subgroups — #45656152

Order 32576.

56 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₅₀₉⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
2	#30654286	C₂	1	1	yes	maximal, characteristic, complemented
3	#30654234	C₂	1	1	yes	maximal, characteristic
4	#22550871	—	2	1	no	—
5	#22550864	—	2	1	no	—
6	#22550860	—	2	1	no	—
7	Dic₂₀₃₆	C₄	1	1	yes	characteristic
8	#22550849	—	2	1	no	—
9	C₈×D₅₀₉	C₂²	1	1	yes	characteristic
10	D₈×C₅₀₉	C₄	1	1	yes	characteristic
11	#16861332	—	4	1	no	—
12	Dic₁₀₁₈	—	2	1	no	—
13	C₂²⋊D₅₀₉	—	4	1	no	—
14	C₂×Dic₅₀₉	—	4	1	no	—
15	C₄×D₅₀₉	D₄	1	1	yes	characteristic
16	#16861325	D₄	1	1	yes	characteristic
17	D₄×C₅₀₉	—	2	1	no	—
18	C₈×C₅₀₉	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₅₀₉	—	4	1	no	—
20	C₂×D₅₀₉	SD₁₆	1	1	yes	characteristic
21	C₄×C₅₀₉	C₂²⋊C₄	1	1	yes	characteristic
22	Dic₅₀₉	—	4	1	no	—
23	Dic₅₀₉	D₈	1	1	yes	characteristic
24	C₂×C₅₀₉	—	8	1	no	—
25	C₂×C₅₀₉	#102	1	1	yes	characteristic
26	D₅₀₉	—	2	1	no	—
27	C₅₀₉	#359	1	1	yes	characteristic, complemented, Sylow
28	#359	—	509	1	no	maximal, Sylow
29	#135	—	509	1	no	—
30	C₂×C₁₆	—	509	1	no	—
31	#108	—	509	1	no	—
32	#41	—	1018	1	no	—
33	Q₁₆	—	509	1	no	—
34	SD₁₆	—	1018	1	no	—
35	D₈	#14523266	1	1	yes	characteristic
36	M₁₆	—	1018	1	no	—
37	C₂×C₈	—	509	1	no	—
38	C₁₆	—	1018	1	no	—
39	Q₈	—	1018	1	no	—
40	D₄	—	2	1	no	—
41	D₄	—	2036	1	no	—
42	C₂×C₄	—	2036	1	no	—
43	C₂×C₄	—	509	1	no	—
44	C₈	—	2036	1	no	—
45	C₈	—	509	1	no	—
46	C₈	C₂×#14523266	1	1	yes	characteristic
47	C₂²	—	4	1	no	—
48	C₂²	—	509	1	no	—
49	C₄	—	2036	1	no	—
50	C₄	—	509	1	no	—
51	C₄	C₂²⋊#14523266	1	1	yes	characteristic
52	C₂	—	8	1	no	—
53	C₂	—	1018	1	no	—
54	C₂	#30654277	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#30654277
commutator	C₈×C₅₀₉	C₂×C₄
Frattini	C₈	C₂×#14523266
Fitting	D₈×C₅₀₉	C₄
socle	C₂×C₅₀₉	#102
prime radical	#30654234	C₂
cosocle	C₈×D₅₀₉	C₂²
hypercenter	D₈	#14523266
hypocenter	C₅₀₉	#359
LCS	G ▷ C₈×C₅₀₉ ▷ C₄×C₅₀₉ ▷ C₂×C₅₀₉ ▷ C₅₀₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ D₈

Complements

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

N	H ≅ G/N	type
1. D₅₀₉⋊C₁₆	C₂	semidirect
2. #30654286	C₂	semidirect
27. C₅₀₉	#359	semidirect

Maximal subgroups

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

subgroup	index
D₅₀₉⋊C₁₆	2
#30654286	2
#30654234	2
#359	509