Subgroups — #45656155

Order 32576.

58 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#30654290	C₂	1	1	yes	maximal, characteristic, complemented
2	#30654283	C₂	1	1	yes	maximal, characteristic, complemented
3	#30654238	C₂	1	1	yes	maximal, characteristic
4	C₄⋊#14523266	—	2	1	no	—
5	#22550864	—	2	1	no	—
6	#22550862	—	2	1	no	—
7	#22550848	—	2	1	no	—
8	D₄₀₇₂	C₄	1	1	yes	characteristic, complemented
9	C₈×D₅₀₉	C₂²	1	1	yes	characteristic
10	Q₁₆×C₅₀₉	C₄	1	1	yes	characteristic, complemented
11	C₂×#14523266	—	4	1	no	—
12	D₂₀₃₆	—	2	1	no	—
13	D₂₀₃₆	—	4	1	no	—
14	C₄×D₅₀₉	—	4	1	no	—
15	C₄×D₅₀₉	D₄	1	1	yes	characteristic
16	#16861325	D₄	1	1	yes	characteristic
17	Q₈×C₅₀₉	—	2	1	no	—
18	C₈×C₅₀₉	C₂×C₄	1	1	yes	characteristic
19	C₂×D₅₀₉	—	4	1	no	—
20	C₂×D₅₀₉	D₈	1	1	yes	characteristic
21	#14523266	—	8	1	no	—
22	C₄×C₅₀₉	—	4	1	no	—
23	C₄×C₅₀₉	C₂²⋊C₄	1	1	yes	characteristic
24	Dic₅₀₉	SD₁₆	1	1	yes	characteristic
25	C₂×C₅₀₉	#102	1	1	yes	characteristic
26	D₅₀₉	—	2	1	no	—
27	D₅₀₉	—	8	1	no	—
28	C₅₀₉	#360	1	1	yes	characteristic, complemented, Sylow
29	#360	—	509	1	no	maximal, Sylow
30	#135	—	509	1	no	—
31	M₃₂	—	509	1	no	—
32	#106	—	509	1	no	—
33	#41	—	1018	1	no	—
34	Q₁₆	#14523266	1	1	yes	characteristic, complemented
35	SD₁₆	—	1018	1	no	—
36	D₈	—	509	1	no	—
37	C₂×C₈	—	509	1	no	—
38	C₄⋊C₄	—	1018	1	no	—
39	C₁₆	—	1018	1	no	—
40	Q₈	—	2	1	no	—
41	D₄	—	2036	1	no	—
42	D₄	—	1018	1	no	—
43	C₂×C₄	—	2036	1	no	—
44	C₂×C₄	—	2036	1	no	—
45	C₂×C₄	—	509	1	no	—
46	C₈	—	509	1	no	—
47	C₈	C₂×#14523266	1	1	yes	characteristic
48	C₂²	—	2036	1	no	—
49	C₂²	—	509	1	no	—
50	C₄	—	4	1	no	—
51	C₄	—	4072	1	no	—
52	C₄	—	509	1	no	—
53	C₄	C₂²⋊#14523266	1	1	yes	characteristic
54	C₂	—	4072	1	no	—
55	C₂	—	1018	1	no	—
56	C₂	#30654277	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#30654277
commutator	C₈×C₅₀₉	C₂×C₄
Frattini	C₈	C₂×#14523266
Fitting	Q₁₆×C₅₀₉	C₄
socle	C₂×C₅₀₉	#102
prime radical	#30654238	C₂
cosocle	C₈×D₅₀₉	C₂²
hypercenter	Q₁₆	#14523266
hypocenter	C₅₀₉	#360
LCS	G ▷ C₈×C₅₀₉ ▷ C₄×C₅₀₉ ▷ C₂×C₅₀₉ ▷ C₅₀₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ Q₁₆

Complements

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

N	H ≅ G/N	type
1. #30654290	C₂	semidirect
2. #30654283	C₂	semidirect
8. D₄₀₇₂	C₄	semidirect
10. Q₁₆×C₅₀₉	C₄	semidirect
28. C₅₀₉	#360	semidirect
34. Q₁₆	#14523266	semidirect

Maximal subgroups

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

subgroup	index
#30654290	2
#30654283	2
#30654238	2
#360	509