Subgroups — #45655635

Order 32576.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#30654214	C₂	1	2	yes	maximal
2	#30654147	C₂	1	1	yes	maximal, characteristic
3	#30654146	C₂	1	2	yes	maximal
4	C₈×Dic₅₀₉	C₂	1	1	yes	maximal, characteristic
5	#106×C₅₀₉	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₁₀₁₈	—	2	2	no	—
7	C₂×#16861325	C₂²	1	1	yes	characteristic
8	C₄⋊Dic₅₀₉	C₂²	1	2	yes	—
9	#22550840	—	2	2	no	—
10	C₄×Dic₅₀₉	C₂²	1	1	yes	characteristic
11	C₂×C₈×C₅₀₉	C₂²	1	1	yes	characteristic
12	C₄⋊C₄×C₅₀₉	C₂²	1	2	yes	—
13	Dic₁₀₁₈	—	4	4	no	—
14	C₂×Dic₅₀₉	—	2	2	no	—
15	C₂×Dic₅₀₉	D₄	1	2	yes	—
16	#16861325	Q₈	1	2	yes	complemented
17	C₂×C₄×C₅₀₉	—	2	2	no	—
18	C₂×C₄×C₅₀₉	C₂³	1	1	yes	characteristic
19	C₈×C₅₀₉	Q₈	1	2	yes	complemented
20	C₂²×C₅₀₉	C₂×D₄	1	1	yes	characteristic
21	C₄×C₅₀₉	—	4	2	no	—
22	C₄×C₅₀₉	C₂×Q₈	1	1	yes	characteristic
23	C₄×C₅₀₉	C₂×Q₈	1	1	yes	characteristic
24	Dic₅₀₉	—	4	2	no	—
25	Dic₅₀₉	SD₁₆	1	4	yes	—
26	C₂×C₅₀₉	C₄⋊Q₈	1	1	yes	characteristic
27	C₂×C₅₀₉	C₂×SD₁₆	1	2	yes	—
28	C₅₀₉	#498	1	1	yes	characteristic, complemented, Sylow
29	#498	—	509	1	no	maximal, Sylow
30	C₄⋊Q₈	—	509	2	no	—
31	#106	—	509	1	no	—
32	#106	—	509	2	no	—
33	#106	D₅₀₉	1	1	yes	characteristic
34	C₄×C₈	—	509	1	no	—
35	C₂×Q₈	—	1018	2	no	—
36	C₂×C₈	—	509	1	no	—
37	C₂×C₈	C₂×D₅₀₉	1	1	yes	characteristic
38	C₄⋊C₄	—	1018	2	no	—
39	C₄⋊C₄	—	509	2	no	—
40	C₄⋊C₄	C₂×D₅₀₉	1	2	yes	—
41	C₄²	—	509	1	no	—
42	Q₈	—	2036	4	no	—
43	C₂×C₄	—	2	2	no	—
44	C₂×C₄	—	1018	2	no	—
45	C₂×C₄	—	509	2	no	—
46	C₂×C₄	C₂²×D₅₀₉	1	1	yes	characteristic
47	C₈	—	509	2	no	—
48	C₈	Dic₁₀₁₈	1	2	yes	complemented
49	C₂²	D₄×D₅₀₉	1	1	yes	characteristic
50	C₄	—	4	2	no	—
51	C₄	—	2036	2	no	—
52	C₄	—	509	4	no	—
53	C₄	Q₈×D₅₀₉	1	1	yes	characteristic
54	C₄	C₂×Dic₁₀₁₈	1	1	yes	characteristic
55	C₂	#30654214	1	1	yes	characteristic
56	C₂	SD₁₆×D₅₀₉	1	2	yes	—

	subgroup	G/H
center	C₂²	D₄×D₅₀₉
commutator, cosocle	C₂×C₄×C₅₀₉	C₂³
Frattini	C₂×C₄	C₂²×D₅₀₉
Fitting	#106×C₅₀₉	C₂
socle, prime radical	C₂²×C₅₀₉	C₂×D₄
hypercenter	#106	D₅₀₉
hypocenter	C₅₀₉	#498
LCS	G ▷ C₂×C₄×C₅₀₉ ▷ C₂×C₅₀₉ ▷ C₅₀₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #106

Complements

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

N	H ≅ G/N	type
16. #16861325	Q₈	semidirect
19. C₈×C₅₀₉	Q₈	semidirect
28. C₅₀₉	#498	semidirect
48. C₈	Dic₁₀₁₈	semidirect

Maximal subgroups

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

subgroup	index
#30654214	2
#30654147	2
#30654146	2
C₈×Dic₅₀₉	2
#106×C₅₀₉	2
#498	509