Subgroups — #45655588

Order 32576.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#30654215	C₂	1	1	yes	maximal, characteristic
2	#30654139	C₂	1	4	yes	maximal
3	C₄×#16861325	C₂	1	1	yes	maximal, characteristic
4	C₄⋊Q₈×C₅₀₉	C₂	1	1	yes	maximal, characteristic
5	C₂×Dic₁₀₁₈	—	2	1	no	—
6	C₂×#16861325	C₂²	1	2	yes	—
7	C₄⋊Dic₅₀₉	—	2	1	no	—
8	C₄⋊Dic₅₀₉	C₂²	1	2	yes	—
9	C₂×Q₈×C₅₀₉	C₂²	1	2	yes	—
10	C₄⋊C₄×C₅₀₉	—	2	1	no	—
11	C₄⋊C₄×C₅₀₉	—	2	1	no	—
12	C₄²×C₅₀₉	C₂²	1	1	yes	characteristic
13	Dic₁₀₁₈	—	4	2	no	—
14	C₂×Dic₅₀₉	—	2	2	no	—
15	#16861325	—	2	2	no	—
16	Q₈×C₅₀₉	—	2	4	no	—
17	C₂×C₄×C₅₀₉	—	2	2	no	—
18	C₂×C₄×C₅₀₉	D₄	1	1	yes	characteristic
19	C₂×C₄×C₅₀₉	D₄	1	1	yes	characteristic
20	C₂×C₄×C₅₀₉	C₂³	1	1	yes	characteristic
21	C₂²×C₅₀₉	C₂×D₄	1	1	yes	characteristic
22	C₄×C₅₀₉	—	4	2	no	—
23	C₄×C₅₀₉	SD₁₆	1	2	yes	—
24	C₄×C₅₀₉	Q₁₆	1	2	yes	—
25	C₄×C₅₀₉	#41	1	2	yes	—
26	Dic₅₀₉	—	4	2	no	—
27	C₂×C₅₀₉	#124	1	1	yes	characteristic
28	C₂×C₅₀₉	C₂×SD₁₆	1	1	yes	characteristic
29	C₂×C₅₀₉	C₂×Q₁₆	1	1	yes	characteristic
30	C₅₀₉	#487	1	1	yes	characteristic, complemented, Sylow
31	#487	—	509	1	no	maximal, Sylow
32	C₄⋊Q₈	—	509	1	no	—
33	C₄⋊Q₈	D₅₀₉	1	1	yes	characteristic
34	#103	—	509	4	no	—
35	C₄×C₈	—	509	1	no	—
36	C₂×Q₈	—	1018	1	no	—
37	C₂×Q₈	C₂×D₅₀₉	1	2	yes	—
38	C₂×C₈	—	509	2	no	—
39	C₄⋊C₄	—	2	1	no	—
40	C₄⋊C₄	—	2	1	no	—
41	C₄⋊C₄	—	1018	1	no	—
42	C₄⋊C₄	—	509	2	no	—
43	C₄²	C₂×D₅₀₉	1	1	yes	characteristic
44	Q₈	—	2	4	no	—
45	Q₈	—	2036	2	no	—
46	C₂×C₄	—	2	2	no	—
47	C₂×C₄	—	1018	2	no	—
48	C₂×C₄	C₂²⋊D₅₀₉	1	1	yes	characteristic
49	C₂×C₄	C₂²⋊D₅₀₉	1	1	yes	characteristic
50	C₂×C₄	C₂²×D₅₀₉	1	1	yes	characteristic
51	C₈	—	1018	2	no	—
52	C₂²	C₂×C₂²⋊D₅₀₉	1	1	yes	characteristic
53	C₄	—	4	2	no	—
54	C₄	—	2036	2	no	—
55	C₄	#22550848	1	2	yes	—
56	C₄	#22550851	1	2	yes	—
57	C₄	#22550860	1	2	yes	—
58	C₂	#30654202	1	1	yes	characteristic
59	C₂	C₂×#22550848	1	1	yes	characteristic
60	C₂	C₂×#22550851	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₅₀₉
commutator, cosocle	C₂×C₄×C₅₀₉	C₂³
Frattini	C₂×C₄	C₂²×D₅₀₉
Fitting	C₄⋊Q₈×C₅₀₉	C₂
socle, prime radical	C₂²×C₅₀₉	C₂×D₄
hypercenter	C₄⋊Q₈	D₅₀₉
hypocenter	C₅₀₉	#487
LCS	G ▷ C₂×C₄×C₅₀₉ ▷ C₂×C₅₀₉ ▷ C₅₀₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
30. C₅₀₉	#487	semidirect

Maximal subgroups

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

subgroup	index
#30654215	2
#30654139	2
C₄×#16861325	2
C₄⋊Q₈×C₅₀₉	2
#487	509