Subgroups — #45655273

Order 32576.

114 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×D₅₀₉	C₂	1	1	yes	maximal, characteristic
2	#30654173	C₂	1	2	yes	maximal
3	#30654148	C₂	1	2	yes	maximal, complemented
4	#30654147	C₂	1	1	yes	maximal, characteristic, complemented
5	#106×C₅₀₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄×D₅₀₉	C₂²	1	1	yes	characteristic
7	C₂×#16861325	C₂²	1	1	yes	characteristic
8	C₈×D₅₀₉	C₄	1	4	yes	complemented
9	C₄⋊Dic₅₀₉	C₂²	1	2	yes	—
10	#22550838	—	2	2	no	—
11	C₄×Dic₅₀₉	—	2	2	no	—
12	C₂×C₈×C₅₀₉	C₂²	1	1	yes	characteristic
13	C₄⋊C₄×C₅₀₉	C₂²	1	2	yes	—
14	C₂²×D₅₀₉	D₄	1	1	yes	characteristic
15	C₂×Dic₅₀₉	—	2	2	no	—
16	C₂×Dic₅₀₉	D₄	1	1	yes	characteristic
17	C₄×D₅₀₉	C₂×C₄	1	2	yes	—
18	C₄×D₅₀₉	Q₈	1	2	yes	—
19	#16861325	C₂×C₄	1	2	yes	—
20	C₂×C₄×C₅₀₉	—	2	2	no	—
21	C₂×C₄×C₅₀₉	C₂³	1	1	yes	characteristic
22	C₈×C₅₀₉	C₂×C₄	1	2	yes	—
23	C₂²×C₅₀₉	C₂×D₄	1	1	yes	characteristic
24	C₂×D₅₀₉	—	2	2	no	—
25	C₂×D₅₀₉	C₄⋊C₄	1	2	yes	—
26	C₄×C₅₀₉	—	4	2	no	—
27	C₄×C₅₀₉	C₂²×C₄	1	1	yes	characteristic
28	C₄×C₅₀₉	C₂×Q₈	1	1	yes	characteristic
29	Dic₅₀₉	—	4	2	no	—
30	Dic₅₀₉	C₄⋊C₄	1	2	yes	—
31	C₂×C₅₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
32	C₂×C₅₀₉	#135	1	2	yes	—
33	D₅₀₉	—	2	2	no	—
34	C₅₀₉	#427	1	1	yes	characteristic, complemented, Sylow
35	#427	—	509	1	no	maximal, Sylow
36	C₂²×C₈	—	509	1	no	—
37	#117	—	509	2	no	—
38	#107	—	509	2	no	—
39	#106	—	509	1	no	—
40	#106	D₅₀₉	1	1	yes	characteristic, complemented
41	C₂²×C₄	—	509	1	no	—
42	C₂×C₈	—	509	1	no	—
43	C₂×C₈	—	509	4	no	—
44	C₂×C₈	C₂×D₅₀₉	1	1	yes	characteristic
45	C₄⋊C₄	—	509	2	no	—
46	C₄⋊C₄	C₂×D₅₀₉	1	2	yes	—
47	C₂²⋊C₄	—	1018	2	no	—
48	C₄²	—	1018	2	no	—
49	C₂³	—	509	1	no	—
50	C₂×C₄	—	2	2	no	—
51	C₂×C₄	—	1018	2	no	—
52	C₂×C₄	—	509	1	no	—
53	C₂×C₄	—	509	2	no	—
54	C₂×C₄	—	509	2	no	—
55	C₂×C₄	C₂²×D₅₀₉	1	1	yes	characteristic
56	C₈	—	509	2	no	—
57	C₈	C₄×D₅₀₉	1	2	yes	—
58	C₂²	—	1018	2	no	—
59	C₂²	—	509	2	no	—
60	C₂²	D₄×D₅₀₉	1	1	yes	characteristic
61	C₄	—	4	2	no	—
62	C₄	—	2036	2	no	—
63	C₄	—	509	2	no	—
64	C₄	C₂×C₄×D₅₀₉	1	1	yes	characteristic
65	C₄	Q₈×D₅₀₉	1	1	yes	characteristic
66	C₂	—	1018	2	no	—
67	C₂	C₄⋊C₄×D₅₀₉	1	1	yes	characteristic
68	C₂	#30654236	1	2	yes	—

	subgroup	G/H
center	C₂²	D₄×D₅₀₉
commutator	C₄×C₅₀₉	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₅₀₉
Fitting	#106×C₅₀₉	C₂
socle	C₂²×C₅₀₉	C₂×D₄
prime radical	C₂²×D₅₀₉	D₄
cosocle	C₂×C₄×C₅₀₉	C₂³
hypercenter	#106	D₅₀₉
hypocenter	C₅₀₉	#427
LCS	G ▷ 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
3. #30654148	C₂	semidirect
4. #30654147	C₂	semidirect
5. #106×C₅₀₉	C₂	semidirect
8. C₈×D₅₀₉	C₄	semidirect
34. C₅₀₉	#427	semidirect
40. #106	D₅₀₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×D₅₀₉	2
#30654173	2
#30654148	2
#30654147	2
#106×C₅₀₉	2
#427	509