Subgroups — C₄×#22550844 #45655188

Order 32576.

142 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#22550844	C₂	1	2	yes	maximal
2	C₄²×D₅₀₉	C₂	1	1	yes	maximal, characteristic
3	#30654125	C₂	1	2	yes	maximal, complemented
4	C₄×#16861325	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×C₈×C₅₀₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄×D₅₀₉	C₄	1	2	yes	—
7	C₂×C₄×D₅₀₉	C₂²	1	1	yes	characteristic
8	#22550844	C₄	1	8	yes	direct product factor
9	C₂×#16861325	C₂²	1	2	yes	—
10	C₄×Dic₅₀₉	C₄	1	2	yes	—
11	C₄×Dic₅₀₉	C₂²	1	1	yes	characteristic
12	C₂×C₈×C₅₀₉	C₂²	1	2	yes	—
13	C₄²×C₅₀₉	C₂²	1	1	yes	characteristic
14	C₂²×D₅₀₉	C₂×C₄	1	1	yes	characteristic
15	C₂×Dic₅₀₉	C₂×C₄	1	2	yes	—
16	C₂×Dic₅₀₉	C₂×C₄	1	1	yes	characteristic
17	C₄×D₅₀₉	—	2	4	no	—
18	C₄×D₅₀₉	C₂×C₄	1	4	yes	—
19	#16861325	C₂×C₄	1	4	yes	complemented
20	C₂×C₄×C₅₀₉	C₂×C₄	1	2	yes	—
21	C₂×C₄×C₅₀₉	C₂³	1	1	yes	characteristic
22	C₈×C₅₀₉	C₂×C₄	1	4	yes	complemented
23	C₂²×C₅₀₉	C₂²×C₄	1	1	yes	characteristic
24	C₂×D₅₀₉	—	2	2	no	—
25	C₂×D₅₀₉	C₄²	1	2	yes	—
26	C₄×C₅₀₉	M₁₆	1	4	yes	complemented
27	C₄×C₅₀₉	C₂²×C₄	1	2	yes	—
28	Dic₅₀₉	—	2	2	no	—
29	Dic₅₀₉	C₄²	1	2	yes	—
30	C₂×C₅₀₉	C₂×C₄²	1	1	yes	characteristic
31	C₂×C₅₀₉	C₂×M₁₆	1	2	yes	—
32	D₅₀₉	—	2	2	no	—
33	C₅₀₉	C₄×M₁₆	1	1	yes	characteristic, complemented, Sylow
34	C₄×M₁₆	—	509	1	no	maximal, Sylow
35	C₂×M₁₆	—	509	2	no	—
36	C₂×C₄²	—	509	1	no	—
37	#97	—	509	2	no	—
38	C₄×C₈	—	509	1	no	—
39	C₄×C₈	D₅₀₉	1	1	yes	characteristic, complemented
40	C₂²×C₄	—	509	2	no	—
41	C₂²×C₄	—	509	1	no	—
42	M₁₆	—	509	8	no	—
43	C₂×C₈	—	509	2	no	—
44	C₂×C₈	C₂×D₅₀₉	1	2	yes	—
45	C₄²	—	509	2	no	—
46	C₄²	—	509	1	no	—
47	C₄²	C₂×D₅₀₉	1	1	yes	characteristic
48	C₂³	—	509	1	no	—
49	C₂×C₄	—	1018	4	no	—
50	C₂×C₄	—	509	2	no	—
51	C₂×C₄	—	509	1	no	—
52	C₂×C₄	—	509	4	no	—
53	C₂×C₄	C₄×D₅₀₉	1	2	yes	—
54	C₂×C₄	C₂²×D₅₀₉	1	1	yes	characteristic
55	C₈	—	509	4	no	—
56	C₈	C₄×D₅₀₉	1	4	yes	complemented
57	C₂²	—	1018	2	no	—
58	C₂²	—	509	2	no	—
59	C₂²	C₂×C₄×D₅₀₉	1	1	yes	characteristic
60	C₄	—	1018	2	no	—
61	C₄	—	509	2	no	—
62	C₄	#22550844	1	4	yes	direct product factor
63	C₄	C₂×C₄×D₅₀₉	1	2	yes	—
64	C₂	—	1018	2	no	—
65	C₂	C₄²×D₅₀₉	1	1	yes	characteristic
66	C₂	C₂×#22550844	1	2	yes	—

	subgroup	G/H
center	C₄²	C₂×D₅₀₉
commutator	C₂×C₅₀₉	C₂×C₄²
Frattini	C₂×C₄	C₂²×D₅₀₉
Fitting	C₄×C₈×C₅₀₉	C₂
socle	C₂²×C₅₀₉	C₂²×C₄
prime radical	C₂²×D₅₀₉	C₂×C₄
cosocle	C₂×C₄×C₅₀₉	C₂³
hypercenter	C₄×C₈	D₅₀₉
hypocenter	C₅₀₉	C₄×M₁₆
LCS	G ▷ C₂×C₅₀₉ ▷ C₅₀₉
UCS	C₁ ◁ C₄² ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
3. #30654125	C₂	semidirect
4. C₄×#16861325	C₂	semidirect
5. C₄×C₈×C₅₀₉	C₂	semidirect
8. #22550844	C₄	direct (N × H)
19. #16861325	C₂×C₄	semidirect
22. C₈×C₅₀₉	C₂×C₄	semidirect
26. C₄×C₅₀₉	M₁₆	semidirect
33. C₅₀₉	C₄×M₁₆	semidirect
39. C₄×C₈	D₅₀₉	semidirect
56. C₈	C₄×D₅₀₉	semidirect
62. C₄	#22550844	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#22550844	2
C₄²×D₅₀₉	2
#30654125	2
C₄×#16861325	2
C₄×C₈×C₅₀₉	2
C₄×M₁₆	509

Subgroups — C4×#22550844 #45655188

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#22550844 #45655188