Subgroups — #126×D₅₀₉ #45655761

Order 32576.

246 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#30654210	C₂	1	3	yes	maximal, complemented
2	#30654209	C₂	1	1	yes	maximal, characteristic, complemented
3	#30654208	C₂	1	3	yes	maximal, complemented
4	C₄⋊C₄×D₅₀₉	C₂	1	3	yes	maximal, complemented
5	C₂²⋊C₄×D₅₀₉	C₂	1	3	yes	maximal
6	C₄²×D₅₀₉	C₂	1	1	yes	maximal, characteristic, complemented
7	#126×C₅₀₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂³×D₅₀₉	C₂²	1	1	yes	characteristic
9	C₂×C₄×D₅₀₉	C₂²	1	3	yes	—
10	C₂×C₄×D₅₀₉	C₂²	1	3	yes	—
11	C₄⋊Dic₅₀₉	C₂²	1	3	yes	complemented
12	#22550840	C₂²	1	3	yes	complemented
13	#22550840	C₂²	1	3	yes	complemented
14	C₂²⋊Dic₅₀₉	C₂²	1	3	yes	—
15	#22550838	C₂²	1	3	yes	—
16	#22550838	C₂²	1	3	yes	—
17	C₄×Dic₅₀₉	C₂²	1	3	yes	complemented
18	C₄⋊C₄×C₅₀₉	C₂²	1	3	yes	complemented
19	C₂²⋊C₄×C₅₀₉	C₂²	1	3	yes	—
20	C₄²×C₅₀₉	C₂²	1	1	yes	characteristic, complemented
21	C₂²×D₅₀₉	—	2	6	no	—
22	C₂²×D₅₀₉	C₂³	1	1	yes	characteristic
23	C₂²×D₅₀₉	C₂³	1	1	yes	characteristic
24	C₂×Dic₅₀₉	C₂³	1	3	yes	—
25	C₂×Dic₅₀₉	C₂³	1	3	yes	—
26	C₄×D₅₀₉	—	2	6	no	—
27	C₄×D₅₀₉	—	2	6	no	—
28	C₂³×C₅₀₉	C₂³	1	1	yes	characteristic
29	C₂×C₄×C₅₀₉	C₂³	1	3	yes	—
30	C₂×C₄×C₅₀₉	C₂³	1	3	yes	—
31	C₂²×C₅₀₉	—	2	3	no	—
32	C₂²×C₅₀₉	C₂⁴	1	1	yes	characteristic
33	C₂×D₅₀₉	—	2	3	no	—
34	C₂×D₅₀₉	—	4	4	no	—
35	C₂×D₅₀₉	#41	1	6	yes	—
36	C₄×C₅₀₉	—	2	3	no	—
37	C₄×C₅₀₉	—	2	3	no	—
38	Dic₅₀₉	—	2	3	no	—
39	Dic₅₀₉	—	2	3	no	—
40	C₂×C₅₀₉	—	4	1	no	—
41	C₂×C₅₀₉	C₂×#41	1	3	yes	—
42	D₅₀₉	—	4	1	no	—
43	D₅₀₉	#126	1	4	yes	direct product factor
44	C₅₀₉	C₂×#126	1	1	yes	characteristic, complemented, Sylow
45	C₂×#126	—	509	1	no	maximal, Sylow
46	#126	—	509	3	no	—
47	#126	—	509	3	no	—
48	#126	—	509	1	no	—
49	#126	D₅₀₉	1	1	yes	characteristic, direct product factor
50	C₂×C₄⋊C₄	—	509	3	no	—
51	C₂×C₂²⋊C₄	—	509	3	no	—
52	C₂×C₄²	—	509	1	no	—
53	C₂⁴	—	509	1	no	—
54	C₂²×C₄	—	509	3	no	—
55	C₂²×C₄	—	509	3	no	—
56	C₄⋊C₄	—	509	3	no	—
57	C₄⋊C₄	—	509	3	no	—
58	C₄⋊C₄	—	509	3	no	—
59	C₄⋊C₄	C₂×D₅₀₉	1	3	yes	complemented
60	C₂²⋊C₄	—	509	3	no	—
61	C₂²⋊C₄	—	509	3	no	—
62	C₂²⋊C₄	—	509	3	no	—
63	C₂²⋊C₄	C₂×D₅₀₉	1	3	yes	—
64	C₄²	—	509	3	no	—
65	C₄²	C₂×D₅₀₉	1	1	yes	characteristic, complemented
66	C₂³	—	1018	6	no	—
67	C₂³	—	509	1	no	—
68	C₂³	—	509	1	no	—
69	C₂³	C₂²×D₅₀₉	1	1	yes	characteristic
70	C₂×C₄	—	1018	6	no	—
71	C₂×C₄	—	1018	6	no	—
72	C₂×C₄	—	509	3	no	—
73	C₂×C₄	—	509	3	no	—
74	C₂×C₄	C₂²×D₅₀₉	1	3	yes	—
75	C₂×C₄	C₂²×D₅₀₉	1	3	yes	—
76	C₂²	—	2	3	no	—
77	C₂²	—	2036	4	no	—
78	C₂²	—	1018	3	no	—
79	C₂²	—	509	6	no	—
80	C₂²	C₂³×D₅₀₉	1	1	yes	characteristic
81	C₄	—	2	3	no	—
82	C₄	—	2	3	no	—
83	C₄	—	1018	3	no	—
84	C₄	—	1018	3	no	—
85	C₂	—	4	1	no	—
86	C₂	—	2036	1	no	—
87	C₂	—	509	4	no	—
88	C₂	#41×D₅₀₉	1	3	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₅₀₉
commutator, socle, cosocle	C₂²×C₅₀₉	C₂⁴
Fitting	#126×C₅₀₉	C₂
prime radical	C₂³×D₅₀₉	C₂²
hypercenter	#126	D₅₀₉
hypocenter	C₅₀₉	C₂×#126
LCS	G ▷ C₂²×C₅₀₉ ▷ C₅₀₉
UCS	C₁ ◁ C₂² ◁ #126

Complements

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

N	H ≅ G/N	type
1. #30654210	C₂	semidirect
2. #30654209	C₂	semidirect
3. #30654208	C₂	semidirect
4. C₄⋊C₄×D₅₀₉	C₂	semidirect
6. C₄²×D₅₀₉	C₂	semidirect
7. #126×C₅₀₉	C₂	semidirect
11. C₄⋊Dic₅₀₉	C₂²	semidirect
12. #22550840	C₂²	semidirect
13. #22550840	C₂²	semidirect
17. C₄×Dic₅₀₉	C₂²	semidirect
18. C₄⋊C₄×C₅₀₉	C₂²	semidirect
20. C₄²×C₅₀₉	C₂²	semidirect
43. D₅₀₉	#126	direct (N × H)
44. C₅₀₉	C₂×#126	semidirect
49. #126	D₅₀₉	direct (N × H)
59. C₄⋊C₄	C₂×D₅₀₉	semidirect
65. C₄²	C₂×D₅₀₉	semidirect

Maximal subgroups

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

subgroup	index
#30654210	2
#30654209	2
#30654208	2
C₄⋊C₄×D₅₀₉	2
C₂²⋊C₄×D₅₀₉	2
C₄²×D₅₀₉	2
#126×C₅₀₉	2
C₂×#126	509

Subgroups — #126×D509 #45655761

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #126×D₅₀₉ #45655761