Subgroups — #45655552

Order 32576.

114 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#30654189	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊Dic₅₀₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#30654149	C₂	1	2	yes	maximal, complemented
4	#30654137	C₂	1	2	yes	maximal, complemented
5	C₂²⋊C₈×C₅₀₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₂₀₃₆	C₂²	1	1	yes	characteristic
7	C₂×C₂²⋊D₅₀₉	—	2	1	no	—
8	C₂²×Dic₅₀₉	—	2	1	no	—
9	C₄⋊Dic₅₀₉	—	2	1	no	—
10	C₄⋊Dic₅₀₉	C₂²	1	2	yes	—
11	C₄⋊Dic₅₀₉	C₂²	1	1	yes	characteristic
12	#22550838	—	2	1	no	—
13	C₂²×C₄×C₅₀₉	C₂²	1	1	yes	characteristic
14	C₂×C₈×C₅₀₉	C₂²	1	2	yes	—
15	C₂²×D₅₀₉	—	2	1	no	—
16	D₂₀₃₆	—	2	2	no	—
17	C₂²⋊D₅₀₉	—	4	2	no	—
18	C₂×Dic₅₀₉	—	2	1	no	—
19	C₂×Dic₅₀₉	—	2	2	no	—
20	C₂×Dic₅₀₉	—	4	2	no	—
21	C₂³×C₅₀₉	D₄	1	1	yes	characteristic
22	C₂×C₄×C₅₀₉	—	2	2	no	—
23	C₂×C₄×C₅₀₉	D₄	1	1	yes	characteristic
24	C₂×C₄×C₅₀₉	C₂³	1	1	yes	characteristic
25	C₈×C₅₀₉	—	2	2	no	—
26	C₂²×C₅₀₉	—	2	1	no	—
27	C₂²×C₅₀₉	—	2	1	no	—
28	C₂²×C₅₀₉	D₈	1	2	yes	—
29	C₂²×C₅₀₉	C₂×D₄	1	1	yes	characteristic
30	C₂×D₅₀₉	—	4	1	no	—
31	C₂×D₅₀₉	—	4	1	no	—
32	C₂×D₅₀₉	—	4	1	no	—
33	C₄×C₅₀₉	—	2	1	no	—
34	C₄×C₅₀₉	#41	1	2	yes	—
35	Dic₅₀₉	—	4	2	no	—
36	Dic₅₀₉	—	4	1	no	—
37	C₂×C₅₀₉	—	2	2	no	—
38	C₂×C₅₀₉	#123	1	1	yes	characteristic
39	C₂×C₅₀₉	C₂×D₈	1	1	yes	characteristic
40	C₂×C₅₀₉	#137	1	1	yes	characteristic
41	D₅₀₉	—	8	1	no	—
42	C₅₀₉	#480	1	1	yes	characteristic, complemented, Sylow
43	#480	—	509	1	no	maximal, Sylow
44	#121	—	509	1	no	—
45	C₂×C₄⋊C₄	—	509	1	no	—
46	#107	—	509	2	no	—
47	#102	—	509	2	no	—
48	C₂²⋊C₈	D₅₀₉	1	1	yes	characteristic, complemented
49	C₂×D₄	—	1018	1	no	—
50	C₂×D₄	—	509	1	no	—
51	C₂²×C₄	—	1018	1	no	—
52	C₂²×C₄	C₂×D₅₀₉	1	1	yes	characteristic
53	C₂×C₈	C₂×D₅₀₉	1	2	yes	—
54	C₄⋊C₄	—	1018	1	no	—
55	C₄⋊C₄	—	509	2	no	—
56	C₄⋊C₄	—	509	1	no	—
57	C₂²⋊C₄	—	1018	1	no	—
58	C₂³	—	1018	1	no	—
59	C₂³	D₂₀₃₆	1	1	yes	characteristic
60	D₄	—	2036	2	no	—
61	D₄	—	1018	2	no	—
62	C₂×C₄	—	2	2	no	—
63	C₂×C₄	—	2036	2	no	—
64	C₂×C₄	—	1018	1	no	—
65	C₂×C₄	—	1018	2	no	—
66	C₂×C₄	D₂₀₃₆	1	1	yes	characteristic
67	C₂×C₄	C₂²×D₅₀₉	1	1	yes	characteristic
68	C₈	—	2	2	no	—
69	C₂²	—	2	1	no	—
70	C₂²	—	2	1	no	—
71	C₂²	—	2036	1	no	—
72	C₂²	—	2036	1	no	—
73	C₂²	—	2036	1	no	—
74	C₂²	D₄₀₇₂	1	2	yes	—
75	C₂²	C₂×D₂₀₃₆	1	1	yes	characteristic
76	C₄	—	2	1	no	—
77	C₄	—	2036	2	no	—
78	C₄	—	2036	1	no	—
79	C₄	#22550860	1	2	yes	—
80	C₂	—	2	2	no	—
81	C₂	—	4072	1	no	—
82	C₂	#30654199	1	1	yes	characteristic
83	C₂	C₂×D₄₀₇₂	1	1	yes	characteristic
84	C₂	#30654248	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₂₀₃₆
commutator, cosocle	C₂×C₄×C₅₀₉	C₂³
Frattini	C₂×C₄	C₂²×D₅₀₉
Fitting	C₂²⋊C₈×C₅₀₉	C₂
socle	C₂²×C₅₀₉	C₂×D₄
prime radical	#30654189	C₂
hypercenter	C₂²⋊C₈	D₅₀₉
hypocenter	C₅₀₉	#480
LCS	G ▷ C₂×C₄×C₅₀₉ ▷ C₂×C₅₀₉ ▷ C₅₀₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²⋊C₈

Complements

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

N	H ≅ G/N	type
2. C₂×C₄⋊Dic₅₀₉	C₂	semidirect
3. #30654149	C₂	semidirect
4. #30654137	C₂	semidirect
5. C₂²⋊C₈×C₅₀₉	C₂	semidirect
42. C₅₀₉	#480	semidirect
48. C₂²⋊C₈	D₅₀₉	semidirect

Maximal subgroups

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

subgroup	index
#30654189	2
C₂×C₄⋊Dic₅₀₉	2
#30654149	2
#30654137	2
C₂²⋊C₈×C₅₀₉	2
#480	509