Subgroups — #45655005

Order 32576.

142 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂²⋊Dic₅₀₉	C₂	1	2	yes	maximal
2	C₂×C₂²⋊C₄×C₅₀₉	C₂	1	1	yes	maximal, characteristic
3	C₂²×Dic₅₀₉	—	2	2	no	—
4	C₂²⋊Dic₅₀₉	—	2	2	no	—
5	C₂²⋊Dic₅₀₉	C₄	1	4	yes	complemented
6	C₂⁴×C₅₀₉	C₂²	1	1	yes	characteristic
7	C₂²×C₄×C₅₀₉	C₄	1	2	yes	complemented
8	C₂²⋊C₄×C₅₀₉	—	2	2	no	—
9	C₂×Dic₅₀₉	—	2	4	no	—
10	C₂×Dic₅₀₉	—	4	4	no	—
11	C₂³×C₅₀₉	—	4	2	no	—
12	C₂³×C₅₀₉	C₂×C₄	1	2	yes	—
13	C₂³×C₅₀₉	C₂×C₄	1	1	yes	characteristic
14	C₂³×C₅₀₉	D₄	1	2	yes	—
15	C₂³×C₅₀₉	D₄	1	1	yes	characteristic
16	C₂³×C₅₀₉	Q₈	1	1	yes	characteristic
17	C₂×C₄×C₅₀₉	—	2	2	no	—
18	C₂×C₄×C₅₀₉	—	4	2	no	—
19	C₂²×C₅₀₉	—	2	2	no	—
20	C₂²×C₅₀₉	—	2	4	no	—
21	C₂²×C₅₀₉	—	4	2	no	—
22	C₂²×C₅₀₉	—	4	1	no	—
23	C₂²×C₅₀₉	—	4	1	no	—
24	C₂²×C₅₀₉	C₄²	1	1	yes	characteristic
25	C₂²×C₅₀₉	C₂²⋊C₄	1	2	yes	—
26	C₂²×C₅₀₉	C₂²⋊C₄	1	1	yes	characteristic
27	C₂²×C₅₀₉	C₄⋊C₄	1	2	yes	—
28	C₂²×C₅₀₉	C₄⋊C₄	1	1	yes	characteristic
29	C₄×C₅₀₉	—	4	2	no	—
30	Dic₅₀₉	—	4	4	no	—
31	C₂×C₅₀₉	—	2	2	no	—
32	C₂×C₅₀₉	—	2	1	no	—
33	C₂×C₅₀₉	—	2	2	no	—
34	C₂×C₅₀₉	—	2	1	no	—
35	C₂×C₅₀₉	#95	1	1	yes	characteristic
36	C₂×C₅₀₉	#99	1	2	yes	—
37	C₅₀₉	#342	1	1	yes	characteristic, complemented, Sylow
38	#342	—	509	1	no	maximal, Sylow
39	C₂×C₂²⋊C₄	—	509	2	no	—
40	C₂×C₂²⋊C₄	D₅₀₉	1	1	yes	characteristic
41	C₂⁴	C₂×D₅₀₉	1	1	yes	characteristic
42	C₂²×C₄	—	1018	2	no	—
43	C₂²×C₄	Dic₅₀₉	1	2	yes	complemented
44	C₂²⋊C₄	—	2	2	no	—
45	C₂²⋊C₄	—	1018	2	no	—
46	C₂²⋊C₄	—	509	4	no	—
47	C₂³	—	4	2	no	—
48	C₂³	C₄×D₅₀₉	1	2	yes	—
49	C₂³	C₂×Dic₅₀₉	1	1	yes	characteristic
50	C₂³	C₂²⋊D₅₀₉	1	2	yes	—
51	C₂³	D₂₀₃₆	1	1	yes	characteristic
52	C₂³	Dic₁₀₁₈	1	1	yes	characteristic
53	C₂×C₄	—	2	2	no	—
54	C₂×C₄	—	4	2	no	—
55	C₂×C₄	—	2036	4	no	—
56	C₂×C₄	—	1018	4	no	—
57	C₂²	—	2	2	no	—
58	C₂²	—	2	4	no	—
59	C₂²	—	4	2	no	—
60	C₂²	—	4	1	no	—
61	C₂²	—	4	1	no	—
62	C₂²	C₄×Dic₅₀₉	1	1	yes	characteristic
63	C₂²	#22550838	1	2	yes	—
64	C₂²	C₂²⋊Dic₅₀₉	1	1	yes	characteristic
65	C₂²	#22550840	1	2	yes	—
66	C₂²	C₄⋊Dic₅₀₉	1	1	yes	characteristic
67	C₄	—	4	2	no	—
68	C₄	—	2036	4	no	—
69	C₂	—	2	2	no	—
70	C₂	—	2	1	no	—
71	C₂	—	2	2	no	—
72	C₂	—	2	1	no	—
73	C₂	#30654122	1	1	yes	characteristic
74	C₂	#30654130	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²⋊Dic₅₀₉
commutator	C₂²×C₅₀₉	C₄²
Frattini	C₂⁴	C₂×D₅₀₉
Fitting	C₂×C₂²⋊C₄×C₅₀₉	C₂
socle	C₂²×C₅₀₉	C₂²⋊C₄
prime radical, cosocle	C₂⁴×C₅₀₉	C₂²
hypercenter	C₂×C₂²⋊C₄	D₅₀₉
hypocenter	C₅₀₉	#342
LCS	G ▷ C₂²×C₅₀₉ ▷ C₂×C₅₀₉ ▷ C₅₀₉
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂×C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
5. C₂²⋊Dic₅₀₉	C₄	semidirect
7. C₂²×C₄×C₅₀₉	C₄	semidirect
37. C₅₀₉	#342	semidirect
43. C₂²×C₄	Dic₅₀₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂²⋊Dic₅₀₉	2
C₂×C₂²⋊C₄×C₅₀₉	2
#342	509