Subgroups — #45656055

Order 32576.

248 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#30654262	C₂	1	2	yes	maximal, complemented
2	#30654248	C₂	1	6	yes	maximal, complemented
3	#30654237	C₂	1	3	yes	maximal, complemented
4	C₂×Dic₂₀₃₆	C₂	1	3	yes	maximal, complemented
5	#131×C₅₀₉	C₂	1	1	yes	maximal, characteristic, complemented
6	#22550861	C₂²	1	6	yes	—
7	#22550860	—	2	6	no	—
8	C₂×Dic₁₀₁₈	C₂²	1	6	yes	—
9	Q₈×D₅₀₉	—	2	2	no	—
10	Dic₂₀₃₆	C₂²	1	3	yes	—
11	Dic₂₀₃₆	C₂²	1	6	yes	—
12	#22550850	C₂²	1	6	yes	—
13	D₄₀₇₂	C₂²	1	1	yes	characteristic
14	#41×C₅₀₉	C₂²	1	1	yes	characteristic
15	M₁₆×C₅₀₉	C₂²	1	3	yes	—
16	C₂×C₈×C₅₀₉	C₂²	1	3	yes	—
17	Dic₁₀₁₈	—	2	6	no	—
18	Dic₁₀₁₈	C₂³	1	6	yes	—
19	D₂₀₃₆	C₂³	1	2	yes	—
20	C₂²⋊D₅₀₉	—	2	6	no	—
21	C₂×Dic₅₀₉	—	2	6	no	—
22	C₄×D₅₀₉	—	2	6	no	—
23	Q₈×C₅₀₉	D₄	1	1	yes	characteristic
24	D₄×C₅₀₉	D₄	1	3	yes	—
25	C₂×C₄×C₅₀₉	C₂³	1	3	yes	—
26	C₈×C₅₀₉	C₂³	1	1	yes	characteristic
27	C₈×C₅₀₉	C₂³	1	3	yes	—
28	C₂²×C₅₀₉	C₂×D₄	1	3	yes	—
29	C₂×D₅₀₉	—	2	2	no	—
30	C₄×C₅₀₉	C₂×D₄	1	3	yes	—
31	C₄×C₅₀₉	C₂⁴	1	1	yes	characteristic
32	Dic₅₀₉	—	2	6	no	—
33	C₂×C₅₀₉	—	2	3	no	—
34	C₂×C₅₀₉	C₂²×D₄	1	1	yes	characteristic
35	D₅₀₉	—	4	2	no	—
36	C₅₀₉	#578	1	1	yes	characteristic, complemented, Sylow
37	#578	—	509	1	no	maximal, Sylow
38	2₋¹⁺⁴	—	509	2	no	—
39	#137	—	509	6	no	—
40	#135	—	509	3	no	—
41	C₂×Q₁₆	—	509	3	no	—
42	#131	D₅₀₉	1	1	yes	characteristic, complemented
43	#41	—	1018	6	no	—
44	#41	—	509	6	no	—
45	#41	C₂×D₅₀₉	1	1	yes	characteristic
46	C₂×Q₈	—	1018	2	no	—
47	C₂×Q₈	—	509	6	no	—
48	Q₁₆	—	509	3	no	—
49	Q₁₆	—	509	6	no	—
50	SD₁₆	—	509	6	no	—
51	D₈	—	509	1	no	—
52	M₁₆	C₂×D₅₀₉	1	3	yes	—
53	C₂×C₈	C₂×D₅₀₉	1	3	yes	—
54	Q₈	—	1018	6	no	—
55	Q₈	—	509	6	no	—
56	Q₈	D₂₀₃₆	1	1	yes	characteristic
57	D₄	—	1018	6	no	—
58	D₄	—	509	2	no	—
59	D₄	D₂₀₃₆	1	3	yes	—
60	C₂×C₄	—	1018	6	no	—
61	C₂×C₄	—	1018	6	no	—
62	C₂×C₄	C₂²×D₅₀₉	1	3	yes	—
63	C₈	C₂²×D₅₀₉	1	1	yes	characteristic
64	C₈	C₂²×D₅₀₉	1	3	yes	—
65	C₂²	—	1018	2	no	—
66	C₂²	C₂×D₂₀₃₆	1	3	yes	—
67	C₄	—	1018	6	no	—
68	C₄	C₂×D₂₀₃₆	1	3	yes	—
69	C₄	C₂³×D₅₀₉	1	1	yes	characteristic
70	C₂	—	2	3	no	—
71	C₂	—	2036	2	no	—
72	C₂	C₂²×D₂₀₃₆	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂²×D₂₀₃₆
commutator, cosocle	C₄×C₅₀₉	C₂⁴
Frattini	C₄	C₂³×D₅₀₉
Fitting	#131×C₅₀₉	C₂
socle	C₂×C₅₀₉	C₂²×D₄
hypercenter	#131	D₅₀₉
hypocenter	C₅₀₉	#578
LCS	G ▷ C₄×C₅₀₉ ▷ C₂×C₅₀₉ ▷ C₅₀₉
UCS	C₁ ◁ C₂ ◁ #41 ◁ #131

Complements

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

N	H ≅ G/N	type
1. #30654262	C₂	semidirect
2. #30654248	C₂	semidirect
3. #30654237	C₂	semidirect
4. C₂×Dic₂₀₃₆	C₂	semidirect
5. #131×C₅₀₉	C₂	semidirect
36. C₅₀₉	#578	semidirect
42. #131	D₅₀₉	semidirect

Maximal subgroups

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

subgroup	index
#30654262	2
#30654248	2
#30654237	2
C₂×Dic₂₀₃₆	2
#131×C₅₀₉	2
#578	509