Subgroups — #45655361

Order 32576.

104 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#30654237	C₂	1	1	yes	maximal, characteristic
2	#30654221	C₂	1	2	yes	maximal, complemented
3	#30654151	C₂	1	1	yes	maximal, characteristic, complemented
4	#30654143	C₂	1	2	yes	maximal, complemented
5	#97×C₅₀₉	C₂	1	1	yes	maximal, characteristic, complemented
6	#22550861	C₂²	1	2	yes	—
7	Dic₂₀₃₆	C₄	1	1	yes	characteristic, complemented
8	#22550850	C₄	1	2	yes	complemented
9	D₄₀₇₂	C₄	1	1	yes	characteristic, complemented
10	#22550845	C₂²	1	2	yes	—
11	#22550844	—	2	2	no	—
12	C₈×D₅₀₉	—	2	2	no	—
13	C₂×C₈×C₅₀₉	C₂²	1	1	yes	characteristic
14	C₂×C₈×C₅₀₉	C₂²	1	1	yes	characteristic
15	C₄²×C₅₀₉	C₂²	1	1	yes	characteristic
16	Dic₁₀₁₈	C₂×C₄	1	2	yes	—
17	D₂₀₃₆	C₂×C₄	1	2	yes	—
18	C₂²⋊D₅₀₉	—	2	2	no	—
19	C₄×D₅₀₉	—	2	2	no	—
20	#16861325	—	2	2	no	—
21	C₂×C₄×C₅₀₉	—	2	1	no	—
22	C₂×C₄×C₅₀₉	C₂³	1	1	yes	characteristic
23	C₈×C₅₀₉	C₂×C₄	1	1	yes	characteristic
24	C₈×C₅₀₉	C₂×C₄	1	1	yes	characteristic
25	C₈×C₅₀₉	D₄	1	2	yes	—
26	C₂²×C₅₀₉	#41	1	1	yes	characteristic
27	C₂×D₅₀₉	—	2	2	no	—
28	C₄×C₅₀₉	—	4	1	no	—
29	C₄×C₅₀₉	C₂²×C₄	1	1	yes	characteristic
30	C₄×C₅₀₉	C₂×D₄	1	1	yes	characteristic
31	Dic₅₀₉	—	2	2	no	—
32	C₂×C₅₀₉	—	2	1	no	—
33	C₂×C₅₀₉	C₄×D₄	1	1	yes	characteristic
34	D₅₀₉	—	4	2	no	—
35	C₅₀₉	#444	1	1	yes	characteristic, complemented, Sylow
36	#444	—	509	1	no	maximal, Sylow
37	#135	—	509	1	no	—
38	#131	—	509	2	no	—
39	#108	—	509	1	no	—
40	#104	—	509	2	no	—
41	#97	D₅₀₉	1	1	yes	characteristic, complemented
42	#41	—	509	2	no	—
43	Q₁₆	—	509	1	no	—
44	SD₁₆	—	509	2	no	—
45	D₈	—	509	1	no	—
46	M₁₆	—	1018	2	no	—
47	M₁₆	—	509	2	no	—
48	C₂×C₈	—	1018	2	no	—
49	C₂×C₈	C₂×D₅₀₉	1	1	yes	characteristic
50	C₂×C₈	C₂×D₅₀₉	1	1	yes	characteristic
51	C₄²	C₂×D₅₀₉	1	1	yes	characteristic
52	Q₈	—	509	2	no	—
53	D₄	—	1018	2	no	—
54	D₄	—	509	2	no	—
55	C₂×C₄	—	2	1	no	—
56	C₂×C₄	—	1018	2	no	—
57	C₂×C₄	C₂²×D₅₀₉	1	1	yes	characteristic
58	C₈	—	1018	2	no	—
59	C₈	C₄×D₅₀₉	1	1	yes	characteristic
60	C₈	C₄×D₅₀₉	1	1	yes	characteristic
61	C₈	D₂₀₃₆	1	2	yes	—
62	C₂²	—	1018	2	no	—
63	C₂²	#22550861	1	1	yes	characteristic
64	C₄	—	4	1	no	—
65	C₄	—	1018	2	no	—
66	C₄	C₂×C₄×D₅₀₉	1	1	yes	characteristic
67	C₄	C₂×D₂₀₃₆	1	1	yes	characteristic
68	C₂	—	2	1	no	—
69	C₂	—	2036	2	no	—
70	C₂	C₄×D₂₀₃₆	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×D₂₀₃₆
commutator	C₄×C₅₀₉	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₅₀₉
Fitting	#97×C₅₀₉	C₂
socle	C₂×C₅₀₉	C₄×D₄
prime radical	#30654237	C₂
cosocle	C₂×C₄×C₅₀₉	C₂³
hypercenter	#97	D₅₀₉
hypocenter	C₅₀₉	#444
LCS	G ▷ C₄×C₅₀₉ ▷ C₂×C₅₀₉ ▷ C₅₀₉
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ #97

Complements

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

N	H ≅ G/N	type
2. #30654221	C₂	semidirect
3. #30654151	C₂	semidirect
4. #30654143	C₂	semidirect
5. #97×C₅₀₉	C₂	semidirect
7. Dic₂₀₃₆	C₄	semidirect
8. #22550850	C₄	semidirect
9. D₄₀₇₂	C₄	semidirect
35. C₅₀₉	#444	semidirect
41. #97	D₅₀₉	semidirect

Maximal subgroups

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

subgroup	index
#30654237	2
#30654221	2
#30654151	2
#30654143	2
#97×C₅₀₉	2
#444	509