Subgroups — #45656285

Order 32576.

104 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#30654302	C₂	1	2	yes	maximal, complemented
2	#30654285	C₂	1	1	yes	maximal, characteristic, complemented
3	#30654279	C₂	1	2	yes	maximal, complemented
4	#30654269	C₂	1	1	yes	maximal, characteristic, complemented
5	#30654234	C₂	1	1	yes	maximal, characteristic
6	#22550871	C₂²	1	2	yes	—
7	#22550870	—	2	2	no	—
8	D₅₀₉⋊C₈	C₂²	1	1	yes	characteristic
9	C₂×#16861332	—	2	2	no	—
10	C₄×#14523266	C₂²	1	1	yes	characteristic
11	#22550860	C₂²	1	2	yes	—
12	Dic₂₀₃₆	C₄	1	1	yes	characteristic, complemented
13	#22550849	C₄	1	2	yes	complemented
14	C₈×D₅₀₉	C₂²	1	1	yes	characteristic
15	D₈×C₅₀₉	C₄	1	1	yes	characteristic, complemented
16	C₂×#14523266	—	2	1	no	—
17	#16861332	—	2	2	no	—
18	#16861332	D₄	1	2	yes	—
19	Dic₁₀₁₈	C₂×C₄	1	2	yes	—
20	C₂²⋊D₅₀₉	—	2	2	no	—
21	C₂×Dic₅₀₉	—	2	2	no	—
22	C₄×D₅₀₉	C₂³	1	1	yes	characteristic
23	#16861325	C₂×C₄	1	1	yes	characteristic
24	D₄×C₅₀₉	C₂×C₄	1	2	yes	—
25	C₈×C₅₀₉	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₅₀₉	—	2	2	no	—
27	C₂×D₅₀₉	#41	1	1	yes	characteristic
28	#14523266	—	4	1	no	—
29	C₄×C₅₀₉	C₂²×C₄	1	1	yes	characteristic
30	Dic₅₀₉	—	2	2	no	—
31	Dic₅₀₉	C₂×D₄	1	1	yes	characteristic
32	C₂×C₅₀₉	—	4	2	no	—
33	C₂×C₅₀₉	C₄×D₄	1	1	yes	characteristic
34	D₅₀₉	—	2	1	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	—	509	1	no	—
42	#41	—	509	2	no	—
43	Q₁₆	—	509	1	no	—
44	SD₁₆	—	509	2	no	—
45	D₈	#14523266	1	1	yes	characteristic, complemented
46	M₁₆	—	1018	2	no	—
47	M₁₆	—	509	2	no	—
48	C₂×C₈	—	1018	2	no	—
49	C₂×C₈	—	509	1	no	—
50	C₂×C₈	—	509	1	no	—
51	C₄²	—	509	1	no	—
52	Q₈	—	509	2	no	—
53	D₄	—	1018	2	no	—
54	D₄	C₂×#14523266	1	2	yes	—
55	C₂×C₄	—	1018	1	no	—
56	C₂×C₄	—	1018	2	no	—
57	C₂×C₄	—	509	1	no	—
58	C₈	—	1018	2	no	—
59	C₈	—	509	2	no	—
60	C₈	—	509	1	no	—
61	C₈	C₂×#14523266	1	1	yes	characteristic
62	C₂²	—	2	2	no	—
63	C₂²	—	509	1	no	—
64	C₄	—	2036	1	no	—
65	C₄	—	1018	2	no	—
66	C₄	—	509	1	no	—
67	C₄	C₂²×#14523266	1	1	yes	characteristic
68	C₂	—	4	2	no	—
69	C₂	—	1018	1	no	—
70	C₂	D₄×#14523266	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₄×#14523266
commutator	C₄×C₅₀₉	C₂²×C₄
Frattini	C₄	C₂²×#14523266
Fitting	D₈×C₅₀₉	C₄
socle	C₂×C₅₀₉	C₄×D₄
prime radical	#30654234	C₂
cosocle	C₄×D₅₀₉	C₂³
hypercenter	D₈	#14523266
hypocenter	C₅₀₉	#444
LCS	G ▷ C₄×C₅₀₉ ▷ C₂×C₅₀₉ ▷ C₅₀₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ D₈

Complements

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

N	H ≅ G/N	type
1. #30654302	C₂	semidirect
2. #30654285	C₂	semidirect
3. #30654279	C₂	semidirect
4. #30654269	C₂	semidirect
12. Dic₂₀₃₆	C₄	semidirect
13. #22550849	C₄	semidirect
15. D₈×C₅₀₉	C₄	semidirect
35. C₅₀₉	#444	semidirect
45. D₈	#14523266	semidirect

Maximal subgroups

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

subgroup	index
#30654302	2
#30654285	2
#30654279	2
#30654269	2
#30654234	2
#444	509