Subgroups — #45656280

Order 32576.

106 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	#30654286	C₂	1	1	yes	maximal, characteristic, complemented
3	#30654279	C₂	1	2	yes	maximal, complemented
4	C₈×#14523266	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	1	yes	characteristic
19	#16861332	D₄	1	1	yes	characteristic
20	Dic₁₀₁₈	C₂×C₄	1	2	yes	—
21	C₂²⋊D₅₀₉	—	2	2	no	—
22	C₂×Dic₅₀₉	—	2	2	no	—
23	C₄×D₅₀₉	C₂³	1	1	yes	characteristic
24	#16861325	C₂×C₄	1	1	yes	characteristic
25	D₄×C₅₀₉	C₂×C₄	1	2	yes	—
26	C₈×C₅₀₉	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₅₀₉	—	2	2	no	—
28	C₂×D₅₀₉	#41	1	1	yes	characteristic
29	#14523266	—	2	2	no	—
30	C₄×C₅₀₉	C₂²×C₄	1	1	yes	characteristic
31	Dic₅₀₉	—	2	2	no	—
32	Dic₅₀₉	C₂×D₄	1	1	yes	characteristic
33	C₂×C₅₀₉	—	4	2	no	—
34	C₂×C₅₀₉	C₄×D₄	1	1	yes	characteristic
35	D₅₀₉	—	2	1	no	—
36	C₅₀₉	#443	1	1	yes	characteristic, complemented, Sylow
37	#443	—	509	1	no	maximal, Sylow
38	#135	—	509	1	no	—
39	#131	—	509	2	no	—
40	#108	—	509	1	no	—
41	#104	—	509	2	no	—
42	C₄×C₈	—	509	1	no	—
43	#41	—	509	2	no	—
44	Q₁₆	—	509	1	no	—
45	SD₁₆	—	509	2	no	—
46	D₈	#14523266	1	1	yes	characteristic, complemented
47	M₁₆	—	1018	2	no	—
48	M₁₆	—	509	2	no	—
49	C₂×C₈	—	1018	2	no	—
50	C₂×C₈	—	509	1	no	—
51	C₂×C₈	—	509	1	no	—
52	C₄²	—	509	1	no	—
53	Q₈	—	509	2	no	—
54	D₄	—	1018	2	no	—
55	D₄	C₂×#14523266	1	2	yes	—
56	C₂×C₄	—	1018	1	no	—
57	C₂×C₄	—	1018	2	no	—
58	C₂×C₄	—	509	1	no	—
59	C₈	—	1018	2	no	—
60	C₈	—	509	1	no	—
61	C₈	—	509	1	no	—
62	C₈	—	509	1	no	—
63	C₈	C₂×#14523266	1	1	yes	characteristic
64	C₂²	—	2	2	no	—
65	C₂²	—	509	1	no	—
66	C₄	—	1018	2	no	—
67	C₄	—	1018	2	no	—
68	C₄	—	509	1	no	—
69	C₄	C₂²×#14523266	1	1	yes	characteristic
70	C₂	—	4	2	no	—
71	C₂	—	1018	1	no	—
72	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	#443
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. #30654286	C₂	semidirect
3. #30654279	C₂	semidirect
4. C₈×#14523266	C₂	semidirect
12. Dic₂₀₃₆	C₄	semidirect
13. #22550849	C₄	semidirect
15. D₈×C₅₀₉	C₄	semidirect
36. C₅₀₉	#443	semidirect
46. 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
#30654286	2
#30654279	2
C₈×#14523266	2
#30654234	2
#443	509