Subgroups — #45656204

Order 32576.

130 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₅₀₉⋊C₈	C₂	1	1	yes	maximal, characteristic
2	#30654294	C₂	1	1	yes	maximal, characteristic
3	#30654269	C₂	1	2	yes	maximal, complemented
4	C₈×#14523266	C₂	1	2	yes	maximal, complemented
5	M₁₆×D₅₀₉	C₂	1	1	yes	maximal, characteristic
6	D₅₀₉⋊C₈	C₄	1	2	yes	complemented
7	D₅₀₉⋊C₈	C₂²	1	1	yes	characteristic
8	D₅₀₉⋊C₈	C₂²	1	1	yes	characteristic
9	C₂×#16861332	C₄	1	2	yes	complemented
10	C₄⋊#14523266	C₄	1	2	yes	—
11	C₂²⋊#14523266	C₄	1	2	yes	—
12	C₄×#14523266	C₂²	1	2	yes	—
13	C₂×C₄×D₅₀₉	C₂²	1	1	yes	characteristic
14	#22550845	C₄	1	1	yes	characteristic, complemented
15	#22550844	C₄	1	2	yes	complemented
16	C₈×D₅₀₉	C₂²	1	2	yes	—
17	M₁₆×C₅₀₉	C₄	1	1	yes	characteristic, complemented
18	C₂×#14523266	C₂×C₄	1	4	yes	—
19	#16861332	C₂×C₄	1	2	yes	—
20	#16861332	C₂×C₄	1	2	yes	—
21	C₂²×D₅₀₉	C₂×C₄	1	1	yes	characteristic
22	C₂×Dic₅₀₉	C₂×C₄	1	1	yes	characteristic
23	C₄×D₅₀₉	C₂×C₄	1	1	yes	characteristic
24	C₄×D₅₀₉	C₂×C₄	1	1	yes	characteristic
25	C₄×D₅₀₉	C₂×C₄	1	1	yes	characteristic
26	C₄×D₅₀₉	C₂³	1	1	yes	characteristic
27	#16861325	C₂×C₄	1	2	yes	—
28	C₂×C₄×C₅₀₉	C₂×C₄	1	1	yes	characteristic
29	C₈×C₅₀₉	C₂×C₄	1	2	yes	—
30	C₂²×C₅₀₉	C₄²	1	1	yes	characteristic
31	C₂×D₅₀₉	—	2	2	no	—
32	C₂×D₅₀₉	C₄²	1	1	yes	characteristic
33	C₂×D₅₀₉	C₂²×C₄	1	1	yes	characteristic
34	#14523266	—	2	4	no	—
35	C₄×C₅₀₉	C₄²	1	1	yes	characteristic
36	C₄×C₅₀₉	C₂²×C₄	1	1	yes	characteristic
37	Dic₅₀₉	C₄²	1	1	yes	characteristic
38	Dic₅₀₉	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₅₀₉	—	2	1	no	—
40	C₂×C₅₀₉	C₂×C₄²	1	1	yes	characteristic
41	D₅₀₉	—	2	1	no	—
42	D₅₀₉	#131	1	2	yes	—
43	C₅₀₉	#405	1	1	yes	characteristic, complemented, Sylow
44	#405	—	509	1	no	maximal, Sylow
45	C₂×M₁₆	—	509	1	no	—
46	C₂²×C₈	—	509	1	no	—
47	#117	—	509	1	no	—
48	#97	—	509	2	no	—
49	C₄×C₈	—	509	2	no	—
50	C₂²×C₄	—	509	1	no	—
51	M₁₆	—	509	1	no	—
52	M₁₆	—	509	2	no	—
53	M₁₆	#14523266	1	1	yes	characteristic, complemented
54	C₂×C₈	—	509	2	no	—
55	C₂×C₈	—	509	2	no	—
56	C₂×C₈	—	509	1	no	—
57	C₂×C₈	—	509	1	no	—
58	C₂×C₈	—	509	2	no	—
59	C₄⋊C₄	—	509	2	no	—
60	C₂²⋊C₄	—	509	2	no	—
61	C₄²	—	509	2	no	—
62	C₂³	—	509	1	no	—
63	C₂×C₄	—	509	4	no	—
64	C₂×C₄	—	509	1	no	—
65	C₂×C₄	—	509	1	no	—
66	C₂×C₄	—	509	1	no	—
67	C₂×C₄	—	509	1	no	—
68	C₂×C₄	—	509	1	no	—
69	C₂×C₄	C₂×#14523266	1	1	yes	characteristic
70	C₈	—	509	2	no	—
71	C₈	—	509	2	no	—
72	C₈	—	509	2	no	—
73	C₈	C₂×#14523266	1	2	yes	—
74	C₂²	—	1018	2	no	—
75	C₂²	—	509	1	no	—
76	C₂²	—	509	1	no	—
77	C₂²	C₄×#14523266	1	1	yes	characteristic
78	C₄	—	1018	4	no	—
79	C₄	—	509	1	no	—
80	C₄	—	509	1	no	—
81	C₄	C₄×#14523266	1	1	yes	characteristic
82	C₄	C₂²×#14523266	1	1	yes	characteristic
83	C₂	—	2	1	no	—
84	C₂	—	1018	1	no	—
85	C₂	—	509	2	no	—
86	C₂	C₂×C₄×#14523266	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄	C₂²×#14523266
commutator, socle	C₂×C₅₀₉	C₂×C₄²
Fitting	M₁₆×C₅₀₉	C₄
prime radical	C₂²×D₅₀₉	C₂×C₄
cosocle	C₄×D₅₀₉	C₂³
hypercenter	M₁₆	#14523266
hypocenter	C₅₀₉	#405
LCS	G ▷ C₂×C₅₀₉ ▷ C₅₀₉
UCS	C₁ ◁ C₄ ◁ M₁₆

Complements

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

N	H ≅ G/N	type
3. #30654269	C₂	semidirect
4. C₈×#14523266	C₂	semidirect
6. D₅₀₉⋊C₈	C₄	semidirect
9. C₂×#16861332	C₄	semidirect
14. #22550845	C₄	semidirect
15. #22550844	C₄	semidirect
17. M₁₆×C₅₀₉	C₄	semidirect
43. C₅₀₉	#405	semidirect
53. M₁₆	#14523266	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₅₀₉⋊C₈	2
#30654294	2
#30654269	2
C₈×#14523266	2
M₁₆×D₅₀₉	2
#405	509