Subgroups — #45656113

Order 32576.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₅₀₉⋊M₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×D₅₀₉⋊C₈	C₂	1	1	yes	maximal, characteristic
3	M₁₆×D₅₀₉	C₂	1	1	yes	maximal, characteristic
4	#22550871	C₄	1	2	yes	—
5	#22550870	—	2	1	no	—
6	D₅₀₉⋊C₈	—	2	1	no	—
7	D₅₀₉⋊C₈	—	2	1	no	—
8	D₅₀₉⋊C₈	C₄	1	2	yes	—
9	C₂×#16861332	—	2	1	no	—
10	C₂×C₄×D₅₀₉	C₂²	1	1	yes	characteristic
11	#22550845	C₄	1	1	yes	characteristic
12	#22550844	—	2	1	no	—
13	C₈×D₅₀₉	—	2	1	no	—
14	M₁₆×C₅₀₉	C₄	1	1	yes	characteristic
15	#16861332	—	2	2	no	—
16	#16861332	—	2	2	no	—
17	C₂²×D₅₀₉	Q₈	1	1	yes	characteristic
18	C₂×Dic₅₀₉	D₄	1	1	yes	characteristic
19	C₄×D₅₀₉	C₂×C₄	1	1	yes	characteristic
20	C₄×D₅₀₉	C₂×C₄	1	1	yes	characteristic
21	C₄×D₅₀₉	D₄	1	1	yes	characteristic
22	C₄×D₅₀₉	D₄	1	1	yes	characteristic
23	#16861325	—	2	1	no	—
24	C₂×C₄×C₅₀₉	C₂×C₄	1	1	yes	characteristic
25	C₈×C₅₀₉	—	2	1	no	—
26	C₂²×C₅₀₉	C₄⋊C₄	1	1	yes	characteristic
27	C₂×D₅₀₉	—	2	2	no	—
28	C₂×D₅₀₉	C₄⋊C₄	1	1	yes	characteristic
29	C₂×D₅₀₉	C₄⋊C₄	1	1	yes	characteristic
30	C₄×C₅₀₉	C₄²	1	1	yes	characteristic
31	C₄×C₅₀₉	C₂²⋊C₄	1	1	yes	characteristic
32	Dic₅₀₉	C₂²⋊C₄	1	1	yes	characteristic
33	Dic₅₀₉	C₂²⋊C₄	1	1	yes	characteristic
34	C₂×C₅₀₉	—	2	1	no	—
35	C₂×C₅₀₉	#95	1	1	yes	characteristic
36	D₅₀₉	—	2	1	no	—
37	D₅₀₉	#108	1	2	yes	—
38	C₅₀₉	#341	1	1	yes	characteristic, complemented, Sylow
39	#341	—	509	1	no	maximal, Sylow
40	C₂×M₁₆	—	509	1	no	—
41	C₂×M₁₆	—	509	1	no	—
42	C₂²×C₈	—	509	1	no	—
43	C₂²×C₄	—	509	1	no	—
44	M₁₆	—	1018	1	no	—
45	M₁₆	—	1018	1	no	—
46	M₁₆	—	509	2	no	—
47	M₁₆	—	509	1	no	—
48	M₁₆	#14523266	1	1	yes	characteristic
49	C₂×C₈	—	1018	1	no	—
50	C₂×C₈	—	1018	1	no	—
51	C₂×C₈	—	1018	1	no	—
52	C₂×C₈	—	1018	1	no	—
53	C₂×C₈	—	509	2	no	—
54	C₂³	—	509	1	no	—
55	C₂×C₄	—	509	1	no	—
56	C₂×C₄	—	509	1	no	—
57	C₂×C₄	—	509	1	no	—
58	C₂×C₄	—	509	1	no	—
59	C₂×C₄	—	509	1	no	—
60	C₂×C₄	C₂×#14523266	1	1	yes	characteristic
61	C₈	—	2	1	no	—
62	C₈	—	1018	2	no	—
63	C₈	—	1018	2	no	—
64	C₈	—	1018	1	no	—
65	C₂²	—	1018	2	no	—
66	C₂²	—	509	1	no	—
67	C₂²	—	509	1	no	—
68	C₂²	C₄⋊#14523266	1	1	yes	characteristic
69	C₄	—	509	1	no	—
70	C₄	—	509	1	no	—
71	C₄	C₄×#14523266	1	1	yes	characteristic
72	C₄	C₂²⋊#14523266	1	1	yes	characteristic
73	C₂	—	2	1	no	—
74	C₂	—	1018	1	no	—
75	C₂	—	509	2	no	—
76	C₂	(C₂×#14523266)⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂²⋊#14523266
commutator	C₄×C₅₀₉	C₄²
Frattini	C₂×C₄	C₂×#14523266
Fitting	M₁₆×C₅₀₉	C₄
socle	C₂×C₅₀₉	#95
prime radical	C₂²×D₅₀₉	Q₈
cosocle	C₂×C₄×D₅₀₉	C₂²
hypercenter	M₁₆	#14523266
hypocenter	C₅₀₉	#341
LCS	G ▷ C₄×C₅₀₉ ▷ C₂×C₅₀₉ ▷ C₅₀₉
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ M₁₆

Complements

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

N	H ≅ G/N	type
38. C₅₀₉	#341	semidirect

Maximal subgroups

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

subgroup	index
D₅₀₉⋊M₁₆	2
C₂×D₅₀₉⋊C₈	2
M₁₆×D₅₀₉	2
#341	509