Subgroups — #45656136

Order 32576.

56 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#30654289	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#22550864	C₂	1	1	yes	maximal, characteristic, complemented
3	#30654222	C₂	1	1	yes	maximal, characteristic
4	#22550864	—	2	1	no	—
5	#22550864	—	2	1	no	—
6	#22550845	—	2	1	no	—
7	#22550845	C₄	1	1	yes	characteristic
8	C₂×#16861325	—	2	1	no	—
9	C₂×#16861325	C₂²	1	1	yes	characteristic
10	#41×C₅₀₉	C₄	1	1	yes	characteristic
11	#16861325	—	2	1	no	—
12	#16861325	D₄	1	1	yes	characteristic
13	#16861325	D₄	1	1	yes	characteristic
14	Q₈×C₅₀₉	C₈	1	1	yes	characteristic
15	D₄×C₅₀₉	—	2	1	no	—
16	D₄×C₅₀₉	C₈	1	1	yes	characteristic
17	C₂×C₄×C₅₀₉	—	2	1	no	—
18	C₂×C₄×C₅₀₉	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₅₀₉	—	2	1	no	—
20	C₂²×C₅₀₉	M₁₆	1	1	yes	characteristic
21	C₄×C₅₀₉	—	2	1	no	—
22	C₄×C₅₀₉	C₂²⋊C₄	1	1	yes	characteristic
23	C₄×C₅₀₉	C₂×C₈	1	1	yes	characteristic
24	C₂×C₅₀₉	—	2	1	no	—
25	C₂×C₅₀₉	—	4	1	no	—
26	C₂×C₅₀₉	C₂²⋊C₈	1	1	yes	characteristic
27	C₅₀₉	#350	1	1	yes	characteristic, complemented, Sylow
28	#350	—	509	1	no	maximal, Sylow
29	#131	—	509	1	no	—
30	M₃₂	—	509	1	no	—
31	C₂×C₁₆	—	509	1	no	—
32	#41	#14523266	1	1	yes	characteristic
33	M₁₆	—	1018	1	no	—
34	M₁₆	—	509	1	no	—
35	C₂×C₈	—	1018	1	no	—
36	C₂×C₈	—	509	1	no	—
37	C₁₆	—	1018	1	no	—
38	C₁₆	—	1018	1	no	—
39	Q₈	#16861332	1	1	yes	characteristic
40	D₄	—	2	1	no	—
41	D₄	#16861332	1	1	yes	characteristic
42	C₂×C₄	—	2	1	no	—
43	C₂×C₄	C₂×#14523266	1	1	yes	characteristic
44	C₈	—	1018	1	no	—
45	C₈	—	509	1	no	—
46	C₈	—	509	1	no	—
47	C₂²	—	2	1	no	—
48	C₂²	#22550870	1	1	yes	characteristic
49	C₄	—	2	1	no	—
50	C₄	C₂²⋊#14523266	1	1	yes	characteristic
51	C₄	C₂×#16861332	1	1	yes	characteristic
52	C₂	—	2	1	no	—
53	C₂	—	4	1	no	—
54	C₂	C₂²⋊#16861332	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂²⋊#14523266
commutator	C₄×C₅₀₉	C₂×C₈
Frattini	C₂×C₄	C₂×#14523266
Fitting, prime radical	#41×C₅₀₉	C₄
socle	C₂×C₅₀₉	C₂²⋊C₈
cosocle	C₂×#16861325	C₂²
hypercenter	#41	#14523266
hypocenter	C₅₀₉	#350
LCS	G ▷ C₄×C₅₀₉ ▷ C₂×C₅₀₉ ▷ C₅₀₉
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ #41

Complements

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

N	H ≅ G/N	type
1. #30654289	C₂	semidirect
2. C₂×#22550864	C₂	semidirect
27. C₅₀₉	#350	semidirect

Maximal subgroups

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

subgroup	index
#30654289	2
C₂×#22550864	2
#30654222	2
#350	509