Subgroups — #45656102

Order 32576.

54 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊#16861332	C₂	1	2	yes	maximal
2	C₈×Dic₅₀₉	C₂	1	1	yes	maximal, characteristic
3	C₂×#16861332	—	2	2	no	—
4	C₂×#16861325	C₄	1	1	yes	characteristic
5	C₄×Dic₅₀₉	C₂²	1	1	yes	characteristic
6	C₂×C₈×C₅₀₉	C₄	1	1	yes	characteristic
7	#16861332	—	4	2	no	—
8	C₂×Dic₅₀₉	D₄	1	1	yes	characteristic
9	C₂×Dic₅₀₉	Q₈	1	1	yes	characteristic
10	#16861325	C₈	1	2	yes	complemented
11	C₂×C₄×C₅₀₉	C₂×C₄	1	1	yes	characteristic
12	C₈×C₅₀₉	—	2	1	no	—
13	C₂²×C₅₀₉	C₄⋊C₄	1	1	yes	characteristic
14	C₄×C₅₀₉	C₂×C₈	1	1	yes	characteristic
15	C₄×C₅₀₉	M₁₆	1	1	yes	characteristic
16	Dic₅₀₉	—	2	1	no	—
17	Dic₅₀₉	D₈	1	1	yes	characteristic
18	Dic₅₀₉	Q₁₆	1	1	yes	characteristic
19	C₂×C₅₀₉	C₄⋊C₈	1	1	yes	characteristic
20	C₂×C₅₀₉	#107	1	1	yes	characteristic
21	C₂×C₅₀₉	#108	1	1	yes	characteristic
22	C₅₀₉	#335	1	1	yes	characteristic, complemented, Sylow
23	#335	—	509	1	no	maximal, Sylow
24	C₄⋊C₈	—	509	2	no	—
25	C₄×C₈	—	509	1	no	—
26	C₂×C₈	—	1018	2	no	—
27	C₂×C₈	—	509	1	no	—
28	C₂×C₈	#14523266	1	1	yes	characteristic
29	C₄²	—	509	1	no	—
30	C₂×C₄	—	509	1	no	—
31	C₂×C₄	—	509	1	no	—
32	C₂×C₄	C₂×#14523266	1	1	yes	characteristic
33	C₈	—	2	1	no	—
34	C₈	—	2036	2	no	—
35	C₈	—	509	2	no	—
36	C₂²	C₄⋊#14523266	1	1	yes	characteristic
37	C₄	—	1018	1	no	—
38	C₄	—	509	1	no	—
39	C₄	—	509	1	no	—
40	C₄	D₅₀₉⋊C₈	1	1	yes	characteristic
41	C₄	#22550870	1	1	yes	characteristic
42	C₂	#30654282	1	1	yes	characteristic
43	C₂	#30654284	1	1	yes	characteristic
44	C₂	#30654285	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₄⋊#14523266
commutator	C₄×C₅₀₉	C₂×C₈
Frattini	C₂×C₄	C₂×#14523266
Fitting	C₂×C₈×C₅₀₉	C₄
socle, prime radical	C₂²×C₅₀₉	C₄⋊C₄
cosocle	C₄×Dic₅₀₉	C₂²
hypercenter	C₂×C₈	#14523266
hypocenter	C₅₀₉	#335
LCS	G ▷ C₄×C₅₀₉ ▷ C₂×C₅₀₉ ▷ C₅₀₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈

Complements

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

N	H ≅ G/N	type
10. #16861325	C₈	semidirect
22. C₅₀₉	#335	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊#16861332	2
C₈×Dic₅₀₉	2
#335	509