Subgroups — #45656157

Order 32576.

54 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#30654290	C₂	1	1	yes	maximal, characteristic
2	#30654285	C₂	1	1	yes	maximal, characteristic
3	Q₁₆×D₅₀₉	C₂	1	1	yes	maximal, characteristic
4	#22550871	—	2	1	no	—
5	#22550864	—	2	1	no	—
6	Q₈×D₅₀₉	—	2	1	no	—
7	Dic₂₀₃₆	C₄	1	1	yes	characteristic
8	#22550851	—	2	1	no	—
9	C₈×D₅₀₉	C₂²	1	1	yes	characteristic
10	Q₁₆×C₅₀₉	C₄	1	1	yes	characteristic
11	#16861332	—	4	1	no	—
12	Dic₁₀₁₈	—	2	1	no	—
13	Dic₁₀₁₈	—	4	1	no	—
14	C₄×D₅₀₉	—	4	1	no	—
15	C₄×D₅₀₉	D₄	1	1	yes	characteristic
16	#16861325	D₄	1	1	yes	characteristic
17	Q₈×C₅₀₉	—	2	1	no	—
18	C₈×C₅₀₉	C₂×C₄	1	1	yes	characteristic
19	C₂×D₅₀₉	SD₁₆	1	1	yes	characteristic
20	C₄×C₅₀₉	—	4	1	no	—
21	C₄×C₅₀₉	C₂²⋊C₄	1	1	yes	characteristic
22	Dic₅₀₉	—	4	1	no	—
23	Dic₅₀₉	D₈	1	1	yes	characteristic
24	C₂×C₅₀₉	#102	1	1	yes	characteristic
25	D₅₀₉	—	2	1	no	—
26	C₅₀₉	#362	1	1	yes	characteristic, complemented, Sylow
27	#362	—	509	1	no	maximal, Sylow
28	C₂×Q₁₆	—	509	1	no	—
29	M₃₂	—	509	1	no	—
30	#108	—	509	1	no	—
31	C₂×Q₈	—	1018	1	no	—
32	Q₁₆	—	1018	1	no	—
33	Q₁₆	—	509	1	no	—
34	Q₁₆	#14523266	1	1	yes	characteristic
35	M₁₆	—	1018	1	no	—
36	C₂×C₈	—	509	1	no	—
37	C₁₆	—	1018	1	no	—
38	Q₈	—	2	1	no	—
39	Q₈	—	2036	1	no	—
40	Q₈	—	1018	1	no	—
41	C₂×C₄	—	2036	1	no	—
42	C₂×C₄	—	509	1	no	—
43	C₈	—	2036	1	no	—
44	C₈	—	509	1	no	—
45	C₈	C₂×#14523266	1	1	yes	characteristic
46	C₂²	—	509	1	no	—
47	C₄	—	4	1	no	—
48	C₄	—	2036	1	no	—
49	C₄	—	509	1	no	—
50	C₄	C₂²⋊#14523266	1	1	yes	characteristic
51	C₂	—	1018	1	no	—
52	C₂	#30654277	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#30654277
commutator	C₈×C₅₀₉	C₂×C₄
Frattini	C₈	C₂×#14523266
Fitting	Q₁₆×C₅₀₉	C₄
socle	C₂×C₅₀₉	#102
prime radical	C₂×D₅₀₉	SD₁₆
cosocle	C₈×D₅₀₉	C₂²
hypercenter	Q₁₆	#14523266
hypocenter	C₅₀₉	#362
LCS	G ▷ C₈×C₅₀₉ ▷ C₄×C₅₀₉ ▷ C₂×C₅₀₉ ▷ C₅₀₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ Q₁₆

Complements

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

N	H ≅ G/N	type
26. C₅₀₉	#362	semidirect

Maximal subgroups

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

subgroup	index
#30654290	2
#30654285	2
Q₁₆×D₅₀₉	2
#362	509