Subgroups — #45656156

Order 32576.

62 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#30654290	C₂	1	1	yes	maximal, characteristic, complemented
2	#30654285	C₂	1	1	yes	maximal, characteristic, complemented
3	D₈×D₅₀₉	C₂	1	1	yes	maximal, characteristic
4	#22550871	—	2	1	no	—
5	#22550864	—	2	1	no	—
6	D₄×D₅₀₉	—	2	1	no	—
7	D₄₀₇₂	C₄	1	1	yes	characteristic
8	#22550846	—	2	1	no	—
9	C₈×D₅₀₉	C₂²	1	1	yes	characteristic
10	D₈×C₅₀₉	C₄	1	1	yes	characteristic
11	#16861332	—	4	1	no	—
12	C₂²×D₅₀₉	—	4	1	no	—
13	D₂₀₃₆	—	2	1	no	—
14	C₂²⋊D₅₀₉	—	4	1	no	—
15	C₄×D₅₀₉	D₄	1	1	yes	characteristic
16	#16861325	D₄	1	1	yes	characteristic
17	D₄×C₅₀₉	—	2	1	no	—
18	C₈×C₅₀₉	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₅₀₉	—	4	1	no	—
20	C₂×D₅₀₉	—	4	1	no	—
21	C₂×D₅₀₉	—	8	2	no	—
22	C₂×D₅₀₉	SD₁₆	1	1	yes	characteristic
23	C₄×C₅₀₉	C₂²⋊C₄	1	1	yes	characteristic
24	Dic₅₀₉	D₈	1	1	yes	characteristic
25	C₂×C₅₀₉	—	8	1	no	—
26	C₂×C₅₀₉	#102	1	1	yes	characteristic
27	D₅₀₉	—	2	1	no	—
28	D₅₀₉	—	8	1	no	—
29	C₅₀₉	#361	1	1	yes	characteristic, complemented, Sylow
30	#361	—	509	1	no	maximal, Sylow
31	C₂×D₈	—	509	1	no	—
32	M₃₂	—	509	1	no	—
33	#108	—	509	1	no	—
34	C₂×D₄	—	1018	1	no	—
35	D₈	—	1018	1	no	—
36	D₈	—	509	1	no	—
37	D₈	#14523266	1	1	yes	characteristic
38	M₁₆	—	1018	1	no	—
39	C₂×C₈	—	509	1	no	—
40	C₁₆	—	1018	1	no	—
41	C₂³	—	2036	1	no	—
42	D₄	—	2	1	no	—
43	D₄	—	2036	1	no	—
44	D₄	—	1018	1	no	—
45	C₂×C₄	—	509	1	no	—
46	C₈	—	2036	1	no	—
47	C₈	—	509	1	no	—
48	C₈	C₂×#14523266	1	1	yes	characteristic
49	C₂²	—	4	1	no	—
50	C₂²	—	4072	2	no	—
51	C₂²	—	2036	1	no	—
52	C₂²	—	509	1	no	—
53	C₄	—	509	1	no	—
54	C₄	C₂²⋊#14523266	1	1	yes	characteristic
55	C₂	—	8	1	no	—
56	C₂	—	4072	1	no	—
57	C₂	—	1018	1	no	—
58	C₂	#30654277	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#30654277
commutator	C₈×C₅₀₉	C₂×C₄
Frattini	C₈	C₂×#14523266
Fitting	D₈×C₅₀₉	C₄
socle	C₂×C₅₀₉	#102
prime radical	D₈×D₅₀₉	C₂
cosocle	C₈×D₅₀₉	C₂²
hypercenter	D₈	#14523266
hypocenter	C₅₀₉	#361
LCS	G ▷ C₈×C₅₀₉ ▷ C₄×C₅₀₉ ▷ C₂×C₅₀₉ ▷ C₅₀₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ D₈

Complements

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

N	H ≅ G/N	type
1. #30654290	C₂	semidirect
2. #30654285	C₂	semidirect
29. C₅₀₉	#361	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
D₈×D₅₀₉	2
#361	509