Subgroups — #45655644

Order 32576.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#30654207	C₂	1	1	yes	maximal, characteristic
2	#30654148	C₂	1	2	yes	maximal
3	#30654146	C₂	1	2	yes	maximal
4	#30654126	C₂	1	1	yes	maximal, characteristic
5	C₄⋊Q₈×C₅₀₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#16861325	C₂²	1	2	yes	—
7	C₄⋊Dic₅₀₉	C₂²	1	1	yes	characteristic
8	C₄⋊Dic₅₀₉	C₂²	1	1	yes	characteristic
9	#22550840	—	2	1	no	—
10	#22550840	—	2	1	no	—
11	C₂×Q₈×C₅₀₉	—	2	1	no	—
12	C₄⋊C₄×C₅₀₉	—	2	1	no	—
13	C₄⋊C₄×C₅₀₉	C₂²	1	2	yes	—
14	C₄²×C₅₀₉	C₂²	1	1	yes	characteristic
15	C₂×Dic₅₀₉	—	2	1	no	—
16	C₂×Dic₅₀₉	—	2	1	no	—
17	#16861325	Q₈	1	4	yes	complemented
18	Q₈×C₅₀₉	—	4	2	no	—
19	C₂×C₄×C₅₀₉	—	2	2	no	—
20	C₂×C₄×C₅₀₉	D₄	1	1	yes	characteristic
21	C₂×C₄×C₅₀₉	D₄	1	1	yes	characteristic
22	C₂×C₄×C₅₀₉	C₂³	1	1	yes	characteristic
23	C₂²×C₅₀₉	C₂×D₄	1	1	yes	characteristic
24	C₄×C₅₀₉	—	2	1	no	—
25	C₄×C₅₀₉	—	2	1	no	—
26	C₄×C₅₀₉	—	4	2	no	—
27	C₄×C₅₀₉	C₂×Q₈	1	2	yes	—
28	Dic₅₀₉	—	4	1	no	—
29	Dic₅₀₉	—	4	1	no	—
30	C₂×C₅₀₉	C₄⋊Q₈	1	1	yes	characteristic
31	C₂×C₅₀₉	AGL(1,ℤ/8)	1	1	yes	characteristic
32	C₂×C₅₀₉	#137	1	1	yes	characteristic
33	C₅₀₉	#501	1	1	yes	characteristic, complemented, Sylow
34	#501	—	509	1	no	maximal, Sylow
35	C₄⋊Q₈	D₅₀₉	1	1	yes	characteristic
36	#125	—	509	1	no	—
37	#107	—	509	2	no	—
38	#106	—	509	2	no	—
39	#97	—	509	1	no	—
40	C₂×Q₈	—	2	1	no	—
41	C₂×C₈	—	509	2	no	—
42	C₄⋊C₄	—	2	1	no	—
43	C₄⋊C₄	—	1018	1	no	—
44	C₄⋊C₄	—	1018	1	no	—
45	C₄⋊C₄	—	509	1	no	—
46	C₄⋊C₄	—	509	1	no	—
47	C₄⋊C₄	C₂×D₅₀₉	1	2	yes	—
48	C₄²	C₂×D₅₀₉	1	1	yes	characteristic
49	Q₈	—	4	2	no	—
50	C₂×C₄	—	2	2	no	—
51	C₂×C₄	—	1018	1	no	—
52	C₂×C₄	—	1018	1	no	—
53	C₂×C₄	C₂²⋊D₅₀₉	1	1	yes	characteristic
54	C₂×C₄	C₂²⋊D₅₀₉	1	1	yes	characteristic
55	C₂×C₄	C₂²×D₅₀₉	1	1	yes	characteristic
56	C₈	—	509	4	no	—
57	C₂²	C₂×C₂²⋊D₅₀₉	1	1	yes	characteristic
58	C₄	—	2	1	no	—
59	C₄	—	2	1	no	—
60	C₄	—	4	2	no	—
61	C₄	—	2036	1	no	—
62	C₄	—	2036	1	no	—
63	C₄	Q₈×D₅₀₉	1	2	yes	—
64	C₂	#30654213	1	1	yes	characteristic
65	C₂	#30654241	1	1	yes	characteristic
66	C₂	#30654246	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₅₀₉
commutator, cosocle	C₂×C₄×C₅₀₉	C₂³
Frattini	C₂×C₄	C₂²×D₅₀₉
Fitting	C₄⋊Q₈×C₅₀₉	C₂
socle, prime radical	C₂²×C₅₀₉	C₂×D₄
hypercenter	C₄⋊Q₈	D₅₀₉
hypocenter	C₅₀₉	#501
LCS	G ▷ C₂×C₄×C₅₀₉ ▷ C₂×C₅₀₉ ▷ C₅₀₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
17. #16861325	Q₈	semidirect
33. C₅₀₉	#501	semidirect

Maximal subgroups

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

subgroup	index
#30654207	2
#30654148	2
#30654146	2
#30654126	2
C₄⋊Q₈×C₅₀₉	2
#501	509