Subgroups — Q₈×#16861325 #45655364

Order 32576.

102 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊#16861325	C₂	1	3	yes	maximal
2	C₄×#16861325	C₂	1	3	yes	maximal
3	C₄×Q₈×C₅₀₉	C₂	1	1	yes	maximal, characteristic
4	C₂×#16861325	C₂²	1	3	yes	—
5	C₂×#16861325	C₂²	1	1	yes	characteristic
6	C₂×Q₈×C₅₀₉	C₄	1	1	yes	characteristic
7	C₄⋊C₄×C₅₀₉	C₄	1	3	yes	—
8	C₄²×C₅₀₉	C₂²	1	3	yes	—
9	#16861325	—	2	3	no	—
10	#16861325	Q₈	1	2	yes	direct product factor
11	Q₈×C₅₀₉	C₈	1	4	yes	complemented
12	C₂×C₄×C₅₀₉	C₂×C₄	1	3	yes	—
13	C₂×C₄×C₅₀₉	C₂×C₄	1	3	yes	—
14	C₂×C₄×C₅₀₉	C₂³	1	1	yes	characteristic
15	C₂²×C₅₀₉	C₂²×C₄	1	1	yes	characteristic
16	C₄×C₅₀₉	—	2	3	no	—
17	C₄×C₅₀₉	C₂×C₈	1	6	yes	—
18	C₄×C₅₀₉	C₂×Q₈	1	1	yes	characteristic
19	C₄×C₅₀₉	#41	1	1	yes	characteristic
20	C₂×C₅₀₉	C₄×Q₈	1	1	yes	characteristic
21	C₂×C₅₀₉	C₂²×C₈	1	1	yes	characteristic
22	C₂×C₅₀₉	#131	1	1	yes	characteristic
23	C₅₀₉	C₈×Q₈	1	1	yes	characteristic, complemented, Sylow
24	C₈×Q₈	—	509	1	no	maximal, Sylow
25	C₄×Q₈	D₅₀₉	1	1	yes	characteristic
26	C₄⋊C₈	—	509	3	no	—
27	C₄×C₈	—	509	3	no	—
28	C₂×Q₈	Dic₅₀₉	1	1	yes	characteristic
29	C₂×C₈	—	509	3	no	—
30	C₂×C₈	—	509	1	no	—
31	C₄⋊C₄	Dic₅₀₉	1	3	yes	—
32	C₄²	C₂×D₅₀₉	1	3	yes	—
33	Q₈	#16861325	1	4	yes	direct product factor
34	C₂×C₄	C₂×Dic₅₀₉	1	3	yes	—
35	C₂×C₄	C₂×Dic₅₀₉	1	3	yes	—
36	C₂×C₄	C₂²×D₅₀₉	1	1	yes	characteristic
37	C₈	—	1018	3	no	—
38	C₈	—	509	2	no	—
39	C₂²	C₂²×Dic₅₀₉	1	1	yes	characteristic
40	C₄	—	2	3	no	—
41	C₄	C₂×#16861325	1	6	yes	—
42	C₄	Q₈×D₅₀₉	1	1	yes	characteristic
43	C₄	#22550862	1	1	yes	characteristic
44	C₂	Q₈×Dic₅₀₉	1	1	yes	characteristic
45	C₂	C₂²×#16861325	1	1	yes	characteristic
46	C₂	#30654222	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₅₀₉
commutator	C₂×C₅₀₉	C₂²×C₈
Fitting	C₄×Q₈×C₅₀₉	C₂
socle, prime radical	C₂²×C₅₀₉	C₂²×C₄
cosocle	C₂×C₄×C₅₀₉	C₂³
hypercenter	C₄×Q₈	D₅₀₉
hypocenter	C₅₀₉	C₈×Q₈
LCS	G ▷ C₂×C₅₀₉ ▷ C₅₀₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
10. #16861325	Q₈	direct (N × H)
11. Q₈×C₅₀₉	C₈	semidirect
23. C₅₀₉	C₈×Q₈	semidirect
33. Q₈	#16861325	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄⋊#16861325	2
C₄×#16861325	2
C₄×Q₈×C₅₀₉	2
C₈×Q₈	509

Subgroups — Q8×#16861325 #45655364

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×#16861325 #45655364