Subgroups — Q₈×#1044 #11717455

Order 800.

178 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×C₅⋊D₅	C₂	1	1	yes	maximal, characteristic
2	C₄⋊#1044	C₂	1	3	yes	maximal
3	C₄×#1044	C₂	1	3	yes	maximal
4	C₂×#1044	C₂²	1	3	yes	—
5	C₂×#1044	C₂²	1	1	yes	characteristic
6	C₅²×Q₈	C₄	1	1	yes	characteristic, complemented
7	C₄×C₅⋊D₅	C₂²	1	3	yes	—
8	#6045	C₄	1	3	yes	complemented
9	Q₈×AGL(1,𝔽₅)	—	5	2	no	maximal
10	C₂×C₅⋊D₅	C₂³	1	1	yes	characteristic
11	#1044	—	2	3	no	—
12	#1044	Q₈	1	2	yes	direct product factor
13	C₄×C₅²	C₂×C₄	1	3	yes	—
14	#1039	C₂×C₄	1	3	yes	—
15	Q₈×D₅	—	5	2	no	—
16	Q₈×D₅	—	10	2	no	—
17	C₄⋊AGL(1,𝔽₅)	—	5	6	no	—
18	C₄×AGL(1,𝔽₅)	—	5	6	no	—
19	C₂×C₅²	C₂²×C₄	1	1	yes	characteristic
20	C₅⋊D₅	C₂×Q₈	1	1	yes	characteristic
21	C₅⋊D₅	#41	1	1	yes	characteristic
22	C₂×AGL(1,𝔽₅)	—	5	2	no	—
23	C₂×AGL(1,𝔽₅)	—	5	6	no	—
24	C₅×Q₈	—	2	2	no	—
25	C₅×Q₈	AGL(1,𝔽₅)	1	2	yes	complemented
26	C₄×D₅	—	5	6	no	—
27	C₄×D₅	—	10	6	no	—
28	Dic₁₀	—	5	6	no	—
29	Dic₁₀	—	10	6	no	—
30	C₄×Q₈	—	25	1	no	Sylow
31	C₅²	C₄×Q₈	1	1	yes	characteristic, complemented, Sylow
32	C₂×D₅	—	5	2	no	—
33	C₂×D₅	—	10	2	no	—
34	AGL(1,𝔽₅)	—	5	4	no	—
35	AGL(1,𝔽₅)	—	10	6	no	—
36	C₄×C₅	—	2	6	no	—
37	C₄×C₅	C₂×AGL(1,𝔽₅)	1	6	yes	—
38	Dic₅	—	5	6	no	—
39	Dic₅	—	10	6	no	—
40	C₂×Q₈	—	25	1	no	—
41	C₄⋊C₄	—	25	3	no	—
42	C₄²	—	25	3	no	—
43	C₂×C₅	—	2	2	no	—
44	C₂×C₅	C₂²×AGL(1,𝔽₅)	1	2	yes	—
45	D₅	—	5	2	no	—
46	D₅	—	5	2	no	—
47	D₅	—	10	2	no	—
48	D₅	—	10	2	no	—
49	Q₈	—	25	3	no	—
50	Q₈	#1044	1	1	yes	characteristic, direct product factor
51	C₂×C₄	—	25	1	no	—
52	C₂×C₄	—	25	3	no	—
53	C₂×C₄	—	25	3	no	—
54	C₅	—	2	2	no	—
55	C₅	Q₈×AGL(1,𝔽₅)	1	2	yes	complemented
56	C₂²	—	25	1	no	—
57	C₄	—	25	2	no	—
58	C₄	—	25	3	no	—
59	C₄	—	50	3	no	—
60	C₄	C₂×#1044	1	3	yes	—
61	C₂	—	25	1	no	—
62	C₂	—	25	1	no	—
63	C₂	C₂²×#1044	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂²×#1044
commutator, socle	C₂×C₅²	C₂²×C₄
Fitting	C₅²×Q₈	C₄
prime radical, cosocle	C₂×C₅⋊D₅	C₂³
hypercenter	Q₈	#1044
hypocenter	C₅²	C₄×Q₈
LCS	G ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂ ◁ Q₈

Complements

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

N	H ≅ G/N	type
6. C₅²×Q₈	C₄	semidirect
8. #6045	C₄	semidirect
12. #1044	Q₈	direct (N × H)
25. C₅×Q₈	AGL(1,𝔽₅)	semidirect
31. C₅²	C₄×Q₈	semidirect
50. Q₈	#1044	direct (N × H)
55. C₅	Q₈×AGL(1,𝔽₅)	semidirect

Maximal subgroups

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

subgroup	index
Q₈×C₅⋊D₅	2
C₄⋊#1044	2
C₄×#1044	2
Q₈×AGL(1,𝔽₅)	5

Subgroups — Q8×#1044 #11717455

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×#1044 #11717455