Subgroups — Q₈×#6016 #13098775

Order 1600.

153 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×Dic₂₅	C₂	1	1	yes	maximal, characteristic
2	C₄⋊#6016	C₂	1	3	yes	maximal
3	C₄×#6016	C₂	1	3	yes	maximal
4	C₂×Q₈×C₂₅	C₄	1	1	yes	characteristic
5	C₂×#6016	C₂²	1	3	yes	—
6	C₂×#6016	C₂²	1	1	yes	characteristic
7	C₄⋊Dic₂₅	C₄	1	3	yes	—
8	C₄×Dic₂₅	C₂²	1	3	yes	—
9	Q₈×#170	—	5	1	no	maximal
10	Q₈×C₂₅	C₈	1	4	yes	complemented
11	C₂×C₄×C₂₅	C₂×C₄	1	3	yes	—
12	C₂×Dic₂₅	C₂×C₄	1	3	yes	—
13	C₂×Dic₂₅	C₂³	1	1	yes	characteristic
14	#6016	—	2	3	no	—
15	#6016	Q₈	1	2	yes	direct product factor
16	Q₈×Dic₅	—	5	1	no	—
17	C₄⋊#170	—	5	3	no	—
18	C₄×#170	—	5	3	no	—
19	C₂²×C₂₅	C₂²×C₄	1	1	yes	characteristic
20	C₄×C₂₅	C₂×C₈	1	6	yes	—
21	Dic₂₅	—	2	3	no	—
22	Dic₂₅	C₂×Q₈	1	1	yes	characteristic
23	Dic₂₅	#41	1	1	yes	characteristic
24	C₂×C₅×Q₈	AGL(1,𝔽₅)	1	1	yes	characteristic
25	C₂×#170	—	5	1	no	—
26	C₂×#170	—	5	3	no	—
27	C₄⋊Dic₅	—	5	3	no	—
28	C₄×Dic₅	—	5	3	no	—
29	C₈×Q₈	—	25	1	no	Sylow
30	C₂×C₂₅	C₄×Q₈	1	1	yes	characteristic
31	C₂×C₂₅	C₂²×C₈	1	1	yes	characteristic
32	C₂×C₂₅	#131	1	1	yes	characteristic
33	C₅×Q₈	#170	1	4	yes	—
34	C₂×C₄×C₅	C₂×AGL(1,𝔽₅)	1	3	yes	—
35	C₂×Dic₅	—	5	1	no	—
36	C₂×Dic₅	—	5	3	no	—
37	#170	—	5	2	no	—
38	#170	—	10	3	no	—
39	C₄×Q₈	—	25	1	no	—
40	C₄⋊C₈	—	25	3	no	—
41	C₄×C₈	—	25	3	no	—
42	C₂₅	C₈×Q₈	1	1	yes	characteristic, complemented, Sylow
43	C₂²×C₅	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
44	C₄×C₅	C₂×#170	1	6	yes	—
45	Dic₅	—	5	1	no	—
46	Dic₅	—	5	1	no	—
47	Dic₅	—	10	3	no	—
48	C₂×Q₈	#1035	1	1	yes	characteristic
49	C₂×C₈	—	25	1	no	—
50	C₂×C₈	—	25	3	no	—
51	C₄⋊C₄	—	25	3	no	—
52	C₄²	—	25	3	no	—
53	C₂×C₅	#4133	1	1	yes	characteristic
54	C₂×C₅	Q₈×AGL(1,𝔽₅)	1	1	yes	characteristic
55	C₂×C₅	C₂²×#170	1	1	yes	characteristic
56	Q₈	#6016	1	4	yes	direct product factor
57	C₂×C₄	—	25	1	no	—
58	C₂×C₄	—	25	3	no	—
59	C₂×C₄	C₂×#1035	1	3	yes	—
60	C₈	—	25	2	no	—
61	C₈	—	50	3	no	—
62	C₅	Q₈×#170	1	1	yes	characteristic
63	C₂²	C₂²×#1035	1	1	yes	characteristic
64	C₄	—	25	1	no	—
65	C₄	—	25	1	no	—
66	C₄	—	50	3	no	—
67	C₄	C₂×#6016	1	6	yes	—
68	C₂	#11716552	1	1	yes	characteristic
69	C₂	Q₈×#1035	1	1	yes	characteristic
70	C₂	C₂²×#6016	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#1035
commutator	C₂×C₂₅	C₂²×C₈
Frattini, socle, prime radical	C₂²×C₅	C₂²×AGL(1,𝔽₅)
Fitting	C₂×Q₈×C₂₅	C₄
cosocle	C₂×Dic₂₅	C₂³
hypercenter	C₂×Q₈	#1035
hypocenter	C₂₅	C₈×Q₈
LCS	G ▷ C₂×C₂₅ ▷ C₂₅
UCS	C₁ ◁ C₂² ◁ C₂×Q₈

Complements

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

N	H ≅ G/N	type
10. Q₈×C₂₅	C₈	semidirect
15. #6016	Q₈	direct (N × H)
42. C₂₅	C₈×Q₈	semidirect
56. Q₈	#6016	direct (N × H)

Maximal subgroups

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

subgroup	index
Q₈×Dic₂₅	2
C₄⋊#6016	2
C₄×#6016	2
Q₈×#170	5

Subgroups — Q8×#6016 #13098775

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×#6016 #13098775