Subgroups — Q₈×#168 #65312

Order 320.

102 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₅×Q₈	C₂	1	1	yes	maximal, characteristic
2	C₄⋊#168	C₂	1	3	yes	maximal
3	C₄×#168	C₂	1	3	yes	maximal
4	C₂×C₅×Q₈	C₄	1	1	yes	characteristic
5	C₅×C₄⋊C₄	C₄	1	3	yes	—
6	C₄²×C₅	C₂²	1	3	yes	—
7	C₂×#168	C₂²	1	3	yes	—
8	C₂×#168	C₂²	1	1	yes	characteristic
9	C₈×Q₈	—	5	1	no	maximal, Sylow
10	C₅×Q₈	C₈	1	4	yes	complemented
11	C₂×C₄×C₅	C₂×C₄	1	3	yes	—
12	C₂×C₄×C₅	C₂×C₄	1	3	yes	—
13	C₂×C₄×C₅	C₂³	1	1	yes	characteristic
14	#168	—	2	3	no	—
15	#168	Q₈	1	2	yes	direct product factor
16	C₄×Q₈	D₅	1	1	yes	characteristic
17	C₄⋊C₈	—	5	3	no	—
18	C₄×C₈	—	5	3	no	—
19	C₂²×C₅	C₂²×C₄	1	1	yes	characteristic
20	C₄×C₅	—	2	3	no	—
21	C₄×C₅	C₂×C₈	1	6	yes	—
22	C₄×C₅	C₂×Q₈	1	1	yes	characteristic
23	C₄×C₅	#41	1	1	yes	characteristic
24	C₂×Q₈	Dic₅	1	1	yes	characteristic
25	C₂×C₈	—	5	1	no	—
26	C₂×C₈	—	5	3	no	—
27	C₄⋊C₄	Dic₅	1	3	yes	—
28	C₄²	C₂×D₅	1	3	yes	—
29	C₂×C₅	C₄×Q₈	1	1	yes	characteristic
30	C₂×C₅	C₂²×C₈	1	1	yes	characteristic
31	C₂×C₅	#131	1	1	yes	characteristic
32	Q₈	#168	1	4	yes	direct product factor
33	C₂×C₄	C₂×Dic₅	1	3	yes	—
34	C₂×C₄	C₂×Dic₅	1	3	yes	—
35	C₂×C₄	C₂²×D₅	1	1	yes	characteristic
36	C₈	—	5	2	no	—
37	C₈	—	10	3	no	—
38	C₅	C₈×Q₈	1	1	yes	characteristic, complemented, Sylow
39	C₂²	C₂²×Dic₅	1	1	yes	characteristic
40	C₄	—	2	3	no	—
41	C₄	C₂×#168	1	6	yes	—
42	C₄	Q₈×D₅	1	1	yes	characteristic
43	C₄	#713	1	1	yes	characteristic
44	C₂	C₂²×#168	1	1	yes	characteristic
45	C₂	Q₈×Dic₅	1	1	yes	characteristic
46	C₂	#4094	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₅
commutator	C₂×C₅	C₂²×C₈
Fitting	C₄×C₅×Q₈	C₂
socle, prime radical	C₂²×C₅	C₂²×C₄
cosocle	C₂×C₄×C₅	C₂³
hypercenter	C₄×Q₈	D₅
hypocenter	C₅	C₈×Q₈
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
10. C₅×Q₈	C₈	semidirect
15. #168	Q₈	direct (N × H)
32. Q₈	#168	direct (N × H)
38. C₅	C₈×Q₈	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₅×Q₈	2
C₄⋊#168	2
C₄×#168	2
C₈×Q₈	5

Subgroups — Q8×#168 #65312

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×#168 #65312