Subgroups — Q₈⋊#168 #64755

Order 320.

70 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	1	yes	maximal, characteristic
3	C₄×#168	C₂	1	1	yes	maximal, characteristic
4	C₂×C₅×Q₈	C₄	1	1	yes	characteristic
5	C₅×C₄⋊C₄	—	2	1	no	—
6	C₅×C₄⋊C₄	C₄	1	1	yes	characteristic
7	C₄²×C₅	—	2	1	no	—
8	C₄²×C₅	C₂²	1	1	yes	characteristic
9	C₂×#168	—	2	1	no	—
10	C₂×#168	—	2	1	no	—
11	Q₈⋊C₈	—	5	1	no	maximal, Sylow
12	C₅×Q₈	—	2	1	no	—
13	C₅×Q₈	C₈	1	2	yes	complemented
14	C₂×C₄×C₅	—	2	1	no	—
15	C₂×C₄×C₅	—	2	1	no	—
16	C₂×C₄×C₅	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄×C₅	D₄	1	1	yes	characteristic
18	C₂×C₄×C₅	D₄	1	1	yes	characteristic
19	#168	—	2	2	no	—
20	#168	—	4	1	no	—
21	C₄×Q₈	D₅	1	1	yes	characteristic
22	C₄⋊C₈	—	5	1	no	—
23	C₄×C₈	—	5	1	no	—
24	C₂²×C₅	C₂²⋊C₄	1	1	yes	characteristic
25	C₄×C₅	—	2	2	no	—
26	C₄×C₅	—	2	1	no	—
27	C₄×C₅	—	4	1	no	—
28	C₄×C₅	C₂×C₈	1	1	yes	characteristic
29	C₄×C₅	M₁₆	1	1	yes	characteristic
30	C₄×C₅	SD₁₆	1	1	yes	characteristic
31	C₄×C₅	Q₁₆	1	1	yes	characteristic
32	C₂×Q₈	Dic₅	1	1	yes	characteristic
33	C₂×C₈	—	10	1	no	—
34	C₂×C₈	—	10	1	no	—
35	C₄⋊C₄	—	2	1	no	—
36	C₄⋊C₄	Dic₅	1	1	yes	characteristic
37	C₄²	—	2	1	no	—
38	C₄²	C₂×D₅	1	1	yes	characteristic
39	C₂×C₅	C₂²⋊C₈	1	1	yes	characteristic
40	C₂×C₅	#103	1	1	yes	characteristic
41	C₂×C₅	#104	1	1	yes	characteristic
42	Q₈	—	2	1	no	—
43	Q₈	#168	1	2	yes	complemented
44	C₂×C₄	—	2	1	no	—
45	C₂×C₄	—	2	1	no	—
46	C₂×C₄	C₂×Dic₅	1	1	yes	characteristic
47	C₂×C₄	C₂²⋊D₅	1	1	yes	characteristic
48	C₂×C₄	C₂²⋊D₅	1	1	yes	characteristic
49	C₈	—	10	2	no	—
50	C₈	—	20	1	no	—
51	C₅	Q₈⋊C₈	1	1	yes	characteristic, complemented, Sylow
52	C₂²	C₂²⋊Dic₅	1	1	yes	characteristic
53	C₄	—	2	2	no	—
54	C₄	—	2	1	no	—
55	C₄	—	4	1	no	—
56	C₄	C₂×#168	1	1	yes	characteristic
57	C₄	#681	1	1	yes	characteristic
58	C₄	#688	1	1	yes	characteristic
59	C₄	#689	1	1	yes	characteristic
60	C₂	C₂²⋊#168	1	1	yes	characteristic
61	C₂	#3967	1	1	yes	characteristic
62	C₂	#3969	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²⋊D₅
commutator	C₄×C₅	C₂×C₈
Frattini	C₄²	C₂×D₅
Fitting	C₄×C₅×Q₈	C₂
socle, prime radical	C₂²×C₅	C₂²⋊C₄
cosocle	C₄²×C₅	C₂²
hypercenter	C₄×Q₈	D₅
hypocenter	C₅	Q₈⋊C₈
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
13. C₅×Q₈	C₈	semidirect
43. Q₈	#168	semidirect
51. C₅	Q₈⋊C₈	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
Q₈⋊C₈	5

Subgroups — Q8⋊#168 #64755

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈⋊#168 #64755