Subgroups — Q₈×#674 #10604447

Order 640.

134 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×#168	C₂	1	1	yes	maximal, characteristic
2	C₄⋊#674	C₂	1	3	yes	maximal
3	C₄×#674	C₂	1	3	yes	maximal
4	C₄×C₅×Q₈	C₄	1	1	yes	characteristic
5	C₂×#674	C₂²	1	3	yes	—
6	C₂×#674	C₂²	1	1	yes	characteristic
7	C₄⋊#168	C₄	1	3	yes	—
8	C₄×#168	C₂²	1	3	yes	—
9	Q₈×C₁₆	—	5	1	no	maximal, Sylow
10	C₂×C₅×Q₈	C₈	1	1	yes	characteristic
11	C₅×C₄⋊C₄	C₈	1	3	yes	—
12	C₄²×C₅	C₂×C₄	1	3	yes	—
13	C₂×#168	C₂×C₄	1	3	yes	—
14	C₂×#168	C₂³	1	1	yes	characteristic
15	#674	—	2	3	no	—
16	#674	Q₈	1	2	yes	direct product factor
17	C₈×Q₈	—	5	1	no	—
18	C₄⋊C₁₆	—	5	3	no	—
19	C₄×C₁₆	—	5	3	no	—
20	C₅×Q₈	C₁₆	1	4	yes	complemented
21	C₂×C₄×C₅	C₂×C₈	1	3	yes	—
22	C₂×C₄×C₅	C₂×C₈	1	3	yes	—
23	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
24	#168	—	2	3	no	—
25	#168	C₂×Q₈	1	1	yes	characteristic
26	#168	#41	1	1	yes	characteristic
27	C₄×Q₈	AGL(1,𝔽₅)	1	1	yes	characteristic
28	C₂×C₁₆	—	5	1	no	—
29	C₂×C₁₆	—	5	3	no	—
30	C₄⋊C₈	—	5	3	no	—
31	C₄×C₈	—	5	3	no	—
32	C₂²×C₅	C₂²×C₈	1	1	yes	characteristic
33	C₄×C₅	—	2	3	no	—
34	C₄×C₅	C₂×C₁₆	1	6	yes	—
35	C₄×C₅	C₄×Q₈	1	1	yes	characteristic
36	C₄×C₅	#131	1	1	yes	characteristic
37	C₂×Q₈	#170	1	1	yes	characteristic
38	C₂×C₈	—	5	1	no	—
39	C₂×C₈	—	5	3	no	—
40	C₄⋊C₄	#170	1	3	yes	—
41	C₄²	C₂×AGL(1,𝔽₅)	1	3	yes	—
42	C₁₆	—	5	2	no	—
43	C₁₆	—	10	3	no	—
44	C₂×C₅	C₈×Q₈	1	1	yes	characteristic
45	C₂×C₅	C₂²×C₁₆	1	1	yes	characteristic
46	C₂×C₅	#504	1	1	yes	characteristic
47	Q₈	#674	1	4	yes	direct product factor
48	C₂×C₄	C₂×#170	1	3	yes	—
49	C₂×C₄	C₂×#170	1	3	yes	—
50	C₂×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
51	C₈	—	5	1	no	—
52	C₈	—	5	1	no	—
53	C₈	—	10	3	no	—
54	C₅	Q₈×C₁₆	1	1	yes	characteristic, complemented, Sylow
55	C₂²	C₂²×#170	1	1	yes	characteristic
56	C₄	—	2	3	no	—
57	C₄	C₂×#674	1	6	yes	—
58	C₄	#4133	1	1	yes	characteristic
59	C₄	Q₈×AGL(1,𝔽₅)	1	1	yes	characteristic
60	C₂	C₂²×#674	1	1	yes	characteristic
61	C₂	Q₈×#170	1	1	yes	characteristic
62	C₂	#65791	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×AGL(1,𝔽₅)
commutator	C₂×C₅	C₂²×C₁₆
Fitting	C₄×C₅×Q₈	C₄
socle, prime radical	C₂²×C₅	C₂²×C₈
cosocle	C₂×#168	C₂³
hypercenter	C₄×Q₈	AGL(1,𝔽₅)
hypocenter	C₅	Q₈×C₁₆
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
16. #674	Q₈	direct (N × H)
20. C₅×Q₈	C₁₆	semidirect
47. Q₈	#674	direct (N × H)
54. C₅	Q₈×C₁₆	semidirect

Maximal subgroups

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

subgroup	index
Q₈×#168	2
C₄⋊#674	2
C₄×#674	2
Q₈×C₁₆	5

Subgroups — Q8×#674 #10604447

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×#674 #10604447