Subgroups — C₂×C₄×C₅×Q₈ #66180

Order 320.

298 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×Q₈	C₂	1	1	yes	maximal, characteristic
2	C₄×C₅×Q₈	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₅×C₄⋊C₄	C₂	1	3	yes	maximal
4	C₂×C₄²×C₅	C₂	1	3	yes	maximal
5	C₂×C₅×Q₈	C₄	1	8	yes	direct product factor
6	C₂×C₅×Q₈	C₂²	1	4	yes	—
7	C₂²×C₄×C₅	C₂²	1	3	yes	—
8	C₂²×C₄×C₅	C₂²	1	3	yes	—
9	C₂²×C₄×C₅	C₂²	1	1	yes	characteristic
10	C₅×C₄⋊C₄	C₂²	1	12	yes	—
11	C₄²×C₅	C₂²	1	12	yes	—
12	C₂×C₄×Q₈	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
13	C₂³×C₅	C₂³	1	1	yes	characteristic
14	C₅×Q₈	C₂×C₄	1	16	yes	direct product factor
15	C₂×C₄×C₅	—	2	6	no	—
16	C₂×C₄×C₅	C₂×C₄	1	12	yes	—
17	C₂×C₄×C₅	Q₈	1	4	yes	direct product factor
18	C₂×C₄×C₅	C₂³	1	6	yes	—
19	C₂×C₄×C₅	C₂³	1	6	yes	—
20	C₂×C₄×C₅	C₂³	1	2	yes	—
21	C₂²×Q₈	C₂×C₅	1	1	yes	characteristic
22	C₄×Q₈	C₂×C₅	1	8	yes	direct product factor
23	C₂×C₄⋊C₄	C₂×C₅	1	3	yes	—
24	C₂×C₄²	C₂×C₅	1	3	yes	—
25	C₂²×C₅	C₂²×C₄	1	2	yes	—
26	C₂²×C₅	C₂×Q₈	1	2	yes	—
27	C₂²×C₅	#41	1	2	yes	—
28	C₂²×C₅	C₂⁴	1	1	yes	characteristic
29	C₄×C₅	—	2	6	no	—
30	C₄×C₅	C₂²×C₄	1	12	yes	—
31	C₄×C₅	C₂×Q₈	1	4	yes	direct product factor
32	C₂×Q₈	C₄×C₅	1	8	yes	direct product factor
33	C₂×Q₈	C₂²×C₅	1	4	yes	—
34	C₂²×C₄	C₂²×C₅	1	3	yes	—
35	C₂²×C₄	C₂²×C₅	1	3	yes	—
36	C₂²×C₄	C₂²×C₅	1	1	yes	characteristic
37	C₄⋊C₄	C₂²×C₅	1	12	yes	—
38	C₄²	C₂²×C₅	1	12	yes	—
39	C₂×C₅	C₄×Q₈	1	4	yes	direct product factor
40	C₂×C₅	C₂³×C₄	1	1	yes	characteristic
41	C₂×C₅	C₂²×Q₈	1	1	yes	characteristic
42	C₂×C₅	C₂×#41	1	1	yes	characteristic
43	C₂³	C₂³×C₅	1	1	yes	characteristic
44	Q₈	C₂×C₄×C₅	1	16	yes	direct product factor
45	C₂×C₄	—	2	6	no	—
46	C₂×C₄	C₂×C₄×C₅	1	12	yes	—
47	C₂×C₄	C₅×Q₈	1	4	yes	direct product factor
48	C₂×C₄	C₂³×C₅	1	6	yes	—
49	C₂×C₄	C₂³×C₅	1	6	yes	—
50	C₂×C₄	C₂³×C₅	1	2	yes	—
51	C₅	C₂×C₄×Q₈	1	1	yes	characteristic, direct product factor, Sylow
52	C₂²	C₂²×C₄×C₅	1	2	yes	—
53	C₂²	C₂×C₅×Q₈	1	2	yes	—
54	C₂²	C₅×#41	1	2	yes	—
55	C₂²	C₂⁴×C₅	1	1	yes	characteristic
56	C₄	—	2	6	no	—
57	C₄	C₂²×C₄×C₅	1	12	yes	—
58	C₄	C₂×C₅×Q₈	1	4	yes	direct product factor
59	C₂	C₄×C₅×Q₈	1	4	yes	direct product factor
60	C₂	C₂³×C₄×C₅	1	1	yes	characteristic
61	C₂	C₂²×C₅×Q₈	1	1	yes	characteristic
62	C₂	C₂×C₅×#41	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₅	C₂²
commutator	C₂	C₂³×C₄×C₅
Frattini, cosocle	C₂²	C₂⁴×C₅
socle, prime radical	C₂³×C₅	C₂³
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄×C₅×Q₈	C₂	direct (N × H)
5. C₂×C₅×Q₈	C₄	direct (N × H)
12. C₂×C₄×Q₈	C₅	direct (N × H)
14. C₅×Q₈	C₂×C₄	direct (N × H)
17. C₂×C₄×C₅	Q₈	direct (N × H)
22. C₄×Q₈	C₂×C₅	direct (N × H)
31. C₄×C₅	C₂×Q₈	direct (N × H)
32. C₂×Q₈	C₄×C₅	direct (N × H)
39. C₂×C₅	C₄×Q₈	direct (N × H)
44. Q₈	C₂×C₄×C₅	direct (N × H)
47. C₂×C₄	C₅×Q₈	direct (N × H)
51. C₅	C₂×C₄×Q₈	direct (N × H)
58. C₄	C₂×C₅×Q₈	direct (N × H)
59. C₂	C₄×C₅×Q₈	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₅×Q₈	2
C₄×C₅×Q₈	2
C₂×C₅×C₄⋊C₄	2
C₂×C₄²×C₅	2
C₂×C₄×Q₈	5

Subgroups — C2×C4×C5×Q8 #66180

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×C₅×Q₈ #66180