Subgroups — C₂²×C₅×Dic₆ #92074

Order 480.

312 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₄×C₅	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₅×Dic₃	C₂	1	2	yes	maximal
3	C₂×C₅×Dic₆	C₂	1	12	yes	maximal, direct product factor
4	C₂²×C₅×Q₈	—	3	1	no	maximal, Hall
5	C₂³×C₃×C₅	C₂²	1	1	yes	characteristic
6	C₂×C₃×C₄×C₅	C₂²	1	6	yes	—
7	C₂×C₅×Dic₃	C₂²	1	12	yes	—
8	C₅×Dic₆	C₂²	1	16	yes	direct product factor
9	C₂²×Dic₆	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
10	C₂×C₅×Q₈	—	3	12	no	—
11	C₂²×C₄×C₅	—	3	2	no	—
12	C₂²×C₄×C₅	D₃	1	1	yes	characteristic
13	C₂²×C₃×C₅	Q₈	1	4	yes	complemented
14	C₂²×C₃×C₅	C₂³	1	3	yes	—
15	C₃×C₄×C₅	C₂³	1	4	yes	—
16	C₅×Dic₃	C₂³	1	8	yes	—
17	C₂²×C₃×C₄	C₂×C₅	1	1	yes	characteristic
18	C₂²×Dic₃	C₂×C₅	1	2	yes	—
19	C₂×Dic₆	C₂×C₅	1	12	yes	direct product factor
20	C₂³×C₅	C₂×D₃	1	1	yes	characteristic
21	C₅×Q₈	—	3	16	no	—
22	C₂×C₄×C₅	—	3	12	no	—
23	C₂×C₄×C₅	C₂×D₃	1	6	yes	—
24	C₂²×Q₈	—	3	1	no	Sylow
25	C₂×C₃×C₅	C₂×Q₈	1	6	yes	complemented
26	C₂×C₃×C₅	C₂⁴	1	1	yes	characteristic
27	C₂³×C₃	C₂²×C₅	1	1	yes	characteristic
28	C₂×C₃×C₄	C₂²×C₅	1	6	yes	—
29	C₂×Dic₃	C₂²×C₅	1	12	yes	—
30	Dic₆	C₂²×C₅	1	16	yes	direct product factor
31	C₂²×C₅	Dic₆	1	4	yes	direct product factor
32	C₂²×C₅	C₂²×D₃	1	3	yes	—
33	C₄×C₅	—	3	8	no	—
34	C₄×C₅	C₂²×D₃	1	4	yes	—
35	C₂×Q₈	—	3	12	no	—
36	C₂²×C₄	—	3	2	no	—
37	C₂²×C₄	C₅×D₃	1	1	yes	characteristic
38	C₃×C₅	C₂²×Q₈	1	1	yes	characteristic, complemented, Hall
39	C₂²×C₃	C₅×Q₈	1	4	yes	complemented
40	C₂²×C₃	C₂³×C₅	1	3	yes	—
41	C₃×C₄	C₂³×C₅	1	4	yes	—
42	Dic₃	C₂³×C₅	1	8	yes	—
43	C₂×C₅	C₂×Dic₆	1	6	yes	direct product factor
44	C₂×C₅	C₂³×D₃	1	1	yes	characteristic
45	C₂³	C₂×C₅×D₃	1	1	yes	characteristic
46	Q₈	—	3	16	no	—
47	C₂×C₄	—	3	12	no	—
48	C₂×C₄	C₂×C₅×D₃	1	6	yes	—
49	C₂×C₃	C₂×C₅×Q₈	1	6	yes	complemented
50	C₂×C₃	C₂⁴×C₅	1	1	yes	characteristic
51	C₅	C₂²×Dic₆	1	1	yes	characteristic, direct product factor, Sylow
52	C₂²	C₅×Dic₆	1	4	yes	direct product factor
53	C₂²	C₂²×C₅×D₃	1	3	yes	—
54	C₄	—	3	8	no	—
55	C₄	C₂²×C₅×D₃	1	4	yes	—
56	C₃	C₂²×C₅×Q₈	1	1	yes	characteristic, complemented, Sylow
57	C₂	C₂×C₅×Dic₆	1	6	yes	direct product factor
58	C₂	C₂³×C₅×D₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₅	C₂×D₃
commutator, cosocle	C₂×C₃	C₂⁴×C₅
Frattini	C₂	C₂³×C₅×D₃
Fitting	C₂²×C₃×C₄×C₅	C₂
socle, prime radical	C₂³×C₃×C₅	C₂²
hypercenter	C₂²×C₄×C₅	D₃
hypocenter	C₃	C₂²×C₅×Q₈
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₅ ◁ C₂²×C₄×C₅

Complements

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

N	H ≅ G/N	type
3. C₂×C₅×Dic₆	C₂	direct (N × H)
8. C₅×Dic₆	C₂²	direct (N × H)
9. C₂²×Dic₆	C₅	direct (N × H)
13. C₂²×C₃×C₅	Q₈	semidirect
19. C₂×Dic₆	C₂×C₅	direct (N × H)
25. C₂×C₃×C₅	C₂×Q₈	semidirect
30. Dic₆	C₂²×C₅	direct (N × H)
31. C₂²×C₅	Dic₆	direct (N × H)
38. C₃×C₅	C₂²×Q₈	semidirect
39. C₂²×C₃	C₅×Q₈	semidirect
43. C₂×C₅	C₂×Dic₆	direct (N × H)
49. C₂×C₃	C₂×C₅×Q₈	semidirect
51. C₅	C₂²×Dic₆	direct (N × H)
52. C₂²	C₅×Dic₆	direct (N × H)
56. C₃	C₂²×C₅×Q₈	semidirect
57. C₂	C₂×C₅×Dic₆	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×C₃×C₄×C₅	2
C₂²×C₅×Dic₃	2
C₂×C₅×Dic₆	2
C₂²×C₅×Q₈	3
C₂²×Dic₆	5

Subgroups — C22×C5×Dic6 #92074

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₅×Dic₆ #92074