Subgroups — C₄×C₅×Dic₆ #91671

Order 480.

140 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×Dic₆	C₂	1	1	yes	maximal, characteristic
2	C₃×C₄²×C₅	C₂	1	1	yes	maximal, characteristic
3	C₅×C₄⋊Dic₃	C₂	1	1	yes	maximal, characteristic
4	C₅×Dic₃⋊C₄	C₂	1	2	yes	maximal
5	C₄×C₅×Dic₃	C₂	1	2	yes	maximal
6	C₄×C₅×Q₈	—	3	1	no	maximal, Hall
7	C₂×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
8	C₂×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
9	C₂×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
10	C₂×C₅×Dic₃	C₂²	1	2	yes	—
11	C₂×C₅×Dic₃	C₂²	1	2	yes	—
12	C₅×Dic₆	C₄	1	4	yes	direct product factor
13	C₄×Dic₆	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
14	C₂×C₅×Q₈	—	3	1	no	—
15	C₅×C₄⋊C₄	—	3	2	no	—
16	C₅×C₄⋊C₄	—	3	1	no	—
17	C₄²×C₅	—	3	2	no	—
18	C₄²×C₅	D₃	1	1	yes	characteristic
19	C₂²×C₃×C₅	C₂³	1	1	yes	characteristic
20	C₃×C₄×C₅	—	2	1	no	—
21	C₃×C₄×C₅	C₂×C₄	1	2	yes	—
22	C₃×C₄×C₅	Q₈	1	2	yes	complemented
23	C₅×Dic₃	—	2	2	no	—
24	C₅×Dic₃	C₂×C₄	1	4	yes	—
25	C₂×Dic₆	C₂×C₅	1	1	yes	characteristic
26	C₃×C₄²	C₂×C₅	1	1	yes	characteristic
27	C₄⋊Dic₃	C₂×C₅	1	1	yes	characteristic
28	Dic₃⋊C₄	C₂×C₅	1	2	yes	—
29	C₄×Dic₃	C₂×C₅	1	2	yes	—
30	C₅×Q₈	—	3	4	no	—
31	C₂×C₄×C₅	—	3	2	no	—
32	C₂×C₄×C₅	—	3	2	no	—
33	C₂×C₄×C₅	C₂×D₃	1	1	yes	characteristic
34	C₂×C₄×C₅	C₂×D₃	1	1	yes	characteristic
35	C₂×C₄×C₅	C₂×D₃	1	1	yes	characteristic
36	C₄×Q₈	—	3	1	no	Sylow
37	C₂×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₃×C₅	C₂×Q₈	1	1	yes	characteristic
39	C₂×C₃×C₅	#41	1	1	yes	characteristic
40	C₂×C₃×C₄	C₂²×C₅	1	1	yes	characteristic
41	C₂×C₃×C₄	C₂²×C₅	1	1	yes	characteristic
42	C₂×C₃×C₄	C₂²×C₅	1	1	yes	characteristic
43	C₂×Dic₃	C₂²×C₅	1	2	yes	—
44	C₂×Dic₃	C₂²×C₅	1	2	yes	—
45	Dic₆	C₄×C₅	1	4	yes	direct product factor
46	C₂²×C₅	C₂²×D₃	1	1	yes	characteristic
47	C₄×C₅	—	2	1	no	—
48	C₄×C₅	—	3	4	no	—
49	C₄×C₅	—	6	2	no	—
50	C₄×C₅	Dic₆	1	2	yes	direct product factor
51	C₄×C₅	C₄×D₃	1	2	yes	—
52	C₂×Q₈	—	3	1	no	—
53	C₄⋊C₄	—	3	2	no	—
54	C₄⋊C₄	—	3	1	no	—
55	C₄²	—	3	2	no	—
56	C₄²	C₅×D₃	1	1	yes	characteristic
57	C₃×C₅	C₄×Q₈	1	1	yes	characteristic, complemented, Hall
58	C₂²×C₃	C₂³×C₅	1	1	yes	characteristic
59	C₃×C₄	—	2	1	no	—
60	C₃×C₄	C₂×C₄×C₅	1	2	yes	—
61	C₃×C₄	C₅×Q₈	1	2	yes	complemented
62	Dic₃	—	2	2	no	—
63	Dic₃	C₂×C₄×C₅	1	4	yes	—
64	C₂×C₅	C₂×Dic₆	1	1	yes	characteristic
65	C₂×C₅	C₂×C₄×D₃	1	1	yes	characteristic
66	C₂×C₅	#235	1	1	yes	characteristic
67	Q₈	—	3	4	no	—
68	C₂×C₄	—	3	2	no	—
69	C₂×C₄	—	3	2	no	—
70	C₂×C₄	C₂×C₅×D₃	1	1	yes	characteristic
71	C₂×C₄	C₂×C₅×D₃	1	1	yes	characteristic
72	C₂×C₄	C₂×C₅×D₃	1	1	yes	characteristic
73	C₂×C₃	C₂²×C₄×C₅	1	1	yes	characteristic
74	C₂×C₃	C₂×C₅×Q₈	1	1	yes	characteristic
75	C₂×C₃	C₅×#41	1	1	yes	characteristic
76	C₅	C₄×Dic₆	1	1	yes	characteristic, direct product factor, Sylow
77	C₂²	C₂²×C₅×D₃	1	1	yes	characteristic
78	C₄	—	2	1	no	—
79	C₄	—	3	4	no	—
80	C₄	—	6	2	no	—
81	C₄	C₅×Dic₆	1	2	yes	direct product factor
82	C₄	C₄×C₅×D₃	1	2	yes	—
83	C₃	C₄×C₅×Q₈	1	1	yes	characteristic, complemented, Sylow
84	C₂	C₂×C₅×Dic₆	1	1	yes	characteristic
85	C₂	C₂×C₄×C₅×D₃	1	1	yes	characteristic
86	C₂	C₅×#235	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅	C₂×D₃
commutator	C₂×C₃	C₂²×C₄×C₅
Frattini	C₂²	C₂²×C₅×D₃
Fitting	C₃×C₄²×C₅	C₂
socle, prime radical	C₂²×C₃×C₅	C₂³
cosocle	C₂²×C₃	C₂³×C₅
hypercenter	C₄²×C₅	D₃
hypocenter	C₃	C₄×C₅×Q₈
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₄²×C₅

Complements

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

N	H ≅ G/N	type
12. C₅×Dic₆	C₄	direct (N × H)
13. C₄×Dic₆	C₅	direct (N × H)
22. C₃×C₄×C₅	Q₈	semidirect
45. Dic₆	C₄×C₅	direct (N × H)
50. C₄×C₅	Dic₆	direct (N × H)
57. C₃×C₅	C₄×Q₈	semidirect
61. C₃×C₄	C₅×Q₈	semidirect
76. C₅	C₄×Dic₆	direct (N × H)
81. C₄	C₅×Dic₆	direct (N × H)
83. C₃	C₄×C₅×Q₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×Dic₆	2
C₃×C₄²×C₅	2
C₅×C₄⋊Dic₃	2
C₅×Dic₃⋊C₄	2
C₄×C₅×Dic₃	2
C₄×C₅×Q₈	3
C₄×Dic₆	5

Subgroups — C4×C5×Dic6 #91671

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₅×Dic₆ #91671