Subgroups — C₃×C₅×D₃×Q₈ #10624256

Order 720.

164 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₅×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×C₄×C₅×D₃	C₂	1	3	yes	maximal
3	C₃×C₅×Dic₆	C₂	1	3	yes	maximal, complemented
4	C₂×C₃×C₅×Q₈	—	3	1	no	maximal
5	C₅×D₃×Q₈	C₃	1	1	yes	maximal, characteristic, direct product factor
6	C₂×C₃×C₅×D₃	C₂²	1	1	yes	characteristic
7	C₃²×C₄×C₅	C₂²	1	3	yes	—
8	C₃×C₅×Dic₃	C₂²	1	3	yes	—
9	C₃×D₃×Q₈	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
10	C₃×C₅×Q₈	—	2	1	no	—
11	C₃×C₅×Q₈	—	3	3	no	—
12	C₃×C₅×Q₈	D₃	1	1	yes	characteristic, direct product factor
13	C₃×C₅×Q₈	C₂×C₃	1	1	yes	characteristic, complemented
14	C₂×C₃×C₄×C₅	—	3	3	no	—
15	C₄×C₅×D₃	C₂×C₃	1	3	yes	—
16	C₅×Dic₆	C₂×C₃	1	3	yes	complemented
17	C₂×C₃²×C₅	C₂³	1	1	yes	characteristic
18	C₃×C₅×D₃	Q₈	1	2	yes	direct product factor
19	C₂×C₅×Q₈	—	3	1	no	Hall
20	C₃²×Q₈	C₂×C₅	1	1	yes	characteristic, complemented
21	C₃×C₄×D₃	C₂×C₅	1	3	yes	—
22	C₃×Dic₆	C₂×C₅	1	3	yes	complemented
23	C₂²×C₃×C₅	—	3	1	no	—
24	C₂×C₅×D₃	C₂²×C₃	1	1	yes	characteristic
25	C₃×C₄×C₅	—	2	3	no	—
26	C₃×C₄×C₅	—	3	3	no	—
27	C₃×C₄×C₅	C₂×D₃	1	3	yes	—
28	C₃×C₄×C₅	C₂²×C₃	1	3	yes	—
29	C₅×Dic₃	C₂²×C₃	1	3	yes	—
30	C₂×C₃×Q₈	—	3	1	no	—
31	D₃×Q₈	C₃×C₅	1	1	yes	characteristic, direct product factor
32	C₃²×C₅	C₂×Q₈	1	1	yes	characteristic, complemented, Hall
33	C₅×Q₈	—	3	3	no	—
34	C₅×Q₈	C₃×D₃	1	1	yes	characteristic, direct product factor
35	C₂×C₄×C₅	—	3	3	no	—
36	C₂×C₃×D₃	C₂²×C₅	1	1	yes	characteristic
37	C₃²×C₄	C₂²×C₅	1	3	yes	—
38	C₃×Dic₃	C₂²×C₅	1	3	yes	—
39	C₂×C₃×C₅	—	2	1	no	—
40	C₂×C₃×C₅	—	3	2	no	—
41	C₂×C₃×C₅	C₂²×D₃	1	1	yes	characteristic
42	C₂×C₃×C₅	C₂³×C₃	1	1	yes	characteristic
43	C₅×D₃	C₃×Q₈	1	2	yes	direct product factor
44	C₃×Q₈	—	2	1	no	—
45	C₃×Q₈	—	3	3	no	—
46	C₃×Q₈	C₅×D₃	1	1	yes	characteristic, direct product factor
47	C₃×Q₈	C₂×C₃×C₅	1	1	yes	characteristic, complemented
48	C₂×C₃×C₄	—	3	3	no	—
49	C₄×D₃	C₂×C₃×C₅	1	3	yes	—
50	Dic₆	C₂×C₃×C₅	1	3	yes	complemented
51	C₂²×C₅	—	3	1	no	—
52	C₄×C₅	—	3	3	no	—
53	C₄×C₅	C₂×C₃×D₃	1	3	yes	—
54	C₂×C₃²	C₂³×C₅	1	1	yes	characteristic
55	C₃×D₃	C₅×Q₈	1	2	yes	direct product factor
56	C₂×Q₈	—	3	1	no	Sylow
57	C₃×C₅	—	2	1	no	—
58	C₃×C₅	D₃×Q₈	1	1	yes	characteristic, direct product factor
59	C₃×C₅	C₂×C₃×Q₈	1	1	yes	characteristic, complemented
60	C₂²×C₃	—	3	1	no	—
61	C₂×D₃	C₂²×C₃×C₅	1	1	yes	characteristic
62	C₃×C₄	—	2	3	no	—
63	C₃×C₄	—	3	3	no	—
64	C₃×C₄	C₂×C₅×D₃	1	3	yes	—
65	C₃×C₄	C₂²×C₃×C₅	1	3	yes	—
66	Dic₃	C₂²×C₃×C₅	1	3	yes	—
67	C₂×C₅	—	3	2	no	—
68	C₂×C₅	C₂²×C₃×D₃	1	1	yes	characteristic
69	C₃²	C₂×C₅×Q₈	1	1	yes	characteristic, complemented, Sylow
70	Q₈	—	3	3	no	—
71	Q₈	C₃×C₅×D₃	1	1	yes	characteristic, direct product factor
72	C₂×C₄	—	3	3	no	—
73	C₂×C₃	—	2	1	no	—
74	C₂×C₃	—	3	2	no	—
75	C₂×C₃	C₂²×C₅×D₃	1	1	yes	characteristic
76	C₂×C₃	C₂³×C₃×C₅	1	1	yes	characteristic
77	D₃	C₃×C₅×Q₈	1	2	yes	direct product factor
78	C₅	C₃×D₃×Q₈	1	1	yes	characteristic, direct product factor, Sylow
79	C₂²	—	3	1	no	—
80	C₄	—	3	3	no	—
81	C₄	C₂×C₃×C₅×D₃	1	3	yes	—
82	C₃	—	2	1	no	—
83	C₃	C₅×D₃×Q₈	1	1	yes	characteristic, direct product factor
84	C₃	C₂×C₃×C₅×Q₈	1	1	yes	characteristic, complemented
85	C₂	—	3	2	no	—
86	C₂	C₂²×C₃×C₅×D₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₅	C₂²×D₃
commutator, cosocle	C₂×C₃	C₂³×C₃×C₅
Frattini	C₂	C₂²×C₃×C₅×D₃
Fitting	C₃²×C₅×Q₈	C₂
socle	C₂×C₃²×C₅	C₂³
prime radical	C₂×C₃×C₅×D₃	C₂²
hypercenter	C₃×C₅×Q₈	D₃
hypocenter	C₃	C₂×C₃×C₅×Q₈
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₃×C₅ ◁ C₃×C₅×Q₈

Complements

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

N	H ≅ G/N	type
1. C₃²×C₅×Q₈	C₂	semidirect
3. C₃×C₅×Dic₆	C₂	semidirect
5. C₅×D₃×Q₈	C₃	direct (N × H)
9. C₃×D₃×Q₈	C₅	direct (N × H)
12. C₃×C₅×Q₈	D₃	direct (N × H)
13. C₃×C₅×Q₈	C₂×C₃	semidirect
16. C₅×Dic₆	C₂×C₃	semidirect
18. C₃×C₅×D₃	Q₈	direct (N × H)
20. C₃²×Q₈	C₂×C₅	semidirect
22. C₃×Dic₆	C₂×C₅	semidirect
31. D₃×Q₈	C₃×C₅	direct (N × H)
32. C₃²×C₅	C₂×Q₈	semidirect
34. C₅×Q₈	C₃×D₃	direct (N × H)
43. C₅×D₃	C₃×Q₈	direct (N × H)
46. C₃×Q₈	C₅×D₃	direct (N × H)
47. C₃×Q₈	C₂×C₃×C₅	semidirect
50. Dic₆	C₂×C₃×C₅	semidirect
55. C₃×D₃	C₅×Q₈	direct (N × H)
58. C₃×C₅	D₃×Q₈	direct (N × H)
59. C₃×C₅	C₂×C₃×Q₈	semidirect
69. C₃²	C₂×C₅×Q₈	semidirect
71. Q₈	C₃×C₅×D₃	direct (N × H)
77. D₃	C₃×C₅×Q₈	direct (N × H)
78. C₅	C₃×D₃×Q₈	direct (N × H)
83. C₃	C₅×D₃×Q₈	direct (N × H)
84. C₃	C₂×C₃×C₅×Q₈	semidirect

Maximal subgroups

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

subgroup	index
C₃²×C₅×Q₈	2
C₃×C₄×C₅×D₃	2
C₃×C₅×Dic₆	2
C₂×C₃×C₅×Q₈	3
C₅×D₃×Q₈	3
C₃×D₃×Q₈	5

Subgroups — C3×C5×D3×Q8 #10624256

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×D₃×Q₈ #10624256