Subgroups — C₃×Q₈×D₁₅ #10624266

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×D₁₅	C₂	1	3	yes	maximal
3	C₃×Dic₃₀	C₂	1	3	yes	maximal, complemented
4	Q₈×D₁₅	C₃	1	1	yes	maximal, characteristic, direct product factor
5	C₃×Q₈×D₅	—	3	1	no	maximal
6	C₂×C₃×D₁₅	C₂²	1	1	yes	characteristic
7	C₃²×C₄×C₅	C₂²	1	3	yes	—
8	C₃×Dic₁₅	C₂²	1	3	yes	—
9	C₃×D₃×Q₈	—	5	1	no	maximal, Hall
10	C₃×C₅×Q₈	—	2	1	no	—
11	C₃×C₅×Q₈	D₃	1	1	yes	characteristic, complemented
12	C₃×C₅×Q₈	C₂×C₃	1	1	yes	characteristic, complemented
13	C₄×D₁₅	C₂×C₃	1	3	yes	—
14	Dic₃₀	C₂×C₃	1	3	yes	complemented
15	C₃×C₄×D₅	—	3	3	no	—
16	C₃×Dic₁₀	—	3	3	no	—
17	C₂×C₃²×C₅	C₂³	1	1	yes	characteristic
18	C₃×D₁₅	Q₈	1	2	yes	direct product factor
19	Q₈×D₅	—	3	1	no	Hall
20	C₃²×Q₈	D₅	1	1	yes	characteristic, complemented
21	C₃×C₄×D₃	—	5	3	no	—
22	C₃×Dic₆	—	5	3	no	—
23	C₂×D₁₅	C₂²×C₃	1	1	yes	characteristic
24	C₂×C₃×D₅	—	3	1	no	—
25	C₃×C₄×C₅	—	2	3	no	—
26	C₃×C₄×C₅	C₂×D₃	1	3	yes	—
27	C₃×C₄×C₅	C₂²×C₃	1	3	yes	—
28	Dic₁₅	C₂²×C₃	1	3	yes	—
29	C₃×Dic₅	—	3	3	no	—
30	C₂×C₃×Q₈	—	15	1	no	—
31	D₃×Q₈	—	5	1	no	—
32	C₃²×C₅	C₂×Q₈	1	1	yes	characteristic, complemented, Hall
33	C₅×Q₈	C₃×D₃	1	1	yes	characteristic, complemented
34	C₄×D₅	—	3	3	no	—
35	Dic₁₀	—	3	3	no	—
36	C₂×C₃×D₃	—	5	1	no	—
37	C₃²×C₄	C₂×D₅	1	3	yes	—
38	C₃×Dic₃	—	5	3	no	—
39	C₂×C₃×C₅	—	2	1	no	—
40	C₂×C₃×C₅	C₂²×D₃	1	1	yes	characteristic
41	C₂×C₃×C₅	C₂³×C₃	1	1	yes	characteristic
42	D₁₅	C₃×Q₈	1	2	yes	direct product factor
43	C₃×D₅	—	3	2	no	—
44	C₃×Q₈	—	2	1	no	—
45	C₃×Q₈	—	15	3	no	—
46	C₃×Q₈	C₃×D₅	1	1	yes	characteristic, complemented
47	C₃×Q₈	D₁₅	1	1	yes	characteristic, direct product factor
48	C₂×C₃×C₄	—	15	3	no	—
49	C₄×D₃	—	5	3	no	—
50	Dic₆	—	5	3	no	—
51	C₂×D₅	—	3	1	no	—
52	C₄×C₅	C₂×C₃×D₃	1	3	yes	—
53	Dic₅	—	3	3	no	—
54	C₂×C₃²	C₂²×D₅	1	1	yes	characteristic
55	C₃×D₃	—	5	2	no	—
56	C₂×Q₈	—	15	1	no	Sylow
57	C₃×C₅	—	2	1	no	—
58	C₃×C₅	D₃×Q₈	1	1	yes	characteristic, complemented
59	C₃×C₅	C₂×C₃×Q₈	1	1	yes	characteristic, complemented
60	C₂²×C₃	—	15	1	no	—
61	C₂×D₃	—	5	1	no	—
62	C₃×C₄	—	2	3	no	—
63	C₃×C₄	—	15	3	no	—
64	C₃×C₄	C₂×C₃×D₅	1	3	yes	—
65	C₃×C₄	C₂×D₁₅	1	3	yes	—
66	Dic₃	—	5	3	no	—
67	C₂×C₅	C₂²×C₃×D₃	1	1	yes	characteristic
68	D₅	—	3	2	no	—
69	C₃²	Q₈×D₅	1	1	yes	characteristic, complemented, Sylow
70	Q₈	—	15	3	no	—
71	Q₈	C₃×D₁₅	1	1	yes	characteristic, direct product factor
72	C₂×C₄	—	15	3	no	—
73	C₂×C₃	—	2	1	no	—
74	C₂×C₃	—	15	2	no	—
75	C₂×C₃	C₂²×C₃×D₅	1	1	yes	characteristic
76	C₂×C₃	C₂²×D₁₅	1	1	yes	characteristic
77	D₃	—	5	2	no	—
78	C₅	C₃×D₃×Q₈	1	1	yes	characteristic, complemented, Sylow
79	C₂²	—	15	1	no	—
80	C₄	—	15	3	no	—
81	C₄	C₂×C₃×D₁₅	1	3	yes	—
82	C₃	—	2	1	no	—
83	C₃	C₃×Q₈×D₅	1	1	yes	characteristic, complemented
84	C₃	Q₈×D₁₅	1	1	yes	characteristic, direct product factor
85	C₂	—	15	2	no	—
86	C₂	C₂²×C₃×D₁₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃	C₂²×D₁₅
commutator, cosocle	C₂×C₃×C₅	C₂³×C₃
Frattini	C₂	C₂²×C₃×D₁₅
Fitting	C₃²×C₅×Q₈	C₂
socle	C₂×C₃²×C₅	C₂³
prime radical	C₂×C₃×D₁₅	C₂²
hypercenter	C₃×Q₈	D₁₅
hypocenter	C₃×C₅	C₂×C₃×Q₈
LCS	G ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₃ ◁ C₃×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×Dic₃₀	C₂	semidirect
4. Q₈×D₁₅	C₃	direct (N × H)
11. C₃×C₅×Q₈	D₃	semidirect
12. C₃×C₅×Q₈	C₂×C₃	semidirect
14. Dic₃₀	C₂×C₃	semidirect
18. C₃×D₁₅	Q₈	direct (N × H)
20. C₃²×Q₈	D₅	semidirect
32. C₃²×C₅	C₂×Q₈	semidirect
33. C₅×Q₈	C₃×D₃	semidirect
42. D₁₅	C₃×Q₈	direct (N × H)
46. C₃×Q₈	C₃×D₅	semidirect
47. C₃×Q₈	D₁₅	direct (N × H)
58. C₃×C₅	D₃×Q₈	semidirect
59. C₃×C₅	C₂×C₃×Q₈	semidirect
69. C₃²	Q₈×D₅	semidirect
71. Q₈	C₃×D₁₅	direct (N × H)
78. C₅	C₃×D₃×Q₈	semidirect
83. C₃	C₃×Q₈×D₅	semidirect
84. C₃	Q₈×D₁₅	direct (N × H)

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×D₁₅	2
C₃×Dic₃₀	2
Q₈×D₁₅	3
C₃×Q₈×D₅	3
C₃×D₃×Q₈	5

Subgroups — C3×Q8×D15 #10624266

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×Q₈×D₁₅ #10624266