Subgroups — C₃×C₅×#215 #10623863

Order 720.

92 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×C₅×D₁₂	C₂	1	1	yes	maximal, characteristic
3	C₃×C₅×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₅×SD₁₆	—	3	1	no	maximal
5	C₅×#215	C₃	1	1	yes	maximal, characteristic, direct product factor
6	C₂×C₃×C₅×D₃	—	2	1	no	—
7	C₃²×C₄×C₅	—	2	1	no	—
8	C₃²×C₄×C₅	C₂²	1	1	yes	characteristic
9	C₃×#215	C₅	1	1	yes	maximal, characteristic, direct product factor, 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	C₃×C₅×D₄	—	3	1	no	—
14	C₅×D₁₂	C₂×C₃	1	1	yes	characteristic
15	C₃×C₅×C₈	—	3	1	no	—
16	C₅×C₃⋊C₈	C₂×C₃	1	1	yes	characteristic, complemented
17	C₂×C₃²×C₅	D₄	1	1	yes	characteristic
18	C₃×C₅×D₃	—	4	1	no	—
19	C₅×SD₁₆	—	3	1	no	Hall
20	C₃²×Q₈	C₂×C₅	1	1	yes	characteristic, complemented
21	C₃×D₁₂	C₂×C₅	1	1	yes	characteristic
22	C₃×C₃⋊C₈	C₂×C₅	1	1	yes	characteristic, complemented
23	C₂²×C₃×C₅	—	6	1	no	—
24	C₂×C₅×D₃	—	2	1	no	—
25	C₃×C₄×C₅	—	2	2	no	—
26	C₃×C₄×C₅	—	2	1	no	—
27	C₃×C₄×C₅	—	2	1	no	—
28	C₃×C₄×C₅	—	2	1	no	—
29	C₃×C₄×C₅	C₂×D₃	1	1	yes	characteristic
30	C₃×C₄×C₅	C₂²×C₃	1	1	yes	characteristic
31	C₃×SD₁₆	—	3	1	no	—
32	#215	C₃×C₅	1	1	yes	characteristic, direct product factor
33	C₃²×C₅	SD₁₆	1	1	yes	characteristic, complemented, Hall
34	C₅×Q₈	C₃×D₃	1	1	yes	characteristic, complemented
35	C₅×D₄	—	3	1	no	—
36	C₅×C₈	—	3	1	no	—
37	C₂×C₃×D₃	—	2	1	no	—
38	C₃²×C₄	—	2	1	no	—
39	C₃²×C₄	C₂²×C₅	1	1	yes	characteristic
40	C₂×C₃×C₅	—	2	1	no	—
41	C₂×C₃×C₅	—	12	1	no	—
42	C₂×C₃×C₅	#67	1	1	yes	characteristic
43	C₂×C₃×C₅	C₃×D₄	1	1	yes	characteristic
44	C₅×D₃	—	4	1	no	—
45	C₃×Q₈	—	2	1	no	—
46	C₃×Q₈	C₅×D₃	1	1	yes	characteristic, complemented
47	C₃×Q₈	C₂×C₃×C₅	1	1	yes	characteristic, complemented
48	C₃×D₄	—	3	1	no	—
49	D₁₂	C₂×C₃×C₅	1	1	yes	characteristic
50	C₃×C₈	—	3	1	no	—
51	C₃⋊C₈	C₂×C₃×C₅	1	1	yes	characteristic, complemented
52	C₂²×C₅	—	6	1	no	—
53	C₄×C₅	—	2	1	no	—
54	C₄×C₅	C₂×C₃×D₃	1	1	yes	characteristic
55	C₂×C₃²	C₅×D₄	1	1	yes	characteristic
56	C₃×D₃	—	4	1	no	—
57	SD₁₆	—	3	1	no	Sylow
58	C₃×C₅	—	2	1	no	—
59	C₃×C₅	#215	1	1	yes	characteristic, direct product factor
60	C₃×C₅	C₃×SD₁₆	1	1	yes	characteristic, complemented
61	C₂²×C₃	—	6	1	no	—
62	C₂×D₃	—	2	1	no	—
63	C₃×C₄	—	2	2	no	—
64	C₃×C₄	—	2	1	no	—
65	C₃×C₄	—	2	1	no	—
66	C₃×C₄	—	2	1	no	—
67	C₃×C₄	C₂×C₅×D₃	1	1	yes	characteristic
68	C₃×C₄	C₂²×C₃×C₅	1	1	yes	characteristic
69	C₂×C₅	—	12	1	no	—
70	C₂×C₅	C₃×#67	1	1	yes	characteristic
71	C₃²	C₅×SD₁₆	1	1	yes	characteristic, complemented, Sylow
72	Q₈	C₃×C₅×D₃	1	1	yes	characteristic, complemented
73	D₄	—	3	1	no	—
74	C₈	—	3	1	no	—
75	C₂×C₃	—	2	1	no	—
76	C₂×C₃	—	12	1	no	—
77	C₂×C₃	C₅×#67	1	1	yes	characteristic
78	C₂×C₃	C₃×C₅×D₄	1	1	yes	characteristic
79	D₃	—	4	1	no	—
80	C₅	C₃×#215	1	1	yes	characteristic, direct product factor, Sylow
81	C₂²	—	6	1	no	—
82	C₄	—	2	1	no	—
83	C₄	C₂×C₃×C₅×D₃	1	1	yes	characteristic
84	C₃	—	2	1	no	—
85	C₃	C₅×#215	1	1	yes	characteristic, direct product factor
86	C₃	C₃×C₅×SD₁₆	1	1	yes	characteristic, complemented
87	C₂	—	12	1	no	—
88	C₂	C₃×C₅×#67	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₅	#67
commutator, cosocle	C₃×C₄	C₂²×C₃×C₅
Frattini	C₄	C₂×C₃×C₅×D₃
Fitting	C₃²×C₅×Q₈	C₂
socle	C₂×C₃²×C₅	D₄
prime radical	C₃×C₅×D₁₂	C₂
hypercenter	C₃×C₅×Q₈	D₃
hypocenter	C₃	C₃×C₅×SD₁₆
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₃×C₅ ◁ C₃×C₄×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×C₃⋊C₈	C₂	semidirect
5. C₅×#215	C₃	direct (N × H)
9. C₃×#215	C₅	direct (N × H)
11. C₃×C₅×Q₈	D₃	semidirect
12. C₃×C₅×Q₈	C₂×C₃	semidirect
16. C₅×C₃⋊C₈	C₂×C₃	semidirect
20. C₃²×Q₈	C₂×C₅	semidirect
22. C₃×C₃⋊C₈	C₂×C₅	semidirect
32. #215	C₃×C₅	direct (N × H)
33. C₃²×C₅	SD₁₆	semidirect
34. C₅×Q₈	C₃×D₃	semidirect
46. C₃×Q₈	C₅×D₃	semidirect
47. C₃×Q₈	C₂×C₃×C₅	semidirect
51. C₃⋊C₈	C₂×C₃×C₅	semidirect
59. C₃×C₅	#215	direct (N × H)
60. C₃×C₅	C₃×SD₁₆	semidirect
71. C₃²	C₅×SD₁₆	semidirect
72. Q₈	C₃×C₅×D₃	semidirect
80. C₅	C₃×#215	direct (N × H)
85. C₃	C₅×#215	direct (N × H)
86. C₃	C₃×C₅×SD₁₆	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×C₅×D₁₂	2
C₃×C₅×C₃⋊C₈	2
C₃×C₅×SD₁₆	3
C₅×#215	3
C₃×#215	5

Subgroups — C3×C5×#215 #10623863

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#215 #10623863