Subgroups — C₃×C₅×Q₈×D₅ #11925681

Order 1200.

176 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅²×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×C₄×C₅×D₅	C₂	1	3	yes	maximal
3	C₃×C₅×Dic₁₀	C₂	1	3	yes	maximal, complemented
4	C₅×Q₈×D₅	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₂×C₃×C₅×D₅	C₂²	1	1	yes	characteristic
6	C₃×C₄×C₅²	C₂²	1	3	yes	—
7	C₃×C₅×Dic₅	C₂²	1	3	yes	—
8	C₂×C₃×C₅×Q₈	—	5	1	no	maximal
9	C₃×Q₈×D₅	C₅	1	1	yes	maximal, characteristic, direct product factor
10	C₅²×Q₈	C₂×C₃	1	1	yes	characteristic, complemented
11	C₄×C₅×D₅	C₂×C₃	1	3	yes	—
12	C₅×Dic₁₀	C₂×C₃	1	3	yes	complemented
13	C₂×C₃×C₅²	C₂³	1	1	yes	characteristic
14	C₃×C₅×D₅	Q₈	1	2	yes	direct product factor
15	C₃×C₅×Q₈	—	2	2	no	—
16	C₃×C₅×Q₈	—	5	3	no	—
17	C₃×C₅×Q₈	D₅	1	1	yes	characteristic, direct product factor
18	C₃×C₅×Q₈	C₂×C₅	1	1	yes	characteristic, complemented
19	C₂×C₃×C₄×C₅	—	5	3	no	—
20	C₃×C₄×D₅	C₂×C₅	1	3	yes	—
21	C₃×Dic₁₀	C₂×C₅	1	3	yes	complemented
22	C₂×C₅×D₅	C₂²×C₃	1	1	yes	characteristic
23	C₄×C₅²	C₂²×C₃	1	3	yes	—
24	C₅×Dic₅	C₂²×C₃	1	3	yes	—
25	C₂×C₅×Q₈	—	5	1	no	—
26	Q₈×D₅	C₃×C₅	1	1	yes	characteristic, direct product factor
27	C₃×C₅²	C₂×Q₈	1	1	yes	characteristic, complemented, Hall
28	C₂²×C₃×C₅	—	5	1	no	—
29	C₂×C₃×D₅	C₂²×C₅	1	1	yes	characteristic
30	C₃×C₄×C₅	—	2	6	no	—
31	C₃×C₄×C₅	—	5	3	no	—
32	C₃×C₄×C₅	C₂×D₅	1	3	yes	—
33	C₃×C₄×C₅	C₂²×C₅	1	3	yes	—
34	C₃×Dic₅	C₂²×C₅	1	3	yes	—
35	C₂×C₅²	C₂³×C₃	1	1	yes	characteristic
36	C₅×D₅	C₃×Q₈	1	2	yes	direct product factor
37	C₂×C₃×Q₈	—	5	1	no	Hall
38	C₅×Q₈	—	2	2	no	—
39	C₅×Q₈	—	5	3	no	—
40	C₅×Q₈	C₃×D₅	1	1	yes	characteristic, direct product factor
41	C₅×Q₈	C₂×C₃×C₅	1	1	yes	characteristic, complemented
42	C₂×C₄×C₅	—	5	3	no	—
43	C₄×D₅	C₂×C₃×C₅	1	3	yes	—
44	Dic₁₀	C₂×C₃×C₅	1	3	yes	complemented
45	C₂×C₃×C₅	—	2	2	no	—
46	C₂×C₃×C₅	—	5	2	no	—
47	C₂×C₃×C₅	C₂²×D₅	1	1	yes	characteristic
48	C₂×C₃×C₅	C₂³×C₅	1	1	yes	characteristic
49	C₃×D₅	C₅×Q₈	1	2	yes	direct product factor
50	C₅²	C₂×C₃×Q₈	1	1	yes	characteristic, complemented, Sylow
51	C₃×Q₈	—	5	3	no	—
52	C₃×Q₈	C₅×D₅	1	1	yes	characteristic, direct product factor
53	C₂×C₃×C₄	—	5	3	no	—
54	C₂²×C₅	—	5	1	no	—
55	C₂×D₅	C₂²×C₃×C₅	1	1	yes	characteristic
56	C₄×C₅	—	2	6	no	—
57	C₄×C₅	—	5	3	no	—
58	C₄×C₅	C₂×C₃×D₅	1	3	yes	—
59	C₄×C₅	C₂²×C₃×C₅	1	3	yes	—
60	Dic₅	C₂²×C₃×C₅	1	3	yes	—
61	C₂×Q₈	—	5	1	no	Sylow
62	C₃×C₅	—	2	2	no	—
63	C₃×C₅	Q₈×D₅	1	1	yes	characteristic, direct product factor
64	C₃×C₅	C₂×C₅×Q₈	1	1	yes	characteristic, complemented
65	C₂²×C₃	—	5	1	no	—
66	C₃×C₄	—	5	3	no	—
67	C₃×C₄	C₂×C₅×D₅	1	3	yes	—
68	C₂×C₅	—	2	2	no	—
69	C₂×C₅	—	5	2	no	—
70	C₂×C₅	C₂²×C₃×D₅	1	1	yes	characteristic
71	C₂×C₅	C₂³×C₃×C₅	1	1	yes	characteristic
72	D₅	C₃×C₅×Q₈	1	2	yes	direct product factor
73	Q₈	—	5	3	no	—
74	Q₈	C₃×C₅×D₅	1	1	yes	characteristic, direct product factor
75	C₂×C₄	—	5	3	no	—
76	C₂×C₃	—	5	2	no	—
77	C₂×C₃	C₂²×C₅×D₅	1	1	yes	characteristic
78	C₅	—	2	2	no	—
79	C₅	C₃×Q₈×D₅	1	1	yes	characteristic, direct product factor
80	C₅	C₂×C₃×C₅×Q₈	1	1	yes	characteristic, complemented
81	C₂²	—	5	1	no	—
82	C₄	—	5	3	no	—
83	C₄	C₂×C₃×C₅×D₅	1	3	yes	—
84	C₃	C₅×Q₈×D₅	1	1	yes	characteristic, direct product factor, Sylow
85	C₂	—	5	2	no	—
86	C₂	C₂²×C₃×C₅×D₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₅	C₂²×D₅
commutator, cosocle	C₂×C₅	C₂³×C₃×C₅
Frattini	C₂	C₂²×C₃×C₅×D₅
Fitting	C₃×C₅²×Q₈	C₂
socle	C₂×C₃×C₅²	C₂³
prime radical	C₂×C₃×C₅×D₅	C₂²
hypercenter	C₃×C₅×Q₈	D₅
hypocenter	C₅	C₂×C₃×C₅×Q₈
LCS	G ▷ C₂×C₅ ▷ C₅
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. C₃×C₅²×Q₈	C₂	semidirect
3. C₃×C₅×Dic₁₀	C₂	semidirect
4. C₅×Q₈×D₅	C₃	direct (N × H)
9. C₃×Q₈×D₅	C₅	direct (N × H)
10. C₅²×Q₈	C₂×C₃	semidirect
12. C₅×Dic₁₀	C₂×C₃	semidirect
14. C₃×C₅×D₅	Q₈	direct (N × H)
17. C₃×C₅×Q₈	D₅	direct (N × H)
18. C₃×C₅×Q₈	C₂×C₅	semidirect
21. C₃×Dic₁₀	C₂×C₅	semidirect
26. Q₈×D₅	C₃×C₅	direct (N × H)
27. C₃×C₅²	C₂×Q₈	semidirect
36. C₅×D₅	C₃×Q₈	direct (N × H)
40. C₅×Q₈	C₃×D₅	direct (N × H)
41. C₅×Q₈	C₂×C₃×C₅	semidirect
44. Dic₁₀	C₂×C₃×C₅	semidirect
49. C₃×D₅	C₅×Q₈	direct (N × H)
50. C₅²	C₂×C₃×Q₈	semidirect
52. C₃×Q₈	C₅×D₅	direct (N × H)
63. C₃×C₅	Q₈×D₅	direct (N × H)
64. C₃×C₅	C₂×C₅×Q₈	semidirect
72. D₅	C₃×C₅×Q₈	direct (N × H)
74. Q₈	C₃×C₅×D₅	direct (N × H)
79. C₅	C₃×Q₈×D₅	direct (N × H)
80. C₅	C₂×C₃×C₅×Q₈	semidirect
84. C₃	C₅×Q₈×D₅	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×C₅²×Q₈	2
C₃×C₄×C₅×D₅	2
C₃×C₅×Dic₁₀	2
C₅×Q₈×D₅	3
C₂×C₃×C₅×Q₈	5
C₃×Q₈×D₅	5

Subgroups — C3×C5×Q8×D5 #11925681

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×Q₈×D₅ #11925681