Subgroups — C₂×C₃×C₅×Dic₁₀ #11925675

Order 1200.

184 conjugacy classes of subgroups.

Conjugacy classes of subgroups

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

	subgroup	G/H
center	C₂²×C₃×C₅	C₂×D₅
commutator, cosocle	C₂×C₅	C₂³×C₃×C₅
Frattini	C₂	C₂²×C₃×C₅×D₅
Fitting	C₂×C₃×C₄×C₅²	C₂
socle, prime radical	C₂²×C₃×C₅²	C₂²
hypercenter	C₂×C₃×C₄×C₅	D₅
hypocenter	C₅	C₂×C₃×C₅×Q₈
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₃×C₅ ◁ C₂×C₃×C₄×C₅

Complements

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

N	H ≅ G/N	type
3. C₃×C₅×Dic₁₀	C₂	direct (N × H)
4. C₂×C₅×Dic₁₀	C₃	direct (N × H)
9. C₂×C₃×Dic₁₀	C₅	direct (N × H)
12. C₅×Dic₁₀	C₂×C₃	direct (N × H)
13. C₂×C₃×C₅²	Q₈	semidirect
21. C₃×Dic₁₀	C₂×C₅	direct (N × H)
26. C₂×Dic₁₀	C₃×C₅	direct (N × H)
27. C₃×C₅²	C₂×Q₈	semidirect
36. C₂×C₅²	C₃×Q₈	semidirect
45. Dic₁₀	C₂×C₃×C₅	direct (N × H)
48. C₂×C₃×C₅	Dic₁₀	direct (N × H)
49. C₂×C₃×C₅	C₅×Q₈	semidirect
52. C₅²	C₂×C₃×Q₈	semidirect
66. C₃×C₅	C₂×Dic₁₀	direct (N × H)
67. C₃×C₅	C₂×C₅×Q₈	semidirect
73. C₂×C₅	C₃×Dic₁₀	direct (N × H)
74. C₂×C₅	C₃×C₅×Q₈	semidirect
80. C₂×C₃	C₅×Dic₁₀	direct (N × H)
83. C₅	C₂×C₃×Dic₁₀	direct (N × H)
84. C₅	C₂×C₃×C₅×Q₈	semidirect
88. C₃	C₂×C₅×Dic₁₀	direct (N × H)
89. C₂	C₃×C₅×Dic₁₀	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃×C₄×C₅²	2
C₂×C₃×C₅×Dic₅	2
C₃×C₅×Dic₁₀	2
C₂×C₅×Dic₁₀	3
C₂×C₃×C₅×Q₈	5
C₂×C₃×Dic₁₀	5

Subgroups — C2×C3×C5×Dic10 #11925675

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×C₅×Dic₁₀ #11925675