Subgroups — C₅×#1203 #10597109

Order 600.

84 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×D₁₅	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₅²×D₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₅×Dic₅	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×C₂²⋊D₅	—	3	1	no	maximal, Hall
5	C₂×C₃×C₅²	C₂²	1	1	yes	characteristic
6	C₅×D₁₅	—	2	1	no	—
7	C₅²×D₃	—	2	1	no	—
8	C₅×D₁₂	—	5	1	no	maximal
9	#1203	C₅	1	1	yes	maximal, characteristic, direct product factor
10	C₂²×C₅²	—	3	1	no	—
11	C₂×C₅×D₅	—	3	1	no	—
12	C₅×Dic₅	D₃	1	1	yes	characteristic, complemented
13	C₃×C₅²	D₄	1	1	yes	characteristic, complemented, Hall
14	C₂×D₁₅	C₂×C₅	1	1	yes	characteristic, complemented
15	C₂×C₅×D₃	—	2	2	no	—
16	C₂×C₅×D₃	—	5	1	no	—
17	C₂×C₅×D₃	D₅	1	1	yes	characteristic, complemented
18	C₂×C₅×D₃	C₂×C₅	1	1	yes	characteristic, complemented
19	C₃×C₄×C₅	—	5	1	no	—
20	C₃×Dic₅	C₂×C₅	1	1	yes	characteristic, complemented
21	C₂×C₅²	—	6	1	no	—
22	C₂×C₅²	C₂×D₃	1	1	yes	characteristic
23	C₅×D₅	—	6	1	no	—
24	C₅×D₄	—	15	1	no	—
25	C₂²⋊D₅	—	3	1	no	—
26	C₂×C₃×C₅	—	2	2	no	—
27	C₂×C₃×C₅	C₂×D₅	1	1	yes	characteristic
28	C₂×C₃×C₅	C₂²×C₅	1	1	yes	characteristic
29	D₁₅	—	2	1	no	—
30	C₅×D₃	—	2	4	no	—
31	C₅×D₃	—	2	1	no	—
32	C₅×D₃	—	2	1	no	—
33	C₅×D₃	—	10	1	no	—
34	C₅²	D₁₂	1	1	yes	characteristic, complemented, Sylow
35	D₁₂	—	5	1	no	Hall
36	C₂²×C₅	—	3	1	no	—
37	C₂²×C₅	—	3	1	no	—
38	C₂²×C₅	—	6	2	no	—
39	C₂²×C₅	—	15	1	no	—
40	C₂×D₅	—	3	1	no	—
41	C₄×C₅	—	5	1	no	—
42	Dic₅	C₅×D₃	1	1	yes	characteristic, complemented
43	C₃×C₅	—	2	2	no	—
44	C₃×C₅	C₂²⋊D₅	1	1	yes	characteristic, complemented
45	C₃×C₅	C₅×D₄	1	1	yes	characteristic, complemented
46	C₂×D₃	—	5	1	no	—
47	C₂×D₃	C₅×D₅	1	1	yes	characteristic, complemented
48	C₃×C₄	—	5	1	no	—
49	C₂×C₅	—	2	2	no	—
50	C₂×C₅	—	6	4	no	—
51	C₂×C₅	—	6	1	no	—
52	C₂×C₅	—	6	1	no	—
53	C₂×C₅	—	30	1	no	—
54	C₂×C₅	D₃×D₅	1	1	yes	characteristic
55	C₂×C₅	C₂×C₅×D₃	1	1	yes	characteristic
56	D₅	—	6	1	no	—
57	D₄	—	15	1	no	Sylow
58	C₂×C₃	C₂×C₅×D₅	1	1	yes	characteristic
59	D₃	—	2	1	no	—
60	D₃	—	10	1	no	—
61	C₅	—	2	2	no	—
62	C₅	#1203	1	1	yes	characteristic, direct product factor
63	C₅	C₅×D₁₂	1	1	yes	characteristic, complemented
64	C₂²	—	3	1	no	—
65	C₂²	—	15	1	no	—
66	C₄	—	5	1	no	—
67	C₃	C₅×C₂²⋊D₅	1	1	yes	characteristic, complemented, Sylow
68	C₂	—	6	1	no	—
69	C₂	—	30	1	no	—
70	C₂	C₅×D₃×D₅	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂×C₅	D₃×D₅
commutator, cosocle	C₂×C₃×C₅	C₂²×C₅
Frattini	C₂	C₅×D₃×D₅
Fitting, socle	C₂×C₃×C₅²	C₂²
hypocenter	C₃×C₅	C₅×D₄
LCS	G ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₅

Complements

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

N	H ≅ G/N	type
1. C₂×C₅×D₁₅	C₂	semidirect
2. C₂×C₅²×D₃	C₂	semidirect
3. C₃×C₅×Dic₅	C₂	semidirect
9. #1203	C₅	direct (N × H)
12. C₅×Dic₅	D₃	semidirect
13. C₃×C₅²	D₄	semidirect
14. C₂×D₁₅	C₂×C₅	semidirect
17. C₂×C₅×D₃	D₅	semidirect
18. C₂×C₅×D₃	C₂×C₅	semidirect
20. C₃×Dic₅	C₂×C₅	semidirect
34. C₅²	D₁₂	semidirect
42. Dic₅	C₅×D₃	semidirect
44. C₃×C₅	C₂²⋊D₅	semidirect
45. C₃×C₅	C₅×D₄	semidirect
47. C₂×D₃	C₅×D₅	semidirect
62. C₅	#1203	direct (N × H)
63. C₅	C₅×D₁₂	semidirect
67. C₃	C₅×C₂²⋊D₅	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×D₁₅	2
C₂×C₅²×D₃	2
C₃×C₅×Dic₅	2
C₅×C₂²⋊D₅	3
C₅×D₁₂	5
#1203	5

Subgroups — C5×#1203 #10597109

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#1203 #10597109