Subgroups — C₅×#67158 #14268288

Order 1800.

144 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅²×C₃⋊Dic₃	C₂	1	1	yes	maximal, characteristic
2	C₃×C₅×Dic₁₅	C₂	1	2	yes	maximal
3	C₅×#1204	—	3	2	no	maximal
4	C₂×C₃²×C₅²	C₂²	1	1	yes	characteristic
5	#67158	C₅	1	1	yes	maximal, characteristic, direct product factor
6	C₅×#627	—	5	1	no	maximal
7	C₅×Dic₁₅	D₃	1	2	yes	—
8	C₅²×Dic₃	—	3	2	no	—
9	C₅²×Dic₃	—	6	1	no	—
10	C₃×C₅×Dic₅	—	3	2	no	—
11	C₃²×C₅²	Q₈	1	1	yes	characteristic, complemented, Hall
12	C₅×Dic₁₀	—	9	1	no	Hall
13	C₅×C₃⋊Dic₃	—	2	2	no	—
14	C₅×C₃⋊Dic₃	D₅	1	1	yes	characteristic
15	C₅×C₃⋊Dic₃	C₂×C₅	1	1	yes	characteristic
16	C₃×Dic₁₅	C₂×C₅	1	2	yes	—
17	C₃×C₅×Dic₃	—	5	2	no	—
18	C₂×C₃×C₅²	—	2	1	no	—
19	C₂×C₃×C₅²	C₂×D₃	1	2	yes	—
20	C₅×Dic₆	—	15	2	no	—
21	#1204	—	3	2	no	—
22	C₄×C₅²	—	9	1	no	—
23	C₅×Dic₅	—	3	2	no	—
24	C₂×C₃²×C₅	—	2	2	no	—
25	C₂×C₃²×C₅	C₂×D₅	1	1	yes	characteristic
26	C₂×C₃²×C₅	C₂²×C₅	1	1	yes	characteristic
27	C₃×C₅²	—	2	1	no	—
28	C₃×C₅²	Dic₆	1	2	yes	complemented
29	#627	—	5	1	no	Hall
30	C₃×C₄×C₅	—	15	2	no	—
31	Dic₁₅	C₅×D₃	1	2	yes	—
32	C₃×Dic₅	—	3	2	no	—
33	C₅×Dic₃	—	3	2	no	—
34	C₅×Dic₃	—	3	2	no	—
35	C₅×Dic₃	—	5	2	no	—
36	C₅×Dic₃	—	6	4	no	—
37	C₅×Dic₃	—	6	1	no	—
38	C₅×Dic₃	—	6	1	no	—
39	C₅×Dic₃	—	6	4	no	—
40	C₂×C₅²	D₃²	1	1	yes	characteristic
41	C₃²×C₅	—	2	2	no	—
42	C₃²×C₅	Dic₁₀	1	1	yes	characteristic, complemented
43	C₃²×C₅	C₅×Q₈	1	1	yes	characteristic, complemented
44	C₅×Q₈	—	45	1	no	—
45	Dic₁₀	—	9	1	no	—
46	C₃⋊Dic₃	C₅×D₅	1	1	yes	characteristic
47	C₃×Dic₃	—	5	2	no	—
48	C₂×C₃×C₅	—	2	4	no	—
49	C₂×C₃×C₅	—	2	1	no	—
50	C₂×C₃×C₅	—	2	1	no	—
51	C₂×C₃×C₅	—	2	4	no	—
52	C₂×C₃×C₅	D₃×D₅	1	2	yes	—
53	C₂×C₃×C₅	C₂×C₅×D₃	1	2	yes	—
54	C₅²	#627	1	1	yes	characteristic, complemented, Sylow
55	Dic₆	—	15	2	no	—
56	C₄×C₅	—	9	1	no	—
57	C₄×C₅	—	9	1	no	—
58	C₄×C₅	—	15	2	no	—
59	C₄×C₅	—	18	2	no	—
60	Dic₅	—	3	2	no	—
61	C₂×C₃²	C₂×C₅×D₅	1	1	yes	characteristic
62	C₃×C₅	—	2	4	no	—
63	C₃×C₅	—	2	1	no	—
64	C₃×C₅	—	2	1	no	—
65	C₃×C₅	—	2	4	no	—
66	C₃×C₅	#1204	1	2	yes	complemented
67	C₃×C₅	C₅×Dic₆	1	2	yes	complemented
68	C₃×C₄	—	15	2	no	—
69	Dic₃	—	3	2	no	—
70	Dic₃	—	5	2	no	—
71	Dic₃	—	6	1	no	—
72	C₂×C₅	—	2	2	no	—
73	C₂×C₅	C₅×D₃²	1	1	yes	characteristic
74	C₂×C₅	(C₃⋊D₃)⋊D₅	1	1	yes	characteristic
75	C₃²	C₅×Dic₁₀	1	1	yes	characteristic, complemented, Sylow
76	Q₈	—	45	1	no	Sylow
77	C₂×C₃	—	2	1	no	—
78	C₂×C₃	C₅×D₃×D₅	1	2	yes	—
79	C₅	—	2	2	no	—
80	C₅	C₅×#627	1	1	yes	characteristic, complemented
81	C₅	#67158	1	1	yes	characteristic, direct product factor
82	C₄	—	9	1	no	—
83	C₄	—	15	2	no	—
84	C₃	—	2	1	no	—
85	C₃	C₅×#1204	1	2	yes	complemented
86	C₂	C₅×(C₃⋊D₃)⋊D₅	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂×C₅	(C₃⋊D₃)⋊D₅
commutator, cosocle	C₂×C₃²×C₅	C₂²×C₅
Frattini	C₂	C₅×(C₃⋊D₃)⋊D₅
Fitting, socle, prime radical	C₂×C₃²×C₅²	C₂²
hypocenter	C₃²×C₅	C₅×Q₈
LCS	G ▷ C₂×C₃²×C₅ ▷ C₃²×C₅
UCS	C₁ ◁ C₂×C₅

Complements

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

N	H ≅ G/N	type
5. #67158	C₅	direct (N × H)
11. C₃²×C₅²	Q₈	semidirect
28. C₃×C₅²	Dic₆	semidirect
42. C₃²×C₅	Dic₁₀	semidirect
43. C₃²×C₅	C₅×Q₈	semidirect
54. C₅²	#627	semidirect
66. C₃×C₅	#1204	semidirect
67. C₃×C₅	C₅×Dic₆	semidirect
75. C₃²	C₅×Dic₁₀	semidirect
80. C₅	C₅×#627	semidirect
81. C₅	#67158	direct (N × H)
85. C₃	C₅×#1204	semidirect

Maximal subgroups

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

subgroup	index
C₅²×C₃⋊Dic₃	2
C₃×C₅×Dic₁₅	2
C₅×#1204	3
#67158	5
C₅×#627	5

Subgroups — C5×#67158 #14268288

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#67158 #14268288