Subgroups — C₅×#6293 #11762072

Order 1080.

192 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×C₃⋊(C₃⋊Dic₃)	C₂	1	1	yes	maximal, characteristic
2	C₃×C₅×C₃⋊Dic₃	C₂	1	1	yes	maximal, characteristic
3	C₃²×C₅×Dic₃	C₂	1	1	yes	maximal, characteristic
4	C₅×#634	—	3	1	no	maximal
5	C₅×#627	—	3	4	no	maximal
6	C₂×C₃³×C₅	C₂²	1	1	yes	characteristic
7	#6293	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
8	C₃²×C₄×C₅	—	3	1	no	—
9	C₅×C₃⋊Dic₃	—	3	1	no	—
10	C₅×C₃⋊Dic₃	—	3	4	no	—
11	C₅×C₃⋊Dic₃	—	6	4	no	—
12	C₅×C₃⋊Dic₃	D₃	1	1	yes	characteristic
13	C₃×C₅×Dic₃	—	3	4	no	—
14	C₃×C₅×Dic₃	D₃	1	4	yes	—
15	C₃³×C₅	Q₈	1	1	yes	characteristic, complemented, Hall
16	C₅×Dic₆	—	9	4	no	—
17	C₅×Dic₆	—	9	1	no	—
18	C₃⋊(C₃⋊Dic₃)	C₂×C₅	1	1	yes	characteristic
19	C₃×C₃⋊Dic₃	C₂×C₅	1	1	yes	characteristic
20	C₃²×Dic₃	C₂×C₅	1	1	yes	characteristic
21	C₂×C₃²×C₅	—	2	4	no	—
22	C₂×C₃²×C₅	C₂×D₃	1	4	yes	—
23	C₂×C₃²×C₅	C₂×D₃	1	1	yes	characteristic
24	#634	—	3	1	no	—
25	#627	—	3	4	no	—
26	C₃×C₄×C₅	—	3	4	no	—
27	C₃×C₄×C₅	—	9	1	no	—
28	C₅×Dic₃	—	3	4	no	—
29	C₅×Dic₃	—	9	4	no	—
30	C₅×Dic₃	—	9	1	no	—
31	C₅×Dic₃	—	18	4	no	—
32	C₅×Dic₃	C₃⋊D₃	1	1	yes	characteristic
33	C₂×C₃³	C₂²×C₅	1	1	yes	characteristic
34	C₃²×C₅	—	2	4	no	—
35	C₃²×C₅	Dic₆	1	4	yes	complemented
36	C₃²×C₅	Dic₆	1	1	yes	characteristic, complemented
37	C₅×Q₈	—	27	1	no	Hall
38	C₃²×C₄	—	3	1	no	—
39	C₃⋊Dic₃	—	3	1	no	—
40	C₃⋊Dic₃	—	3	4	no	—
41	C₃⋊Dic₃	—	6	4	no	—
42	C₃⋊Dic₃	C₅×D₃	1	1	yes	characteristic
43	C₃×Dic₃	—	3	4	no	—
44	C₃×Dic₃	C₅×D₃	1	4	yes	—
45	C₂×C₃×C₅	—	2	4	no	—
46	C₂×C₃×C₅	D₃²	1	4	yes	—
47	C₂×C₃×C₅	C₂×C₃⋊D₃	1	1	yes	characteristic
48	C₃³	C₅×Q₈	1	1	yes	characteristic, complemented, Sylow
49	Dic₆	—	9	4	no	—
50	Dic₆	—	9	1	no	—
51	C₄×C₅	—	3	1	no	—
52	C₄×C₅	—	9	1	no	—
53	C₄×C₅	—	27	1	no	—
54	C₂×C₃²	—	2	4	no	—
55	C₂×C₃²	C₂×C₅×D₃	1	4	yes	—
56	C₂×C₃²	C₂×C₅×D₃	1	1	yes	characteristic
57	C₃×C₅	—	2	4	no	—
58	C₃×C₅	#627	1	4	yes	complemented
59	C₃×C₅	#634	1	1	yes	characteristic, complemented
60	C₃×C₄	—	3	4	no	—
61	C₃×C₄	—	9	1	no	—
62	Dic₃	—	3	4	no	—
63	Dic₃	—	9	4	no	—
64	Dic₃	—	9	1	no	—
65	Dic₃	—	18	4	no	—
66	Dic₃	C₅×C₃⋊D₃	1	1	yes	characteristic
67	C₂×C₅	D₃×C₃⋊D₃	1	1	yes	characteristic
68	C₃²	—	2	4	no	—
69	C₃²	C₅×Dic₆	1	4	yes	complemented
70	C₃²	C₅×Dic₆	1	1	yes	characteristic, complemented
71	Q₈	—	27	1	no	Sylow
72	C₂×C₃	—	2	4	no	—
73	C₂×C₃	C₅×D₃²	1	4	yes	—
74	C₂×C₃	C₂×C₅×C₃⋊D₃	1	1	yes	characteristic
75	C₅	#6293	1	1	yes	characteristic, direct product factor, Sylow
76	C₄	—	3	1	no	—
77	C₄	—	9	1	no	—
78	C₄	—	27	1	no	—
79	C₃	—	2	4	no	—
80	C₃	C₅×#627	1	4	yes	complemented
81	C₃	C₅×#634	1	1	yes	characteristic, complemented
82	C₂	C₅×D₃×C₃⋊D₃	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂×C₅	D₃×C₃⋊D₃
commutator, cosocle	C₂×C₃³	C₂²×C₅
Frattini	C₂	C₅×D₃×C₃⋊D₃
Fitting, socle, prime radical	C₂×C₃³×C₅	C₂²
hypocenter	C₃³	C₅×Q₈
LCS	G ▷ C₂×C₃³ ▷ C₃³
UCS	C₁ ◁ C₂×C₅

Complements

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

N	H ≅ G/N	type
7. #6293	C₅	direct (N × H)
15. C₃³×C₅	Q₈	semidirect
35. C₃²×C₅	Dic₆	semidirect
36. C₃²×C₅	Dic₆	semidirect
48. C₃³	C₅×Q₈	semidirect
58. C₃×C₅	#627	semidirect
59. C₃×C₅	#634	semidirect
69. C₃²	C₅×Dic₆	semidirect
70. C₃²	C₅×Dic₆	semidirect
75. C₅	#6293	direct (N × H)
80. C₃	C₅×#627	semidirect
81. C₃	C₅×#634	semidirect

Maximal subgroups

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

subgroup	index
C₅×C₃⋊(C₃⋊Dic₃)	2
C₃×C₅×C₃⋊Dic₃	2
C₃²×C₅×Dic₃	2
C₅×#634	3
C₅×#627	3
#6293	5

Subgroups — C5×#6293 #11762072

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#6293 #11762072