Subgroups — C₃×C₅×S₅ #14268365

Order 1800.

84 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×A₅	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×S₅	C₃	1	1	yes	maximal, characteristic, direct product factor
3	C₃×C₅×S₄	—	5	1	no	maximal
4	C₃×S₅	C₅	1	1	yes	maximal, characteristic, direct product factor
5	C₃×C₅×AGL(1,𝔽₅)	—	6	1	no	maximal
6	C₅×A₅	C₂×C₃	1	1	yes	characteristic, complemented
7	C₂×C₃×C₅×D₃	—	10	1	no	maximal
8	C₃×C₅×A₄	—	5	1	no	—
9	C₃×A₅	C₂×C₅	1	1	yes	characteristic, complemented
10	C₃×C₅×D₅	—	6	1	no	—
11	C₅×S₄	—	5	1	no	—
12	S₅	C₃×C₅	1	1	yes	characteristic, direct product factor
13	C₃×C₅×D₄	—	15	1	no	—
14	C₅×AGL(1,𝔽₅)	—	6	1	no	—
15	C₂×C₃²×C₅	—	10	1	no	—
16	C₃×C₅×D₃	—	10	1	no	—
17	C₃×C₅×D₃	—	10	1	no	—
18	C₃×C₅²	—	6	1	no	—
19	C₃×S₄	—	5	1	no	Hall
20	C₂²×C₃×C₅	—	5	1	no	—
21	C₂²×C₃×C₅	—	15	1	no	—
22	C₂×C₅×D₃	—	10	1	no	—
23	C₅×A₄	—	5	1	no	—
24	C₅×A₄	—	10	1	no	—
25	C₃×AGL(1,𝔽₅)	—	6	1	no	—
26	A₅	C₂×C₃×C₅	1	1	yes	characteristic, complemented
27	C₃×C₄×C₅	—	15	1	no	—
28	C₅×D₅	—	6	1	no	—
29	C₃²×C₅	—	10	1	no	—
30	C₅×D₄	—	15	1	no	—
31	C₂×C₃×D₃	—	10	1	no	—
32	C₃×A₄	—	5	1	no	—
33	C₂×C₃×C₅	—	10	1	no	—
34	C₂×C₃×C₅	—	10	1	no	—
35	C₂×C₃×C₅	—	15	1	no	—
36	C₂×C₃×C₅	—	20	1	no	—
37	C₃×D₅	—	6	1	no	—
38	C₅×D₃	—	10	1	no	—
39	C₅×D₃	—	10	1	no	—
40	C₅²	—	6	1	no	Sylow
41	S₄	—	5	1	no	—
42	C₃×D₄	—	15	1	no	—
43	C₂²×C₅	—	5	1	no	—
44	C₂²×C₅	—	15	1	no	—
45	AGL(1,𝔽₅)	—	6	1	no	—
46	C₄×C₅	—	15	1	no	—
47	C₂×C₃²	—	10	1	no	—
48	C₃×D₃	—	10	1	no	—
49	C₃×D₃	—	10	1	no	—
50	C₃×C₅	—	6	1	no	—
51	C₃×C₅	—	10	1	no	—
52	C₃×C₅	—	20	1	no	—
53	C₃×C₅	—	24	1	no	—
54	C₃×C₅	S₅	1	1	yes	characteristic, direct product factor
55	C₂²×C₃	—	5	1	no	—
56	C₂²×C₃	—	15	1	no	—
57	C₂×D₃	—	10	1	no	—
58	A₄	—	5	1	no	—
59	A₄	—	10	1	no	—
60	C₃×C₄	—	15	1	no	—
61	C₂×C₅	—	10	1	no	—
62	C₂×C₅	—	15	1	no	—
63	D₅	—	6	1	no	—
64	C₃²	—	10	1	no	Sylow
65	D₄	—	15	1	no	Sylow
66	C₂×C₃	—	10	1	no	—
67	C₂×C₃	—	10	1	no	—
68	C₂×C₃	—	15	1	no	—
69	C₂×C₃	—	20	1	no	—
70	D₃	—	10	1	no	—
71	D₃	—	10	1	no	—
72	C₅	—	6	1	no	—
73	C₅	—	24	1	no	—
74	C₅	C₃×S₅	1	1	yes	characteristic, direct product factor
75	C₂²	—	5	1	no	—
76	C₂²	—	15	1	no	—
77	C₄	—	15	1	no	—
78	C₃	—	10	1	no	—
79	C₃	—	20	1	no	—
80	C₃	C₅×S₅	1	1	yes	characteristic, direct product factor
81	C₂	—	10	1	no	—
82	C₂	—	15	1	no	—

	subgroup	G/H
center, Fitting, hypercenter	C₃×C₅	S₅
commutator, cosocle, hypocenter	A₅	C₂×C₃×C₅
Frattini	C₁	G
socle	C₃×C₅×A₅	C₂
LCS	G ▷ A₅
UCS	C₁ ◁ C₃×C₅

Complements

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

N	H ≅ G/N	type
1. C₃×C₅×A₅	C₂	semidirect
2. C₅×S₅	C₃	direct (N × H)
4. C₃×S₅	C₅	direct (N × H)
6. C₅×A₅	C₂×C₃	semidirect
9. C₃×A₅	C₂×C₅	semidirect
12. S₅	C₃×C₅	direct (N × H)
26. A₅	C₂×C₃×C₅	semidirect
54. C₃×C₅	S₅	direct (N × H)
74. C₅	C₃×S₅	direct (N × H)
80. C₃	C₅×S₅	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×C₅×A₅	2
C₅×S₅	3
C₃×C₅×S₄	5
C₃×S₅	5
C₃×C₅×AGL(1,𝔽₅)	6
C₂×C₃×C₅×D₃	10

Subgroups — C3×C5×S5 #14268365

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×S₅ #14268365