Subgroups — C₃×C₅⋊(#126⋊C₃) #13086222

Order 1440.

104 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×C₄²⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×#4022	C₃	1	1	yes	maximal, characteristic, complemented
3	C₅⋊(#126⋊C₃)	C₃	1	3	yes	maximal, direct product factor
4	C₃×D₅×A₄	—	4	1	no	maximal
5	C₃×#126⋊C₃	—	5	1	no	maximal, Hall
6	C₃×C₄²×C₅	C₂×C₃	1	1	yes	characteristic, complemented
7	C₃×#685	—	3	1	no	—
8	C₃×#683	—	3	1	no	—
9	C₅×C₄²⋊C₃	C₂×C₃	1	3	yes	complemented
10	C₃×C₅×A₄	—	4	1	no	—
11	#4022	C₃²	1	1	yes	characteristic, complemented, Hall
12	C₃×C₄²⋊C₃	D₅	1	1	yes	characteristic, complemented
13	C₂²×C₃×D₅	A₄	1	1	yes	characteristic
14	D₅×A₄	—	4	3	no	—
15	C₂×C₃×C₄×C₅	—	3	1	no	—
16	C₂×C₃×Dic₅	—	3	1	no	—
17	C₃×#126	—	5	1	no	—
18	#126⋊C₃	—	5	3	no	—
19	C₃²×D₅	—	16	1	no	—
20	C₄²×C₅	C₂×C₃²	1	1	yes	characteristic, complemented
21	#685	—	3	1	no	—
22	#683	—	3	1	no	—
23	C₂×C₃×A₄	—	20	1	no	—
24	C₂²×C₃×C₅	C₂×A₄	1	1	yes	characteristic
25	C₂×C₃×D₅	—	6	1	no	—
26	C₅×A₄	—	4	3	no	—
27	C₃×C₄×C₅	—	6	1	no	—
28	C₃×Dic₅	—	6	1	no	—
29	C₃×C₄⋊C₄	—	15	1	no	—
30	C₃×C₂²⋊C₄	—	15	1	no	—
31	C₃×C₄²	C₃×D₅	1	1	yes	characteristic, complemented
32	C₄²⋊C₃	C₃×D₅	1	3	yes	complemented
33	C₃²×C₅	—	16	1	no	Hall
34	C₂²×D₅	C₃×A₄	1	1	yes	characteristic
35	C₂×C₄×C₅	—	3	1	no	—
36	C₂×Dic₅	—	3	1	no	—
37	C₃×A₄	—	4	1	no	—
38	#126	—	5	1	no	Sylow
39	C₂×C₃×C₅	—	3	1	no	—
40	C₃×D₅	—	4	1	no	—
41	C₃×D₅	—	16	3	no	—
42	C₂³×C₃	—	5	1	no	—
43	C₂×A₄	—	20	3	no	—
44	C₂×C₃×C₄	—	3	1	no	—
45	C₂×C₃×C₄	—	15	1	no	—
46	C₂²×C₅	C₂×C₃×A₄	1	1	yes	characteristic
47	C₂×D₅	—	6	1	no	—
48	C₄×C₅	—	6	1	no	—
49	Dic₅	—	6	1	no	—
50	C₂×C₃²	—	80	1	no	—
51	C₄⋊C₄	—	15	1	no	—
52	C₂²⋊C₄	—	15	1	no	—
53	C₄²	C₃²×D₅	1	1	yes	characteristic, complemented
54	C₃×C₅	—	16	3	no	—
55	C₃×C₅	#126⋊C₃	1	1	yes	characteristic, complemented
56	C₂²×C₃	—	30	1	no	—
57	C₂²×C₃	D₅×A₄	1	1	yes	characteristic
58	A₄	—	4	3	no	—
59	C₃×C₄	—	6	1	no	—
60	C₃×C₄	—	30	1	no	—
61	C₂×C₅	—	3	1	no	—
62	D₅	—	4	1	no	—
63	C₃²	—	16	1	no	Sylow
64	C₂³	—	5	1	no	—
65	C₂×C₄	—	3	1	no	—
66	C₂×C₄	—	15	1	no	—
67	C₂×C₃	—	3	1	no	—
68	C₂×C₃	—	20	1	no	—
69	C₂×C₃	—	80	3	no	—
70	C₅	C₃×#126⋊C₃	1	1	yes	characteristic, complemented, Sylow
71	C₂²	—	30	1	no	—
72	C₂²	C₃×D₅×A₄	1	1	yes	characteristic
73	C₄	—	6	1	no	—
74	C₄	—	30	1	no	—
75	C₃	—	16	3	no	—
76	C₃	C₅⋊(#126⋊C₃)	1	1	yes	characteristic, direct product factor
77	C₂	—	3	1	no	—
78	C₂	—	20	1	no	—

	subgroup	G/H
center, hypercenter	C₃	C₅⋊(#126⋊C₃)
commutator, cosocle, hypocenter	C₄²×C₅	C₂×C₃²
Frattini	C₂²	C₃×D₅×A₄
Fitting	C₃×C₄²×C₅	C₂×C₃
socle	C₂²×C₃×C₅	C₂×A₄
LCS	G ▷ C₄²×C₅
UCS	C₁ ◁ C₃

Complements

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

N	H ≅ G/N	type
1. C₃×C₅×C₄²⋊C₃	C₂	semidirect
2. C₃×#4022	C₃	semidirect
3. C₅⋊(#126⋊C₃)	C₃	direct (N × H)
6. C₃×C₄²×C₅	C₂×C₃	semidirect
9. C₅×C₄²⋊C₃	C₂×C₃	semidirect
11. #4022	C₃²	semidirect
12. C₃×C₄²⋊C₃	D₅	semidirect
20. C₄²×C₅	C₂×C₃²	semidirect
31. C₃×C₄²	C₃×D₅	semidirect
32. C₄²⋊C₃	C₃×D₅	semidirect
53. C₄²	C₃²×D₅	semidirect
55. C₃×C₅	#126⋊C₃	semidirect
70. C₅	C₃×#126⋊C₃	semidirect
76. C₃	C₅⋊(#126⋊C₃)	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×C₄²⋊C₃	2
C₃×#4022	3
C₅⋊(#126⋊C₃)	3
C₃×D₅×A₄	4
C₃×#126⋊C₃	5

Subgroups — C3×C5⋊(#126⋊C3) #13086222

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅⋊(#126⋊C₃) #13086222