Subgroups — C₃⋊(C₅⋊(#126⋊C₃)) #13086236

Order 1440.

86 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	#91765	C₃	1	1	yes	maximal, characteristic, complemented
3	C₅⋊(#126⋊C₃)	—	3	1	no	maximal
4	A₄×D₁₅	—	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	#6714	—	3	1	no	—
8	#6712	—	3	1	no	—
9	C₅×C₄²⋊C₃	—	2	1	no	—
10	C₅×C₄²⋊C₃	D₃	1	1	yes	characteristic, complemented
11	C₃×C₅×A₄	—	4	1	no	—
12	#4022	—	3	1	no	Hall
13	C₃×C₄²⋊C₃	D₅	1	1	yes	characteristic, complemented
14	C₂²×D₁₅	A₄	1	1	yes	characteristic
15	D₅×A₄	—	12	1	no	—
16	C₂×C₃×C₄×C₅	—	3	1	no	—
17	C₂×Dic₁₅	—	3	1	no	—
18	#876	—	5	1	no	—
19	#126⋊C₃	—	15	1	no	—
20	C₃×D₁₅	—	16	1	no	—
21	C₄²×C₅	C₃×D₃	1	1	yes	characteristic, complemented
22	#685	—	9	1	no	—
23	#683	—	9	1	no	—
24	D₃×A₄	—	20	1	no	—
25	C₂²×C₃×C₅	C₂×A₄	1	1	yes	characteristic
26	C₂×D₁₅	—	6	1	no	—
27	C₅×A₄	—	4	1	no	—
28	C₅×A₄	—	8	1	no	—
29	C₃×C₄×C₅	—	6	1	no	—
30	Dic₁₅	—	6	1	no	—
31	C₃×C₄²	C₃×D₅	1	1	yes	characteristic, complemented
32	(C₂×D₃)⋊C₄	—	15	1	no	—
33	Dic₃⋊C₄	—	15	1	no	—
34	C₄²⋊C₃	—	2	1	no	—
35	C₄²⋊C₃	D₁₅	1	1	yes	characteristic, complemented
36	C₃²×C₅	—	16	1	no	Hall
37	C₂²×D₅	—	3	1	no	—
38	C₂×C₄×C₅	—	3	1	no	—
39	C₂×Dic₅	—	9	1	no	—
40	C₃×A₄	—	4	1	no	—
41	#126	—	15	1	no	Sylow
42	C₂×C₃×C₅	—	3	1	no	—
43	D₁₅	—	4	1	no	—
44	C₃×D₅	—	48	1	no	—
45	C₂²×D₃	—	5	1	no	—
46	C₂×A₄	—	60	1	no	—
47	C₂×C₃×C₄	—	3	1	no	—
48	C₂×Dic₃	—	15	1	no	—
49	C₂²×C₅	D₃×A₄	1	1	yes	characteristic
50	C₂×D₅	—	18	1	no	—
51	C₄×C₅	—	6	1	no	—
52	Dic₅	—	18	1	no	—
53	C₃×D₃	—	80	1	no	—
54	C₄⋊C₄	—	45	1	no	—
55	C₂²⋊C₄	—	45	1	no	—
56	C₄²	C₃×D₁₅	1	1	yes	characteristic, complemented
57	C₃×C₅	—	16	1	no	—
58	C₃×C₅	—	32	1	no	—
59	C₃×C₅	#126⋊C₃	1	1	yes	characteristic, complemented
60	C₂²×C₃	D₅×A₄	1	1	yes	characteristic
61	C₂×D₃	—	30	1	no	—
62	A₄	—	4	1	no	—
63	A₄	—	8	1	no	—
64	C₃×C₄	—	6	1	no	—
65	Dic₃	—	30	1	no	—
66	C₂×C₅	—	3	1	no	—
67	D₅	—	12	1	no	—
68	C₃²	—	16	1	no	Sylow
69	C₂³	—	15	1	no	—
70	C₂×C₄	—	3	1	no	—
71	C₂×C₄	—	45	1	no	—
72	C₂×C₃	—	3	1	no	—
73	C₂×C₃	—	240	1	no	—
74	D₃	—	20	1	no	—
75	C₅	C₃⋊(#126⋊C₃)	1	1	yes	characteristic, complemented, Sylow
76	C₂²	—	90	1	no	—
77	C₂²	A₄×D₁₅	1	1	yes	characteristic
78	C₄	—	6	1	no	—
79	C₄	—	90	1	no	—
80	C₃	—	16	1	no	—
81	C₃	—	32	1	no	—
82	C₃	C₅⋊(#126⋊C₃)	1	1	yes	characteristic, complemented
83	C₂	—	3	1	no	—
84	C₂	—	60	1	no	—

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₃×C₄²×C₅	C₂×C₃
Frattini	C₂²	A₄×D₁₅
socle	C₂²×C₃×C₅	C₂×A₄
LCS	G ▷ C₃×C₄²×C₅
UCS	C₁

Complements

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

N	H ≅ G/N	type
1. C₃×C₅×C₄²⋊C₃	C₂	semidirect
2. #91765	C₃	semidirect
6. C₃×C₄²×C₅	C₂×C₃	semidirect
10. C₅×C₄²⋊C₃	D₃	semidirect
13. C₃×C₄²⋊C₃	D₅	semidirect
21. C₄²×C₅	C₃×D₃	semidirect
31. C₃×C₄²	C₃×D₅	semidirect
35. C₄²⋊C₃	D₁₅	semidirect
56. C₄²	C₃×D₁₅	semidirect
59. C₃×C₅	#126⋊C₃	semidirect
75. C₅	C₃⋊(#126⋊C₃)	semidirect
82. C₃	C₅⋊(#126⋊C₃)	semidirect

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
#91765	3
C₅⋊(#126⋊C₃)	3
A₄×D₁₅	4
C₃⋊(#126⋊C₃)	5

Subgroups — C3⋊(C5⋊(#126⋊C3)) #13086236

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃⋊(C₅⋊(#126⋊C₃)) #13086236