Subgroups — C₃²×C₅×#126 #13088235

Order 1440.

360 conjugacy classes of subgroups.

Conjugacy classes of subgroups

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

	subgroup	G/H
center, socle	C₂²×C₃²×C₅	C₂³
commutator, Frattini, cosocle	C₂²	C₂³×C₃²×C₅
prime radical	C₂³×C₃²×C₅	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃²×C₅×C₄⋊C₄	C₂	semidirect
3. C₃²×C₄²×C₅	C₂	semidirect
4. C₃×C₅×#126	C₃	direct (N × H)
8. C₃²×#126	C₅	direct (N × H)
9. C₃×C₅×C₄⋊C₄	C₂×C₃	semidirect
11. C₃×C₄²×C₅	C₂×C₃	semidirect
16. C₅×#126	C₃²	direct (N × H)
17. C₃²×C₄⋊C₄	C₂×C₅	semidirect
19. C₃²×C₄²	C₂×C₅	semidirect
23. C₃×#126	C₃×C₅	direct (N × H)
26. C₅×C₄⋊C₄	C₂×C₃²	semidirect
28. C₄²×C₅	C₂×C₃²	semidirect
36. C₃×C₄⋊C₄	C₂×C₃×C₅	semidirect
38. C₃×C₄²	C₂×C₃×C₅	semidirect
39. C₃²×C₅	#126	direct (N × H)
47. #126	C₃²×C₅	direct (N × H)
59. C₄⋊C₄	C₂×C₃²×C₅	semidirect
61. C₄²	C₂×C₃²×C₅	semidirect
62. C₃×C₅	C₃×#126	direct (N × H)
69. C₃²	C₅×#126	direct (N × H)
75. C₅	C₃²×#126	direct (N × H)
80. C₃	C₃×C₅×#126	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃²×C₅×C₄⋊C₄	2
C₃²×C₅×C₂²⋊C₄	2
C₃²×C₄²×C₅	2
C₃×C₅×#126	3
C₃²×#126	5

Subgroups — C32×C5×#126 #13088235

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×C₅×#126 #13088235