Subgroups — C₃²×C₅×#127 #13088236

Order 1440.

648 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×C₅×D₄	C₂	1	6	yes	maximal, complemented
2	C₃²×C₄²×C₅	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₅×#127	C₃	1	4	yes	maximal, direct product factor
4	C₂³×C₃²×C₅	C₂²	1	4	yes	—
5	C₃²×C₅×D₄	—	2	12	no	—
6	C₂×C₃²×C₄×C₅	C₂²	1	3	yes	—
7	C₃²×#127	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
8	C₂×C₃×C₅×D₄	C₂×C₃	1	24	yes	complemented
9	C₃×C₄²×C₅	C₂×C₃	1	4	yes	complemented
10	C₂²×C₃²×C₅	—	2	12	no	—
11	C₂²×C₃²×C₅	C₂³	1	1	yes	characteristic
12	C₃²×C₄×C₅	D₄	1	6	yes	complemented
13	C₅×#127	C₃²	1	1	yes	characteristic, direct product factor, Hall
14	C₂×C₃²×D₄	C₂×C₅	1	6	yes	complemented
15	C₃²×C₄²	C₂×C₅	1	1	yes	characteristic, complemented
16	C₂³×C₃×C₅	C₂²×C₃	1	16	yes	—
17	C₃×C₅×D₄	—	2	48	no	—
18	C₂×C₃×C₄×C₅	C₂²×C₃	1	12	yes	—
19	C₃×#127	C₃×C₅	1	4	yes	direct product factor
20	C₂×C₃²×C₅	—	4	4	no	—
21	C₂×C₃²×C₅	C₂×D₄	1	3	yes	—
22	C₂×C₅×D₄	C₂×C₃²	1	6	yes	complemented
23	C₄²×C₅	C₂×C₃²	1	1	yes	characteristic, complemented
24	C₂³×C₃²	C₂²×C₅	1	4	yes	—
25	C₃²×D₄	—	2	12	no	—
26	C₂×C₃²×C₄	C₂²×C₅	1	3	yes	—
27	C₂²×C₃×C₅	—	2	48	no	—
28	C₂²×C₃×C₅	C₂³×C₃	1	4	yes	—
29	C₃×C₄×C₅	C₃×D₄	1	24	yes	complemented
30	C₂×C₃×D₄	C₂×C₃×C₅	1	24	yes	complemented
31	C₃×C₄²	C₂×C₃×C₅	1	4	yes	complemented
32	C₃²×C₅	#127	1	1	yes	characteristic, direct product factor, Hall
33	C₂³×C₅	C₂²×C₃²	1	4	yes	—
34	C₅×D₄	—	2	12	no	—
35	C₂×C₄×C₅	C₂²×C₃²	1	3	yes	—
36	C₂²×C₃²	—	2	12	no	—
37	C₂²×C₃²	C₂³×C₅	1	1	yes	characteristic
38	C₃²×C₄	C₅×D₄	1	6	yes	complemented
39	#127	C₃²×C₅	1	1	yes	characteristic, direct product factor, Sylow
40	C₂×C₃×C₅	—	4	16	no	—
41	C₂×C₃×C₅	C₂×C₃×D₄	1	12	yes	—
42	C₂³×C₃	C₂²×C₃×C₅	1	16	yes	—
43	C₃×D₄	—	2	48	no	—
44	C₂×C₃×C₄	C₂²×C₃×C₅	1	12	yes	—
45	C₂²×C₅	—	2	12	no	—
46	C₂²×C₅	C₂³×C₃²	1	1	yes	characteristic
47	C₄×C₅	C₃²×D₄	1	6	yes	complemented
48	C₂×C₃²	—	4	4	no	—
49	C₂×C₃²	C₂×C₅×D₄	1	3	yes	—
50	C₂×D₄	C₂×C₃²×C₅	1	6	yes	complemented
51	C₄²	C₂×C₃²×C₅	1	1	yes	characteristic, complemented
52	C₃×C₅	C₃×#127	1	4	yes	direct product factor
53	C₂²×C₃	—	2	48	no	—
54	C₂²×C₃	C₂³×C₃×C₅	1	4	yes	—
55	C₃×C₄	C₃×C₅×D₄	1	24	yes	complemented
56	C₂×C₅	—	4	4	no	—
57	C₂×C₅	C₂×C₃²×D₄	1	3	yes	—
58	C₃²	C₅×#127	1	1	yes	characteristic, direct product factor, Sylow
59	C₂³	C₂²×C₃²×C₅	1	4	yes	—
60	D₄	—	2	12	no	—
61	C₂×C₄	C₂²×C₃²×C₅	1	3	yes	—
62	C₂×C₃	—	4	16	no	—
63	C₂×C₃	C₂×C₃×C₅×D₄	1	12	yes	—
64	C₅	C₃²×#127	1	1	yes	characteristic, direct product factor, Sylow
65	C₂²	—	2	12	no	—
66	C₂²	C₂³×C₃²×C₅	1	1	yes	characteristic
67	C₄	C₃²×C₅×D₄	1	6	yes	complemented
68	C₃	C₃×C₅×#127	1	4	yes	direct product factor
69	C₂	—	4	4	no	—
70	C₂	C₂×C₃²×C₅×D₄	1	3	yes	—

	subgroup	G/H
center, socle	C₂²×C₃²×C₅	C₂³
commutator, Frattini, cosocle	C₂²	C₂³×C₃²×C₅
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. C₂×C₃²×C₅×D₄	C₂	semidirect
2. C₃²×C₄²×C₅	C₂	semidirect
3. C₃×C₅×#127	C₃	direct (N × H)
7. C₃²×#127	C₅	direct (N × H)
8. C₂×C₃×C₅×D₄	C₂×C₃	semidirect
9. C₃×C₄²×C₅	C₂×C₃	semidirect
12. C₃²×C₄×C₅	D₄	semidirect
13. C₅×#127	C₃²	direct (N × H)
14. C₂×C₃²×D₄	C₂×C₅	semidirect
15. C₃²×C₄²	C₂×C₅	semidirect
19. C₃×#127	C₃×C₅	direct (N × H)
22. C₂×C₅×D₄	C₂×C₃²	semidirect
23. C₄²×C₅	C₂×C₃²	semidirect
29. C₃×C₄×C₅	C₃×D₄	semidirect
30. C₂×C₃×D₄	C₂×C₃×C₅	semidirect
31. C₃×C₄²	C₂×C₃×C₅	semidirect
32. C₃²×C₅	#127	direct (N × H)
38. C₃²×C₄	C₅×D₄	semidirect
39. #127	C₃²×C₅	direct (N × H)
47. C₄×C₅	C₃²×D₄	semidirect
50. C₂×D₄	C₂×C₃²×C₅	semidirect
51. C₄²	C₂×C₃²×C₅	semidirect
52. C₃×C₅	C₃×#127	direct (N × H)
55. C₃×C₄	C₃×C₅×D₄	semidirect
58. C₃²	C₅×#127	direct (N × H)
64. C₅	C₃²×#127	direct (N × H)
67. C₄	C₃²×C₅×D₄	semidirect
68. C₃	C₃×C₅×#127	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃²×C₅×D₄	2
C₃²×C₄²×C₅	2
C₃×C₅×#127	3
C₃²×#127	5

Subgroups — C32×C5×#127 #13088236

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×C₅×#127 #13088236