Subgroups — C₃²×C₄³×C₅² #72192569

Order 14400.

6192 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×C₄²×C₅²	C₂	1	7	yes	maximal
2	C₃×C₄³×C₅²	C₃	1	4	yes	maximal, direct product factor
3	C₂²×C₃²×C₄×C₅²	C₂²	1	7	yes	—
4	C₃²×C₄²×C₅²	C₄	1	28	yes	direct product factor
5	C₃²×C₄³×C₅	C₅	1	6	yes	maximal, direct product factor
6	C₂×C₃×C₄²×C₅²	C₂×C₃	1	28	yes	—
7	C₂³×C₃²×C₅²	C₂³	1	1	yes	characteristic
8	C₂×C₃²×C₄×C₅²	C₂×C₄	1	42	yes	—
9	C₄³×C₅²	C₃²	1	1	yes	characteristic, direct product factor, Hall
10	C₂×C₃²×C₄²×C₅	C₂×C₅	1	42	yes	—
11	C₂²×C₃×C₄×C₅²	C₂²×C₃	1	28	yes	—
12	C₃×C₄²×C₅²	C₃×C₄	1	112	yes	direct product factor
13	C₃×C₄³×C₅	C₃×C₅	1	24	yes	direct product factor
14	C₂²×C₃²×C₅²	C₂²×C₄	1	7	yes	—
15	C₃²×C₄×C₅²	C₄²	1	28	yes	direct product factor
16	C₂×C₄²×C₅²	C₂×C₃²	1	7	yes	—
17	C₂²×C₃²×C₄×C₅	C₂²×C₅	1	42	yes	—
18	C₃²×C₄²×C₅	C₄×C₅	1	168	yes	direct product factor
19	C₂³×C₃×C₅²	C₂³×C₃	1	4	yes	—
20	C₂×C₃×C₄×C₅²	C₂×C₃×C₄	1	168	yes	—
21	C₃²×C₄³	C₅²	1	1	yes	characteristic, direct product factor, Hall
22	C₂×C₃×C₄²×C₅	C₂×C₃×C₅	1	168	yes	—
23	C₂×C₃²×C₅²	C₂×C₄²	1	7	yes	—
24	C₂²×C₄×C₅²	C₂²×C₃²	1	7	yes	—
25	C₄²×C₅²	C₃²×C₄	1	28	yes	direct product factor
26	C₂³×C₃²×C₅	C₂³×C₅	1	6	yes	—
27	C₂×C₃²×C₄×C₅	C₂×C₄×C₅	1	252	yes	—
28	C₄³×C₅	C₃²×C₅	1	6	yes	direct product factor
29	C₂²×C₃×C₅²	C₂²×C₃×C₄	1	28	yes	—
30	C₃×C₄×C₅²	C₃×C₄²	1	112	yes	direct product factor
31	C₂×C₃²×C₄²	C₂×C₅²	1	7	yes	—
32	C₂²×C₃×C₄×C₅	C₂²×C₃×C₅	1	168	yes	—
33	C₃×C₄²×C₅	C₃×C₄×C₅	1	672	yes	direct product factor
34	C₃²×C₅²	C₄³	1	1	yes	characteristic, direct product factor, Hall
35	C₂³×C₅²	C₂³×C₃²	1	1	yes	characteristic
36	C₂×C₄×C₅²	C₂×C₃²×C₄	1	42	yes	—
37	C₃×C₄³	C₃×C₅²	1	4	yes	direct product factor
38	C₂²×C₃²×C₅	C₂²×C₄×C₅	1	42	yes	—
39	C₃²×C₄×C₅	C₄²×C₅	1	168	yes	direct product factor
40	C₂×C₄²×C₅	C₂×C₃²×C₅	1	42	yes	—
41	C₂×C₃×C₅²	C₂×C₃×C₄²	1	28	yes	—
42	C₂²×C₃²×C₄	C₂²×C₅²	1	7	yes	—
43	C₃²×C₄²	C₄×C₅²	1	28	yes	direct product factor
44	C₂³×C₃×C₅	C₂³×C₃×C₅	1	24	yes	—
45	C₂×C₃×C₄×C₅	C₂×C₃×C₄×C₅	1	1008	yes	—
46	C₂²×C₅²	C₂²×C₃²×C₄	1	7	yes	—
47	C₄×C₅²	C₃²×C₄²	1	28	yes	direct product factor
48	C₂×C₃×C₄²	C₂×C₃×C₅²	1	28	yes	—
49	C₂×C₃²×C₅	C₂×C₄²×C₅	1	42	yes	—
50	C₂²×C₄×C₅	C₂²×C₃²×C₅	1	42	yes	—
51	C₄²×C₅	C₃²×C₄×C₅	1	168	yes	direct product factor
52	C₃×C₅²	C₃×C₄³	1	4	yes	direct product factor
53	C₂³×C₃²	C₂³×C₅²	1	1	yes	characteristic
54	C₂×C₃²×C₄	C₂×C₄×C₅²	1	42	yes	—
55	C₄³	C₃²×C₅²	1	1	yes	characteristic, direct product factor, Sylow
56	C₂²×C₃×C₅	C₂²×C₃×C₄×C₅	1	168	yes	—
57	C₃×C₄×C₅	C₃×C₄²×C₅	1	672	yes	direct product factor
58	C₂×C₅²	C₂×C₃²×C₄²	1	7	yes	—
59	C₂²×C₃×C₄	C₂²×C₃×C₅²	1	28	yes	—
60	C₃×C₄²	C₃×C₄×C₅²	1	112	yes	direct product factor
61	C₃²×C₅	C₄³×C₅	1	6	yes	direct product factor
62	C₂³×C₅	C₂³×C₃²×C₅	1	6	yes	—
63	C₂×C₄×C₅	C₂×C₃²×C₄×C₅	1	252	yes	—
64	C₂²×C₃²	C₂²×C₄×C₅²	1	7	yes	—
65	C₃²×C₄	C₄²×C₅²	1	28	yes	direct product factor
66	C₂×C₄²	C₂×C₃²×C₅²	1	7	yes	—
67	C₂×C₃×C₅	C₂×C₃×C₄²×C₅	1	168	yes	—
68	C₅²	C₃²×C₄³	1	1	yes	characteristic, direct product factor, Sylow
69	C₂³×C₃	C₂³×C₃×C₅²	1	4	yes	—
70	C₂×C₃×C₄	C₂×C₃×C₄×C₅²	1	168	yes	—
71	C₂²×C₅	C₂²×C₃²×C₄×C₅	1	42	yes	—
72	C₄×C₅	C₃²×C₄²×C₅	1	168	yes	direct product factor
73	C₂×C₃²	C₂×C₄²×C₅²	1	7	yes	—
74	C₂²×C₄	C₂²×C₃²×C₅²	1	7	yes	—
75	C₄²	C₃²×C₄×C₅²	1	28	yes	direct product factor
76	C₃×C₅	C₃×C₄³×C₅	1	24	yes	direct product factor
77	C₂²×C₃	C₂²×C₃×C₄×C₅²	1	28	yes	—
78	C₃×C₄	C₃×C₄²×C₅²	1	112	yes	direct product factor
79	C₂×C₅	C₂×C₃²×C₄²×C₅	1	42	yes	—
80	C₃²	C₄³×C₅²	1	1	yes	characteristic, direct product factor, Sylow
81	C₂³	C₂³×C₃²×C₅²	1	1	yes	characteristic
82	C₂×C₄	C₂×C₃²×C₄×C₅²	1	42	yes	—
83	C₂×C₃	C₂×C₃×C₄²×C₅²	1	28	yes	—
84	C₅	C₃²×C₄³×C₅	1	6	yes	direct product factor
85	C₂²	C₂²×C₃²×C₄×C₅²	1	7	yes	—
86	C₄	C₃²×C₄²×C₅²	1	28	yes	direct product factor
87	C₃	C₃×C₄³×C₅²	1	4	yes	direct product factor
88	C₂	C₂×C₃²×C₄²×C₅²	1	7	yes	—

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂³	C₂³×C₃²×C₅²
socle, prime radical	C₂³×C₃²×C₅²	C₂³
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃×C₄³×C₅²	C₃	direct (N × H)
4. C₃²×C₄²×C₅²	C₄	direct (N × H)
5. C₃²×C₄³×C₅	C₅	direct (N × H)
9. C₄³×C₅²	C₃²	direct (N × H)
12. C₃×C₄²×C₅²	C₃×C₄	direct (N × H)
13. C₃×C₄³×C₅	C₃×C₅	direct (N × H)
15. C₃²×C₄×C₅²	C₄²	direct (N × H)
18. C₃²×C₄²×C₅	C₄×C₅	direct (N × H)
21. C₃²×C₄³	C₅²	direct (N × H)
25. C₄²×C₅²	C₃²×C₄	direct (N × H)
28. C₄³×C₅	C₃²×C₅	direct (N × H)
30. C₃×C₄×C₅²	C₃×C₄²	direct (N × H)
33. C₃×C₄²×C₅	C₃×C₄×C₅	direct (N × H)
34. C₃²×C₅²	C₄³	direct (N × H)
37. C₃×C₄³	C₃×C₅²	direct (N × H)
39. C₃²×C₄×C₅	C₄²×C₅	direct (N × H)
43. C₃²×C₄²	C₄×C₅²	direct (N × H)
47. C₄×C₅²	C₃²×C₄²	direct (N × H)
51. C₄²×C₅	C₃²×C₄×C₅	direct (N × H)
52. C₃×C₅²	C₃×C₄³	direct (N × H)
55. C₄³	C₃²×C₅²	direct (N × H)
57. C₃×C₄×C₅	C₃×C₄²×C₅	direct (N × H)
60. C₃×C₄²	C₃×C₄×C₅²	direct (N × H)
61. C₃²×C₅	C₄³×C₅	direct (N × H)
65. C₃²×C₄	C₄²×C₅²	direct (N × H)
68. C₅²	C₃²×C₄³	direct (N × H)
72. C₄×C₅	C₃²×C₄²×C₅	direct (N × H)
75. C₄²	C₃²×C₄×C₅²	direct (N × H)
76. C₃×C₅	C₃×C₄³×C₅	direct (N × H)
78. C₃×C₄	C₃×C₄²×C₅²	direct (N × H)
80. C₃²	C₄³×C₅²	direct (N × H)
84. C₅	C₃²×C₄³×C₅	direct (N × H)
86. C₄	C₃²×C₄²×C₅²	direct (N × H)
87. C₃	C₃×C₄³×C₅²	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×C₄²×C₅²	2
C₃×C₄³×C₅²	3
C₃²×C₄³×C₅	5

Subgroups — C32×C43×C52 #72192569

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×C₄³×C₅² #72192569