Subgroups — C₃²×C₅²×C₈² #72192557

Order 14400.

1776 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₄×C₅²×C₈	C₂	1	3	yes	maximal
2	C₃×C₅²×C₈²	C₃	1	4	yes	maximal, direct product factor
3	C₃²×C₄²×C₅²	C₂²	1	1	yes	characteristic
4	C₂×C₃²×C₅²×C₈	C₄	1	6	yes	—
5	C₃²×C₅×C₈²	C₅	1	6	yes	maximal, direct product factor
6	C₃×C₄×C₅²×C₈	C₂×C₃	1	12	yes	—
7	C₂×C₃²×C₄×C₅²	C₂×C₄	1	3	yes	—
8	C₃²×C₅²×C₈	C₈	1	12	yes	direct product factor
9	C₅²×C₈²	C₃²	1	1	yes	characteristic, direct product factor, Hall
10	C₃²×C₄×C₅×C₈	C₂×C₅	1	18	yes	—
11	C₂×C₃×C₅²×C₈	C₃×C₄	1	24	yes	—
12	C₃×C₄²×C₅²	C₂²×C₃	1	4	yes	—
13	C₃×C₅×C₈²	C₃×C₅	1	24	yes	direct product factor
14	C₂²×C₃²×C₅²	C₄²	1	1	yes	characteristic
15	C₃²×C₄×C₅²	C₂×C₈	1	6	yes	—
16	C₄×C₅²×C₈	C₂×C₃²	1	3	yes	—
17	C₂×C₃²×C₅×C₈	C₄×C₅	1	36	yes	—
18	C₃²×C₄²×C₅	C₂²×C₅	1	6	yes	—
19	C₂×C₃×C₄×C₅²	C₂×C₃×C₄	1	12	yes	—
20	C₃×C₅²×C₈	C₃×C₈	1	48	yes	direct product factor
21	C₃²×C₈²	C₅²	1	1	yes	characteristic, direct product factor, Hall
22	C₃×C₄×C₅×C₈	C₂×C₃×C₅	1	72	yes	—
23	C₂×C₃²×C₅²	C₄×C₈	1	3	yes	—
24	C₂×C₅²×C₈	C₃²×C₄	1	6	yes	—
25	C₄²×C₅²	C₂²×C₃²	1	1	yes	characteristic
26	C₂×C₃²×C₄×C₅	C₂×C₄×C₅	1	18	yes	—
27	C₃²×C₅×C₈	C₅×C₈	1	72	yes	direct product factor
28	C₅×C₈²	C₃²×C₅	1	6	yes	direct product factor
29	C₂²×C₃×C₅²	C₃×C₄²	1	4	yes	—
30	C₃×C₄×C₅²	C₂×C₃×C₈	1	24	yes	—
31	C₃²×C₄×C₈	C₂×C₅²	1	3	yes	—
32	C₂×C₃×C₅×C₈	C₃×C₄×C₅	1	144	yes	—
33	C₃×C₄²×C₅	C₂²×C₃×C₅	1	24	yes	—
34	C₃²×C₅²	C₈²	1	1	yes	characteristic, direct product factor, Hall
35	C₂×C₄×C₅²	C₂×C₃²×C₄	1	3	yes	—
36	C₅²×C₈	C₃²×C₈	1	12	yes	direct product factor
37	C₃×C₈²	C₃×C₅²	1	4	yes	direct product factor
38	C₂²×C₃²×C₅	C₄²×C₅	1	6	yes	—
39	C₃²×C₄×C₅	C₂×C₅×C₈	1	36	yes	—
40	C₄×C₅×C₈	C₂×C₃²×C₅	1	18	yes	—
41	C₂×C₃×C₅²	C₃×C₄×C₈	1	12	yes	—
42	C₂×C₃²×C₈	C₄×C₅²	1	6	yes	—
43	C₃²×C₄²	C₂²×C₅²	1	1	yes	characteristic
44	C₂×C₃×C₄×C₅	C₂×C₃×C₄×C₅	1	72	yes	—
45	C₃×C₅×C₈	C₃×C₅×C₈	1	288	yes	direct product factor
46	C₂²×C₅²	C₃²×C₄²	1	1	yes	characteristic
47	C₄×C₅²	C₂×C₃²×C₈	1	6	yes	—
48	C₃×C₄×C₈	C₂×C₃×C₅²	1	12	yes	—
49	C₂×C₃²×C₅	C₄×C₅×C₈	1	18	yes	—
50	C₂×C₅×C₈	C₃²×C₄×C₅	1	36	yes	—
51	C₄²×C₅	C₂²×C₃²×C₅	1	6	yes	—
52	C₃×C₅²	C₃×C₈²	1	4	yes	direct product factor
53	C₂×C₃²×C₄	C₂×C₄×C₅²	1	3	yes	—
54	C₃²×C₈	C₅²×C₈	1	12	yes	direct product factor
55	C₈²	C₃²×C₅²	1	1	yes	characteristic, direct product factor, Sylow
56	C₂²×C₃×C₅	C₃×C₄²×C₅	1	24	yes	—
57	C₃×C₄×C₅	C₂×C₃×C₅×C₈	1	144	yes	—
58	C₂×C₅²	C₃²×C₄×C₈	1	3	yes	—
59	C₂×C₃×C₈	C₃×C₄×C₅²	1	24	yes	—
60	C₃×C₄²	C₂²×C₃×C₅²	1	4	yes	—
61	C₃²×C₅	C₅×C₈²	1	6	yes	direct product factor
62	C₂×C₄×C₅	C₂×C₃²×C₄×C₅	1	18	yes	—
63	C₅×C₈	C₃²×C₅×C₈	1	72	yes	direct product factor
64	C₂²×C₃²	C₄²×C₅²	1	1	yes	characteristic
65	C₃²×C₄	C₂×C₅²×C₈	1	6	yes	—
66	C₄×C₈	C₂×C₃²×C₅²	1	3	yes	—
67	C₂×C₃×C₅	C₃×C₄×C₅×C₈	1	72	yes	—
68	C₅²	C₃²×C₈²	1	1	yes	characteristic, direct product factor, Sylow
69	C₂×C₃×C₄	C₂×C₃×C₄×C₅²	1	12	yes	—
70	C₃×C₈	C₃×C₅²×C₈	1	48	yes	direct product factor
71	C₂²×C₅	C₃²×C₄²×C₅	1	6	yes	—
72	C₄×C₅	C₂×C₃²×C₅×C₈	1	36	yes	—
73	C₂×C₃²	C₄×C₅²×C₈	1	3	yes	—
74	C₂×C₈	C₃²×C₄×C₅²	1	6	yes	—
75	C₄²	C₂²×C₃²×C₅²	1	1	yes	characteristic
76	C₃×C₅	C₃×C₅×C₈²	1	24	yes	direct product factor
77	C₂²×C₃	C₃×C₄²×C₅²	1	4	yes	—
78	C₃×C₄	C₂×C₃×C₅²×C₈	1	24	yes	—
79	C₂×C₅	C₃²×C₄×C₅×C₈	1	18	yes	—
80	C₃²	C₅²×C₈²	1	1	yes	characteristic, direct product factor, Sylow
81	C₂×C₄	C₂×C₃²×C₄×C₅²	1	3	yes	—
82	C₈	C₃²×C₅²×C₈	1	12	yes	direct product factor
83	C₂×C₃	C₃×C₄×C₅²×C₈	1	12	yes	—
84	C₅	C₃²×C₅×C₈²	1	6	yes	direct product factor
85	C₂²	C₃²×C₄²×C₅²	1	1	yes	characteristic
86	C₄	C₂×C₃²×C₅²×C₈	1	6	yes	—
87	C₃	C₃×C₅²×C₈²	1	4	yes	direct product factor
88	C₂	C₃²×C₄×C₅²×C₈	1	3	yes	—

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₄²	C₂²×C₃²×C₅²
socle, prime radical	C₂²×C₃²×C₅²	C₄²
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃×C₅²×C₈²	C₃	direct (N × H)
5. C₃²×C₅×C₈²	C₅	direct (N × H)
8. C₃²×C₅²×C₈	C₈	direct (N × H)
9. C₅²×C₈²	C₃²	direct (N × H)
13. C₃×C₅×C₈²	C₃×C₅	direct (N × H)
20. C₃×C₅²×C₈	C₃×C₈	direct (N × H)
21. C₃²×C₈²	C₅²	direct (N × H)
27. C₃²×C₅×C₈	C₅×C₈	direct (N × H)
28. C₅×C₈²	C₃²×C₅	direct (N × H)
34. C₃²×C₅²	C₈²	direct (N × H)
36. C₅²×C₈	C₃²×C₈	direct (N × H)
37. C₃×C₈²	C₃×C₅²	direct (N × H)
45. C₃×C₅×C₈	C₃×C₅×C₈	direct (N × H)
52. C₃×C₅²	C₃×C₈²	direct (N × H)
54. C₃²×C₈	C₅²×C₈	direct (N × H)
55. C₈²	C₃²×C₅²	direct (N × H)
61. C₃²×C₅	C₅×C₈²	direct (N × H)
63. C₅×C₈	C₃²×C₅×C₈	direct (N × H)
68. C₅²	C₃²×C₈²	direct (N × H)
70. C₃×C₈	C₃×C₅²×C₈	direct (N × H)
76. C₃×C₅	C₃×C₅×C₈²	direct (N × H)
80. C₃²	C₅²×C₈²	direct (N × H)
82. C₈	C₃²×C₅²×C₈	direct (N × H)
84. C₅	C₃²×C₅×C₈²	direct (N × H)
87. C₃	C₃×C₅²×C₈²	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×C₄×C₅²×C₈	2
C₃×C₅²×C₈²	3
C₃²×C₅×C₈²	5

Subgroups — C32×C52×C82 #72192557

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×C₅²×C₈² #72192557