Subgroups — C₃×C₅×C₈³ #72186605

Order 7680.

3208 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄×C₅×C₈²	C₂	1	7	yes	maximal
2	C₅×C₈³	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
3	C₃×C₈³	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
4	C₃×C₄²×C₅×C₈	C₂²	1	7	yes	—
5	C₂×C₃×C₅×C₈²	C₄	1	28	yes	—
6	C₄×C₅×C₈²	C₂×C₃	1	7	yes	—
7	C₂×C₃×C₄×C₅×C₈	C₂×C₄	1	42	yes	—
8	C₃×C₄³×C₅	C₂³	1	1	yes	characteristic
9	C₃×C₅×C₈²	C₈	1	112	yes	direct product factor
10	C₃×C₄×C₈²	C₂×C₅	1	7	yes	—
11	C₄²×C₅×C₈	C₂²×C₃	1	7	yes	—
12	C₂×C₅×C₈²	C₃×C₄	1	28	yes	—
13	C₈³	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
14	C₂²×C₃×C₅×C₈	C₄²	1	28	yes	—
15	C₂×C₃×C₄²×C₅	C₂²×C₄	1	7	yes	—
16	C₃×C₄×C₅×C₈	C₂×C₈	1	168	yes	—
17	C₃×C₄²×C₈	C₂²×C₅	1	7	yes	—
18	C₂×C₃×C₈²	C₄×C₅	1	28	yes	—
19	C₂×C₄×C₅×C₈	C₂×C₃×C₄	1	42	yes	—
20	C₄³×C₅	C₂³×C₃	1	1	yes	characteristic
21	C₅×C₈²	C₃×C₈	1	112	yes	direct product factor
22	C₄×C₈²	C₂×C₃×C₅	1	7	yes	—
23	C₂²×C₃×C₄×C₅	C₂×C₄²	1	7	yes	—
24	C₂×C₃×C₅×C₈	C₄×C₈	1	168	yes	—
25	C₃×C₄²×C₅	C₂²×C₈	1	28	yes	—
26	C₂×C₃×C₄×C₈	C₂×C₄×C₅	1	42	yes	—
27	C₃×C₄³	C₂³×C₅	1	1	yes	characteristic
28	C₃×C₈²	C₅×C₈	1	112	yes	direct product factor
29	C₂²×C₅×C₈	C₃×C₄²	1	28	yes	—
30	C₂×C₄²×C₅	C₂²×C₃×C₄	1	7	yes	—
31	C₄×C₅×C₈	C₂×C₃×C₈	1	168	yes	—
32	C₄²×C₈	C₂²×C₃×C₅	1	7	yes	—
33	C₂×C₈²	C₃×C₄×C₅	1	28	yes	—
34	C₂³×C₃×C₅	C₄³	1	1	yes	characteristic
35	C₂×C₃×C₄×C₅	C₂×C₄×C₈	1	42	yes	—
36	C₃×C₅×C₈	C₈²	1	112	yes	direct product factor
37	C₂²×C₃×C₈	C₄²×C₅	1	28	yes	—
38	C₂×C₃×C₄²	C₂²×C₄×C₅	1	7	yes	—
39	C₃×C₄×C₈	C₂×C₅×C₈	1	168	yes	—
40	C₂²×C₄×C₅	C₂×C₃×C₄²	1	7	yes	—
41	C₂×C₅×C₈	C₃×C₄×C₈	1	168	yes	—
42	C₄²×C₅	C₂²×C₃×C₈	1	28	yes	—
43	C₂×C₄×C₈	C₂×C₃×C₄×C₅	1	42	yes	—
44	C₄³	C₂³×C₃×C₅	1	1	yes	characteristic
45	C₈²	C₃×C₅×C₈	1	112	yes	direct product factor
46	C₂²×C₃×C₅	C₄²×C₈	1	7	yes	—
47	C₃×C₄×C₅	C₂×C₈²	1	28	yes	—
48	C₂²×C₃×C₄	C₂×C₄²×C₅	1	7	yes	—
49	C₂×C₃×C₈	C₄×C₅×C₈	1	168	yes	—
50	C₃×C₄²	C₂²×C₅×C₈	1	28	yes	—
51	C₂³×C₅	C₃×C₄³	1	1	yes	characteristic
52	C₂×C₄×C₅	C₂×C₃×C₄×C₈	1	42	yes	—
53	C₅×C₈	C₃×C₈²	1	112	yes	direct product factor
54	C₂²×C₈	C₃×C₄²×C₅	1	28	yes	—
55	C₂×C₄²	C₂²×C₃×C₄×C₅	1	7	yes	—
56	C₄×C₈	C₂×C₃×C₅×C₈	1	168	yes	—
57	C₂×C₃×C₅	C₄×C₈²	1	7	yes	—
58	C₂³×C₃	C₄³×C₅	1	1	yes	characteristic
59	C₂×C₃×C₄	C₂×C₄×C₅×C₈	1	42	yes	—
60	C₃×C₈	C₅×C₈²	1	112	yes	direct product factor
61	C₂²×C₅	C₃×C₄²×C₈	1	7	yes	—
62	C₄×C₅	C₂×C₃×C₈²	1	28	yes	—
63	C₂²×C₄	C₂×C₃×C₄²×C₅	1	7	yes	—
64	C₂×C₈	C₃×C₄×C₅×C₈	1	168	yes	—
65	C₄²	C₂²×C₃×C₅×C₈	1	28	yes	—
66	C₃×C₅	C₈³	1	1	yes	characteristic, direct product factor, Hall
67	C₂²×C₃	C₄²×C₅×C₈	1	7	yes	—
68	C₃×C₄	C₂×C₅×C₈²	1	28	yes	—
69	C₂×C₅	C₃×C₄×C₈²	1	7	yes	—
70	C₂³	C₃×C₄³×C₅	1	1	yes	characteristic
71	C₂×C₄	C₂×C₃×C₄×C₅×C₈	1	42	yes	—
72	C₈	C₃×C₅×C₈²	1	112	yes	direct product factor
73	C₂×C₃	C₄×C₅×C₈²	1	7	yes	—
74	C₅	C₃×C₈³	1	1	yes	characteristic, direct product factor, Sylow
75	C₂²	C₃×C₄²×C₅×C₈	1	7	yes	—
76	C₄	C₂×C₃×C₅×C₈²	1	28	yes	—
77	C₃	C₅×C₈³	1	1	yes	characteristic, direct product factor, Sylow
78	C₂	C₃×C₄×C₅×C₈²	1	7	yes	—

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₄³	C₂³×C₃×C₅
socle, prime radical	C₂³×C₃×C₅	C₄³
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅×C₈³	C₃	direct (N × H)
3. C₃×C₈³	C₅	direct (N × H)
9. C₃×C₅×C₈²	C₈	direct (N × H)
13. C₈³	C₃×C₅	direct (N × H)
21. C₅×C₈²	C₃×C₈	direct (N × H)
28. C₃×C₈²	C₅×C₈	direct (N × H)
36. C₃×C₅×C₈	C₈²	direct (N × H)
45. C₈²	C₃×C₅×C₈	direct (N × H)
53. C₅×C₈	C₃×C₈²	direct (N × H)
60. C₃×C₈	C₅×C₈²	direct (N × H)
66. C₃×C₅	C₈³	direct (N × H)
72. C₈	C₃×C₅×C₈²	direct (N × H)
74. C₅	C₃×C₈³	direct (N × H)
77. C₃	C₅×C₈³	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×C₄×C₅×C₈²	2
C₅×C₈³	3
C₃×C₈³	5

Subgroups — C3×C5×C83 #72186605

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×C₈³ #72186605