Subgroups — C₄×C₇×C₈×C₉ #72182000

Order 2016.

132 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×C₈×C₉	C₂	1	2	yes	maximal
2	C₄²×C₇×C₉	C₂	1	1	yes	maximal, characteristic
3	C₃×C₄×C₇×C₈	C₃	1	1	yes	maximal, characteristic
4	C₂×C₄×C₇×C₉	C₄	1	2	yes	—
5	C₂×C₄×C₇×C₉	C₂²	1	1	yes	characteristic
6	C₇×C₈×C₉	C₄	1	4	yes	direct product factor
7	C₂×C₃×C₇×C₈	C₂×C₃	1	2	yes	—
8	C₃×C₄²×C₇	C₂×C₃	1	1	yes	characteristic
9	C₄×C₈×C₉	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
10	C₂²×C₇×C₉	C₂×C₄	1	1	yes	characteristic
11	C₄×C₇×C₉	C₈	1	4	yes	direct product factor
12	C₄×C₇×C₉	C₂×C₄	1	2	yes	—
13	C₄×C₇×C₈	C₉	1	1	yes	characteristic, direct product factor, Hall
14	C₂×C₃×C₄×C₇	C₃×C₄	1	2	yes	—
15	C₂×C₃×C₄×C₇	C₂²×C₃	1	1	yes	characteristic
16	C₃×C₇×C₈	C₃×C₄	1	4	yes	—
17	C₂×C₈×C₉	C₂×C₇	1	2	yes	—
18	C₄²×C₉	C₂×C₇	1	1	yes	characteristic
19	C₂×C₇×C₉	C₄²	1	1	yes	characteristic
20	C₂×C₇×C₉	C₂×C₈	1	2	yes	—
21	C₂×C₇×C₈	C₂×C₉	1	2	yes	—
22	C₄²×C₇	C₂×C₉	1	1	yes	characteristic
23	C₃×C₄×C₈	C₃×C₇	1	1	yes	characteristic
24	C₂²×C₃×C₇	C₂×C₃×C₄	1	1	yes	characteristic
25	C₃×C₄×C₇	C₃×C₈	1	4	yes	—
26	C₃×C₄×C₇	C₂×C₃×C₄	1	2	yes	—
27	C₂×C₄×C₉	C₄×C₇	1	2	yes	—
28	C₂×C₄×C₉	C₂²×C₇	1	1	yes	characteristic
29	C₈×C₉	C₄×C₇	1	4	yes	direct product factor
30	C₇×C₉	C₄×C₈	1	1	yes	characteristic, direct product factor, Hall
31	C₂×C₄×C₇	C₄×C₉	1	2	yes	—
32	C₂×C₄×C₇	C₂²×C₉	1	1	yes	characteristic
33	C₇×C₈	C₄×C₉	1	4	yes	direct product factor
34	C₂×C₃×C₈	C₂×C₃×C₇	1	2	yes	—
35	C₃×C₄²	C₂×C₃×C₇	1	1	yes	characteristic
36	C₂×C₃×C₇	C₃×C₄²	1	1	yes	characteristic
37	C₂×C₃×C₇	C₂×C₃×C₈	1	2	yes	—
38	C₂²×C₉	C₂×C₄×C₇	1	1	yes	characteristic
39	C₄×C₉	C₇×C₈	1	4	yes	direct product factor
40	C₄×C₉	C₂×C₄×C₇	1	2	yes	—
41	C₄×C₈	C₇×C₉	1	1	yes	characteristic, direct product factor, Sylow
42	C₂²×C₇	C₂×C₄×C₉	1	1	yes	characteristic
43	C₄×C₇	C₈×C₉	1	4	yes	direct product factor
44	C₄×C₇	C₂×C₄×C₉	1	2	yes	—
45	C₂×C₃×C₄	C₃×C₄×C₇	1	2	yes	—
46	C₂×C₃×C₄	C₂²×C₃×C₇	1	1	yes	characteristic
47	C₃×C₈	C₃×C₄×C₇	1	4	yes	—
48	C₃×C₇	C₃×C₄×C₈	1	1	yes	characteristic
49	C₂×C₉	C₄²×C₇	1	1	yes	characteristic
50	C₂×C₉	C₂×C₇×C₈	1	2	yes	—
51	C₂×C₈	C₂×C₇×C₉	1	2	yes	—
52	C₄²	C₂×C₇×C₉	1	1	yes	characteristic
53	C₂×C₇	C₄²×C₉	1	1	yes	characteristic
54	C₂×C₇	C₂×C₈×C₉	1	2	yes	—
55	C₂²×C₃	C₂×C₃×C₄×C₇	1	1	yes	characteristic
56	C₃×C₄	C₃×C₇×C₈	1	4	yes	—
57	C₃×C₄	C₂×C₃×C₄×C₇	1	2	yes	—
58	C₉	C₄×C₇×C₈	1	1	yes	characteristic, direct product factor, Sylow
59	C₂×C₄	C₄×C₇×C₉	1	2	yes	—
60	C₂×C₄	C₂²×C₇×C₉	1	1	yes	characteristic
61	C₈	C₄×C₇×C₉	1	4	yes	direct product factor
62	C₇	C₄×C₈×C₉	1	1	yes	characteristic, direct product factor, Sylow
63	C₂×C₃	C₃×C₄²×C₇	1	1	yes	characteristic
64	C₂×C₃	C₂×C₃×C₇×C₈	1	2	yes	—
65	C₂²	C₂×C₄×C₇×C₉	1	1	yes	characteristic
66	C₄	C₇×C₈×C₉	1	4	yes	direct product factor
67	C₄	C₂×C₄×C₇×C₉	1	2	yes	—
68	C₃	C₃×C₄×C₇×C₈	1	1	yes	characteristic
69	C₂	C₄²×C₇×C₉	1	1	yes	characteristic
70	C₂	C₂×C₇×C₈×C₉	1	2	yes	—

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂×C₃×C₄	C₂²×C₃×C₇
socle, prime radical	C₂²×C₃×C₇	C₂×C₃×C₄
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
6. C₇×C₈×C₉	C₄	direct (N × H)
9. C₄×C₈×C₉	C₇	direct (N × H)
11. C₄×C₇×C₉	C₈	direct (N × H)
13. C₄×C₇×C₈	C₉	direct (N × H)
29. C₈×C₉	C₄×C₇	direct (N × H)
30. C₇×C₉	C₄×C₈	direct (N × H)
33. C₇×C₈	C₄×C₉	direct (N × H)
39. C₄×C₉	C₇×C₈	direct (N × H)
41. C₄×C₈	C₇×C₉	direct (N × H)
43. C₄×C₇	C₈×C₉	direct (N × H)
58. C₉	C₄×C₇×C₈	direct (N × H)
61. C₈	C₄×C₇×C₉	direct (N × H)
62. C₇	C₄×C₈×C₉	direct (N × H)
66. C₄	C₇×C₈×C₉	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×C₇×C₈×C₉	2
C₄²×C₇×C₉	2
C₃×C₄×C₇×C₈	3
C₄×C₈×C₉	7

Subgroups — C4×C7×C8×C9 #72182000

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₇×C₈×C₉ #72182000