Subgroups — C₄×C₅×C₈×C₁₆₉ #72204101

Order 27040.

132 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄²×C₅×C₁₆₉	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×C₈×C₁₆₉	C₂	1	2	yes	maximal
3	C₂×C₄×C₅×C₁₆₉	C₄	1	2	yes	—
4	C₂×C₄×C₅×C₁₆₉	C₂²	1	1	yes	characteristic
5	C₅×C₈×C₁₆₉	C₄	1	4	yes	direct product factor
6	C₄×C₈×C₁₆₉	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₄²×C₁₆₉	C₂×C₅	1	1	yes	characteristic
8	C₂×C₈×C₁₆₉	C₂×C₅	1	2	yes	—
9	C₄×C₅×C₈×C₁₃	C₁₃	1	1	yes	maximal, characteristic
10	C₂²×C₅×C₁₆₉	C₂×C₄	1	1	yes	characteristic
11	C₄×C₅×C₁₆₉	C₈	1	4	yes	direct product factor
12	C₄×C₅×C₁₆₉	C₂×C₄	1	2	yes	—
13	C₂×C₅×C₁₆₉	C₄²	1	1	yes	characteristic
14	C₂×C₅×C₁₆₉	C₂×C₈	1	2	yes	—
15	C₂×C₄×C₁₆₉	C₄×C₅	1	2	yes	—
16	C₂×C₄×C₁₆₉	C₂²×C₅	1	1	yes	characteristic
17	C₈×C₁₆₉	C₄×C₅	1	4	yes	direct product factor
18	C₂×C₅×C₈×C₁₃	C₂×C₁₃	1	2	yes	—
19	C₄²×C₅×C₁₃	C₂×C₁₃	1	1	yes	characteristic
20	C₅×C₁₆₉	C₄×C₈	1	1	yes	characteristic, direct product factor, Hall
21	C₂²×C₁₆₉	C₂×C₄×C₅	1	1	yes	characteristic
22	C₄×C₁₆₉	C₅×C₈	1	4	yes	direct product factor
23	C₄×C₁₆₉	C₂×C₄×C₅	1	2	yes	—
24	C₂×C₄×C₅×C₁₃	C₄×C₁₃	1	2	yes	—
25	C₂×C₄×C₅×C₁₃	C₂²×C₁₃	1	1	yes	characteristic
26	C₅×C₈×C₁₃	C₄×C₁₃	1	4	yes	—
27	C₄×C₈×C₁₃	C₅×C₁₃	1	1	yes	characteristic
28	C₂×C₁₆₉	C₄²×C₅	1	1	yes	characteristic
29	C₂×C₁₆₉	C₂×C₅×C₈	1	2	yes	—
30	C₂²×C₅×C₁₃	C₂×C₄×C₁₃	1	1	yes	characteristic
31	C₄×C₅×C₁₃	C₈×C₁₃	1	4	yes	—
32	C₄×C₅×C₁₃	C₂×C₄×C₁₃	1	2	yes	—
33	C₂×C₈×C₁₃	C₂×C₅×C₁₃	1	2	yes	—
34	C₄²×C₁₃	C₂×C₅×C₁₃	1	1	yes	characteristic
35	C₁₆₉	C₄×C₅×C₈	1	1	yes	characteristic, direct product factor, Sylow
36	C₄×C₅×C₈	C₁₆₉	1	1	yes	characteristic, direct product factor, Hall
37	C₂×C₅×C₁₃	C₄²×C₁₃	1	1	yes	characteristic
38	C₂×C₅×C₁₃	C₂×C₈×C₁₃	1	2	yes	—
39	C₂×C₄×C₁₃	C₄×C₅×C₁₃	1	2	yes	—
40	C₂×C₄×C₁₃	C₂²×C₅×C₁₃	1	1	yes	characteristic
41	C₈×C₁₃	C₄×C₅×C₁₃	1	4	yes	—
42	C₂×C₅×C₈	C₂×C₁₆₉	1	2	yes	—
43	C₄²×C₅	C₂×C₁₆₉	1	1	yes	characteristic
44	C₅×C₁₃	C₄×C₈×C₁₃	1	1	yes	characteristic
45	C₂²×C₁₃	C₂×C₄×C₅×C₁₃	1	1	yes	characteristic
46	C₄×C₁₃	C₅×C₈×C₁₃	1	4	yes	—
47	C₄×C₁₃	C₂×C₄×C₅×C₁₃	1	2	yes	—
48	C₂×C₄×C₅	C₄×C₁₆₉	1	2	yes	—
49	C₂×C₄×C₅	C₂²×C₁₆₉	1	1	yes	characteristic
50	C₅×C₈	C₄×C₁₆₉	1	4	yes	direct product factor
51	C₄×C₈	C₅×C₁₆₉	1	1	yes	characteristic, direct product factor, Sylow
52	C₂×C₁₃	C₄²×C₅×C₁₃	1	1	yes	characteristic
53	C₂×C₁₃	C₂×C₅×C₈×C₁₃	1	2	yes	—
54	C₂²×C₅	C₂×C₄×C₁₆₉	1	1	yes	characteristic
55	C₄×C₅	C₈×C₁₆₉	1	4	yes	direct product factor
56	C₄×C₅	C₂×C₄×C₁₆₉	1	2	yes	—
57	C₂×C₈	C₂×C₅×C₁₆₉	1	2	yes	—
58	C₄²	C₂×C₅×C₁₆₉	1	1	yes	characteristic
59	C₁₃	C₄×C₅×C₈×C₁₃	1	1	yes	characteristic
60	C₂×C₅	C₂×C₈×C₁₆₉	1	2	yes	—
61	C₂×C₅	C₄²×C₁₆₉	1	1	yes	characteristic
62	C₂×C₄	C₄×C₅×C₁₆₉	1	2	yes	—
63	C₂×C₄	C₂²×C₅×C₁₆₉	1	1	yes	characteristic
64	C₈	C₄×C₅×C₁₆₉	1	4	yes	direct product factor
65	C₅	C₄×C₈×C₁₆₉	1	1	yes	characteristic, direct product factor, Sylow
66	C₂²	C₂×C₄×C₅×C₁₆₉	1	1	yes	characteristic
67	C₄	C₅×C₈×C₁₆₉	1	4	yes	direct product factor
68	C₄	C₂×C₄×C₅×C₁₆₉	1	2	yes	—
69	C₂	C₂×C₅×C₈×C₁₆₉	1	2	yes	—
70	C₂	C₄²×C₅×C₁₆₉	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂×C₄×C₁₃	C₂²×C₅×C₁₃
socle, prime radical	C₂²×C₅×C₁₃	C₂×C₄×C₁₃
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
5. C₅×C₈×C₁₆₉	C₄	direct (N × H)
6. C₄×C₈×C₁₆₉	C₅	direct (N × H)
11. C₄×C₅×C₁₆₉	C₈	direct (N × H)
17. C₈×C₁₆₉	C₄×C₅	direct (N × H)
20. C₅×C₁₆₉	C₄×C₈	direct (N × H)
22. C₄×C₁₆₉	C₅×C₈	direct (N × H)
35. C₁₆₉	C₄×C₅×C₈	direct (N × H)
36. C₄×C₅×C₈	C₁₆₉	direct (N × H)
50. C₅×C₈	C₄×C₁₆₉	direct (N × H)
51. C₄×C₈	C₅×C₁₆₉	direct (N × H)
55. C₄×C₅	C₈×C₁₆₉	direct (N × H)
64. C₈	C₄×C₅×C₁₆₉	direct (N × H)
65. C₅	C₄×C₈×C₁₆₉	direct (N × H)
67. C₄	C₅×C₈×C₁₆₉	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄²×C₅×C₁₆₉	2
C₂×C₅×C₈×C₁₆₉	2
C₄×C₈×C₁₆₉	5
C₄×C₅×C₈×C₁₃	13

Subgroups — C4×C5×C8×C169 #72204101

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₅×C₈×C₁₆₉ #72204101