Subgroups — C₃×C₈×C₉×C₁₂₁ #72203296

Order 26136.

120 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄×C₉×C₁₂₁	C₂	1	1	yes	maximal, characteristic
2	C₃²×C₈×C₁₂₁	C₃	1	1	yes	maximal, characteristic
3	C₈×C₉×C₁₂₁	C₃	1	3	yes	maximal, direct product factor
4	C₂×C₃×C₉×C₁₂₁	C₄	1	1	yes	characteristic
5	C₃×C₉×C₁₂₁	C₈	1	1	yes	characteristic, direct product factor, Hall
6	C₃×C₈×C₁₂₁	C₉	1	3	yes	direct product factor
7	C₃×C₈×C₁₂₁	C₃²	1	1	yes	characteristic
8	C₃×C₈×C₉×C₁₁	C₁₁	1	1	yes	maximal, characteristic
9	C₃²×C₄×C₁₂₁	C₂×C₃	1	1	yes	characteristic
10	C₄×C₉×C₁₂₁	C₂×C₃	1	3	yes	—
11	C₂×C₃²×C₁₂₁	C₃×C₄	1	1	yes	characteristic
12	C₂×C₉×C₁₂₁	C₃×C₄	1	3	yes	—
13	C₃×C₄×C₁₂₁	C₂×C₉	1	3	yes	—
14	C₃×C₄×C₁₂₁	C₂×C₃²	1	1	yes	characteristic
15	C₃×C₄×C₉×C₁₁	C₂×C₁₁	1	1	yes	characteristic
16	C₃²×C₁₂₁	C₃×C₈	1	1	yes	characteristic
17	C₉×C₁₂₁	C₃×C₈	1	3	yes	direct product factor
18	C₈×C₁₂₁	C₃×C₉	1	1	yes	characteristic, direct product factor, Hall
19	C₃²×C₈×C₁₁	C₃×C₁₁	1	1	yes	characteristic
20	C₈×C₉×C₁₁	C₃×C₁₁	1	3	yes	—
21	C₂×C₃×C₁₂₁	C₄×C₉	1	3	yes	—
22	C₂×C₃×C₁₂₁	C₃²×C₄	1	1	yes	characteristic
23	C₂×C₃×C₉×C₁₁	C₄×C₁₁	1	1	yes	characteristic
24	C₄×C₁₂₁	C₂×C₃×C₉	1	1	yes	characteristic
25	C₃²×C₄×C₁₁	C₂×C₃×C₁₁	1	1	yes	characteristic
26	C₄×C₉×C₁₁	C₂×C₃×C₁₁	1	3	yes	—
27	C₃×C₁₂₁	C₈×C₉	1	3	yes	direct product factor
28	C₃×C₁₂₁	C₃²×C₈	1	1	yes	characteristic
29	C₃×C₉×C₁₁	C₈×C₁₁	1	1	yes	characteristic
30	C₃×C₈×C₁₁	C₉×C₁₁	1	3	yes	—
31	C₃×C₈×C₁₁	C₃²×C₁₁	1	1	yes	characteristic
32	C₂×C₁₂₁	C₃×C₄×C₉	1	1	yes	characteristic
33	C₃×C₈×C₉	C₁₂₁	1	1	yes	characteristic, direct product factor, Hall
34	C₂×C₃²×C₁₁	C₃×C₄×C₁₁	1	1	yes	characteristic
35	C₂×C₉×C₁₁	C₃×C₄×C₁₁	1	3	yes	—
36	C₃×C₄×C₁₁	C₂×C₉×C₁₁	1	3	yes	—
37	C₃×C₄×C₁₁	C₂×C₃²×C₁₁	1	1	yes	characteristic
38	C₁₂₁	C₃×C₈×C₉	1	1	yes	characteristic, direct product factor, Sylow
39	C₃×C₄×C₉	C₂×C₁₂₁	1	1	yes	characteristic
40	C₃²×C₁₁	C₃×C₈×C₁₁	1	1	yes	characteristic
41	C₉×C₁₁	C₃×C₈×C₁₁	1	3	yes	—
42	C₈×C₁₁	C₃×C₉×C₁₁	1	1	yes	characteristic
43	C₃²×C₈	C₃×C₁₂₁	1	1	yes	characteristic
44	C₈×C₉	C₃×C₁₂₁	1	3	yes	direct product factor
45	C₂×C₃×C₁₁	C₄×C₉×C₁₁	1	3	yes	—
46	C₂×C₃×C₁₁	C₃²×C₄×C₁₁	1	1	yes	characteristic
47	C₂×C₃×C₉	C₄×C₁₂₁	1	1	yes	characteristic
48	C₄×C₁₁	C₂×C₃×C₉×C₁₁	1	1	yes	characteristic
49	C₃²×C₄	C₂×C₃×C₁₂₁	1	1	yes	characteristic
50	C₄×C₉	C₂×C₃×C₁₂₁	1	3	yes	—
51	C₃×C₁₁	C₈×C₉×C₁₁	1	3	yes	—
52	C₃×C₁₁	C₃²×C₈×C₁₁	1	1	yes	characteristic
53	C₃×C₉	C₈×C₁₂₁	1	1	yes	characteristic, direct product factor, Sylow
54	C₃×C₈	C₉×C₁₂₁	1	3	yes	direct product factor
55	C₃×C₈	C₃²×C₁₂₁	1	1	yes	characteristic
56	C₂×C₁₁	C₃×C₄×C₉×C₁₁	1	1	yes	characteristic
57	C₂×C₃²	C₃×C₄×C₁₂₁	1	1	yes	characteristic
58	C₂×C₉	C₃×C₄×C₁₂₁	1	3	yes	—
59	C₃×C₄	C₂×C₉×C₁₂₁	1	3	yes	—
60	C₃×C₄	C₂×C₃²×C₁₂₁	1	1	yes	characteristic
61	C₁₁	C₃×C₈×C₉×C₁₁	1	1	yes	characteristic
62	C₃²	C₃×C₈×C₁₂₁	1	1	yes	characteristic
63	C₉	C₃×C₈×C₁₂₁	1	3	yes	direct product factor
64	C₈	C₃×C₉×C₁₂₁	1	1	yes	characteristic, direct product factor, Sylow
65	C₂×C₃	C₄×C₉×C₁₂₁	1	3	yes	—
66	C₂×C₃	C₃²×C₄×C₁₂₁	1	1	yes	characteristic
67	C₄	C₂×C₃×C₉×C₁₂₁	1	1	yes	characteristic
68	C₃	C₈×C₉×C₁₂₁	1	3	yes	direct product factor
69	C₃	C₃²×C₈×C₁₂₁	1	1	yes	characteristic
70	C₂	C₃×C₄×C₉×C₁₂₁	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₃×C₄×C₁₁	C₂×C₃²×C₁₁
socle, prime radical	C₂×C₃²×C₁₁	C₃×C₄×C₁₁
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₈×C₉×C₁₂₁	C₃	direct (N × H)
5. C₃×C₉×C₁₂₁	C₈	direct (N × H)
6. C₃×C₈×C₁₂₁	C₉	direct (N × H)
17. C₉×C₁₂₁	C₃×C₈	direct (N × H)
18. C₈×C₁₂₁	C₃×C₉	direct (N × H)
27. C₃×C₁₂₁	C₈×C₉	direct (N × H)
33. C₃×C₈×C₉	C₁₂₁	direct (N × H)
38. C₁₂₁	C₃×C₈×C₉	direct (N × H)
44. C₈×C₉	C₃×C₁₂₁	direct (N × H)
53. C₃×C₉	C₈×C₁₂₁	direct (N × H)
54. C₃×C₈	C₉×C₁₂₁	direct (N × H)
63. C₉	C₃×C₈×C₁₂₁	direct (N × H)
64. C₈	C₃×C₉×C₁₂₁	direct (N × H)
68. C₃	C₈×C₉×C₁₂₁	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×C₉×C₁₂₁	2
C₃²×C₈×C₁₂₁	3
C₈×C₉×C₁₂₁	3
C₃×C₈×C₉×C₁₁	11

Subgroups — C3×C8×C9×C121 #72203296

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₈×C₉×C₁₂₁ #72203296