Subgroups — C₂×C₃×C₄×C₉×C₁₁₃ #72201700

Order 24408.

160 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₉×C₁₁₃	C₂	1	1	yes	maximal, characteristic
2	C₃×C₄×C₉×C₁₁₃	C₂	1	2	yes	maximal, direct product factor
3	C₂×C₃²×C₄×C₁₁₃	C₃	1	1	yes	maximal, characteristic
4	C₂×C₄×C₉×C₁₁₃	C₃	1	3	yes	maximal, direct product factor
5	C₂×C₃×C₉×C₁₁₃	C₄	1	2	yes	direct product factor
6	C₂×C₃×C₉×C₁₁₃	C₂²	1	1	yes	characteristic
7	C₂×C₃×C₄×C₁₁₃	C₉	1	3	yes	direct product factor
8	C₂×C₃×C₄×C₁₁₃	C₃²	1	1	yes	characteristic
9	C₂²×C₃²×C₁₁₃	C₂×C₃	1	1	yes	characteristic
10	C₃²×C₄×C₁₁₃	C₂×C₃	1	2	yes	—
11	C₂²×C₉×C₁₁₃	C₂×C₃	1	3	yes	—
12	C₄×C₉×C₁₁₃	C₂×C₃	1	6	yes	direct product factor
13	C₃×C₉×C₁₁₃	C₂×C₄	1	1	yes	characteristic, direct product factor, Hall
14	C₂×C₃²×C₁₁₃	C₃×C₄	1	2	yes	—
15	C₂×C₃²×C₁₁₃	C₂²×C₃	1	1	yes	characteristic
16	C₂×C₉×C₁₁₃	C₃×C₄	1	6	yes	direct product factor
17	C₂×C₉×C₁₁₃	C₂²×C₃	1	3	yes	—
18	C₂²×C₃×C₁₁₃	C₂×C₉	1	3	yes	—
19	C₂²×C₃×C₁₁₃	C₂×C₃²	1	1	yes	characteristic
20	C₃×C₄×C₁₁₃	C₂×C₉	1	6	yes	direct product factor
21	C₃×C₄×C₁₁₃	C₂×C₃²	1	2	yes	—
22	C₃²×C₁₁₃	C₂×C₃×C₄	1	1	yes	characteristic
23	C₉×C₁₁₃	C₂×C₃×C₄	1	3	yes	direct product factor
24	C₂×C₄×C₁₁₃	C₃×C₉	1	1	yes	characteristic, direct product factor, Hall
25	C₂×C₃×C₁₁₃	C₄×C₉	1	6	yes	direct product factor
26	C₂×C₃×C₁₁₃	C₂²×C₉	1	3	yes	—
27	C₂×C₃×C₁₁₃	C₃²×C₄	1	2	yes	—
28	C₂×C₃×C₁₁₃	C₂²×C₃²	1	1	yes	characteristic
29	C₂²×C₁₁₃	C₂×C₃×C₉	1	1	yes	characteristic
30	C₄×C₁₁₃	C₂×C₃×C₉	1	2	yes	direct product factor
31	C₃×C₁₁₃	C₂×C₄×C₉	1	3	yes	direct product factor
32	C₃×C₁₁₃	C₂×C₃²×C₄	1	1	yes	characteristic
33	C₂×C₁₁₃	C₃×C₄×C₉	1	2	yes	direct product factor
34	C₂×C₁₁₃	C₂²×C₃×C₉	1	1	yes	characteristic
35	C₂×C₃×C₄×C₉	C₁₁₃	1	1	yes	maximal, characteristic, direct product factor, Hall
36	C₁₁₃	C₂×C₃×C₄×C₉	1	1	yes	characteristic, direct product factor, Sylow
37	C₂²×C₃×C₉	C₂×C₁₁₃	1	1	yes	characteristic
38	C₃×C₄×C₉	C₂×C₁₁₃	1	2	yes	direct product factor
39	C₂×C₃²×C₄	C₃×C₁₁₃	1	1	yes	characteristic
40	C₂×C₄×C₉	C₃×C₁₁₃	1	3	yes	direct product factor
41	C₂×C₃×C₉	C₄×C₁₁₃	1	2	yes	direct product factor
42	C₂×C₃×C₉	C₂²×C₁₁₃	1	1	yes	characteristic
43	C₂²×C₃²	C₂×C₃×C₁₁₃	1	1	yes	characteristic
44	C₃²×C₄	C₂×C₃×C₁₁₃	1	2	yes	—
45	C₂²×C₉	C₂×C₃×C₁₁₃	1	3	yes	—
46	C₄×C₉	C₂×C₃×C₁₁₃	1	6	yes	direct product factor
47	C₃×C₉	C₂×C₄×C₁₁₃	1	1	yes	characteristic, direct product factor, Sylow
48	C₂×C₃×C₄	C₉×C₁₁₃	1	3	yes	direct product factor
49	C₂×C₃×C₄	C₃²×C₁₁₃	1	1	yes	characteristic
50	C₂×C₃²	C₃×C₄×C₁₁₃	1	2	yes	—
51	C₂×C₃²	C₂²×C₃×C₁₁₃	1	1	yes	characteristic
52	C₂×C₉	C₃×C₄×C₁₁₃	1	6	yes	direct product factor
53	C₂×C₉	C₂²×C₃×C₁₁₃	1	3	yes	—
54	C₂²×C₃	C₂×C₉×C₁₁₃	1	3	yes	—
55	C₂²×C₃	C₂×C₃²×C₁₁₃	1	1	yes	characteristic
56	C₃×C₄	C₂×C₉×C₁₁₃	1	6	yes	direct product factor
57	C₃×C₄	C₂×C₃²×C₁₁₃	1	2	yes	—
58	C₃²	C₂×C₃×C₄×C₁₁₃	1	1	yes	characteristic
59	C₉	C₂×C₃×C₄×C₁₁₃	1	3	yes	direct product factor
60	C₂×C₄	C₃×C₉×C₁₁₃	1	1	yes	characteristic, direct product factor, Sylow
61	C₂×C₃	C₄×C₉×C₁₁₃	1	6	yes	direct product factor
62	C₂×C₃	C₂²×C₉×C₁₁₃	1	3	yes	—
63	C₂×C₃	C₃²×C₄×C₁₁₃	1	2	yes	—
64	C₂×C₃	C₂²×C₃²×C₁₁₃	1	1	yes	characteristic
65	C₂²	C₂×C₃×C₉×C₁₁₃	1	1	yes	characteristic
66	C₄	C₂×C₃×C₉×C₁₁₃	1	2	yes	direct product factor
67	C₃	C₂×C₄×C₉×C₁₁₃	1	3	yes	direct product factor
68	C₃	C₂×C₃²×C₄×C₁₁₃	1	1	yes	characteristic
69	C₂	C₃×C₄×C₉×C₁₁₃	1	2	yes	direct product factor
70	C₂	C₂²×C₃×C₉×C₁₁₃	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂×C₃	C₂²×C₃²×C₁₁₃
socle, prime radical	C₂²×C₃²×C₁₁₃	C₂×C₃
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃×C₄×C₉×C₁₁₃	C₂	direct (N × H)
4. C₂×C₄×C₉×C₁₁₃	C₃	direct (N × H)
5. C₂×C₃×C₉×C₁₁₃	C₄	direct (N × H)
7. C₂×C₃×C₄×C₁₁₃	C₉	direct (N × H)
12. C₄×C₉×C₁₁₃	C₂×C₃	direct (N × H)
13. C₃×C₉×C₁₁₃	C₂×C₄	direct (N × H)
16. C₂×C₉×C₁₁₃	C₃×C₄	direct (N × H)
20. C₃×C₄×C₁₁₃	C₂×C₉	direct (N × H)
23. C₉×C₁₁₃	C₂×C₃×C₄	direct (N × H)
24. C₂×C₄×C₁₁₃	C₃×C₉	direct (N × H)
25. C₂×C₃×C₁₁₃	C₄×C₉	direct (N × H)
30. C₄×C₁₁₃	C₂×C₃×C₉	direct (N × H)
31. C₃×C₁₁₃	C₂×C₄×C₉	direct (N × H)
33. C₂×C₁₁₃	C₃×C₄×C₉	direct (N × H)
35. C₂×C₃×C₄×C₉	C₁₁₃	direct (N × H)
36. C₁₁₃	C₂×C₃×C₄×C₉	direct (N × H)
38. C₃×C₄×C₉	C₂×C₁₁₃	direct (N × H)
40. C₂×C₄×C₉	C₃×C₁₁₃	direct (N × H)
41. C₂×C₃×C₉	C₄×C₁₁₃	direct (N × H)
46. C₄×C₉	C₂×C₃×C₁₁₃	direct (N × H)
47. C₃×C₉	C₂×C₄×C₁₁₃	direct (N × H)
48. C₂×C₃×C₄	C₉×C₁₁₃	direct (N × H)
52. C₂×C₉	C₃×C₄×C₁₁₃	direct (N × H)
56. C₃×C₄	C₂×C₉×C₁₁₃	direct (N × H)
59. C₉	C₂×C₃×C₄×C₁₁₃	direct (N × H)
60. C₂×C₄	C₃×C₉×C₁₁₃	direct (N × H)
61. C₂×C₃	C₄×C₉×C₁₁₃	direct (N × H)
66. C₄	C₂×C₃×C₉×C₁₁₃	direct (N × H)
67. C₃	C₂×C₄×C₉×C₁₁₃	direct (N × H)
69. C₂	C₃×C₄×C₉×C₁₁₃	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃×C₉×C₁₁₃	2
C₃×C₄×C₉×C₁₁₃	2
C₂×C₃²×C₄×C₁₁₃	3
C₂×C₄×C₉×C₁₁₃	3
C₂×C₃×C₄×C₉	113

Subgroups — C2×C3×C4×C9×C113 #72201700

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×C₄×C₉×C₁₁₃ #72201700