Subgroups — C₂×C₄²×C₉×C₁₁₃ #72209253

Order 32544.

324 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×C₉×C₁₁₃	C₂	1	3	yes	maximal
2	C₄²×C₉×C₁₁₃	C₂	1	4	yes	maximal, direct product factor
3	C₂×C₃×C₄²×C₁₁₃	C₃	1	1	yes	maximal, characteristic
4	C₂³×C₉×C₁₁₃	C₂²	1	1	yes	characteristic
5	C₂×C₄×C₉×C₁₁₃	C₄	1	12	yes	direct product factor
6	C₂×C₄×C₉×C₁₁₃	C₂²	1	6	yes	—
7	C₂²×C₃×C₄×C₁₁₃	C₂×C₃	1	3	yes	—
8	C₃×C₄²×C₁₁₃	C₂×C₃	1	4	yes	—
9	C₂³×C₃×C₁₁₃	C₂²×C₃	1	1	yes	characteristic
10	C₂×C₃×C₄×C₁₁₃	C₃×C₄	1	12	yes	—
11	C₂×C₃×C₄×C₁₁₃	C₂²×C₃	1	6	yes	—
12	C₂²×C₉×C₁₁₃	C₂×C₄	1	6	yes	—
13	C₂²×C₉×C₁₁₃	C₂³	1	1	yes	characteristic
14	C₄×C₉×C₁₁₃	C₂×C₄	1	12	yes	direct product factor
15	C₂×C₄²×C₁₁₃	C₉	1	1	yes	characteristic, direct product factor, Hall
16	C₂×C₉×C₁₁₃	C₄²	1	4	yes	direct product factor
17	C₂×C₉×C₁₁₃	C₂²×C₄	1	3	yes	—
18	C₂²×C₄×C₁₁₃	C₂×C₉	1	3	yes	—
19	C₄²×C₁₁₃	C₂×C₉	1	4	yes	direct product factor
20	C₂²×C₃×C₁₁₃	C₂×C₃×C₄	1	6	yes	—
21	C₂²×C₃×C₁₁₃	C₂³×C₃	1	1	yes	characteristic
22	C₃×C₄×C₁₁₃	C₂×C₃×C₄	1	12	yes	—
23	C₉×C₁₁₃	C₂×C₄²	1	1	yes	characteristic, direct product factor, Hall
24	C₂³×C₁₁₃	C₂²×C₉	1	1	yes	characteristic
25	C₂×C₄×C₁₁₃	C₄×C₉	1	12	yes	direct product factor
26	C₂×C₄×C₁₁₃	C₂²×C₉	1	6	yes	—
27	C₂×C₃×C₁₁₃	C₃×C₄²	1	4	yes	—
28	C₂×C₃×C₁₁₃	C₂²×C₃×C₄	1	3	yes	—
29	C₂²×C₁₁₃	C₂×C₄×C₉	1	6	yes	—
30	C₂²×C₁₁₃	C₂³×C₉	1	1	yes	characteristic
31	C₄×C₁₁₃	C₂×C₄×C₉	1	12	yes	direct product factor
32	C₃×C₁₁₃	C₂×C₃×C₄²	1	1	yes	characteristic
33	C₂×C₄²×C₉	C₁₁₃	1	1	yes	maximal, characteristic, direct product factor, Hall
34	C₂×C₁₁₃	C₄²×C₉	1	4	yes	direct product factor
35	C₂×C₁₁₃	C₂²×C₄×C₉	1	3	yes	—
36	C₂²×C₄×C₉	C₂×C₁₁₃	1	3	yes	—
37	C₄²×C₉	C₂×C₁₁₃	1	4	yes	direct product factor
38	C₁₁₃	C₂×C₄²×C₉	1	1	yes	characteristic, direct product factor, Sylow
39	C₂×C₃×C₄²	C₃×C₁₁₃	1	1	yes	characteristic
40	C₂³×C₉	C₂²×C₁₁₃	1	1	yes	characteristic
41	C₂×C₄×C₉	C₄×C₁₁₃	1	12	yes	direct product factor
42	C₂×C₄×C₉	C₂²×C₁₁₃	1	6	yes	—
43	C₂²×C₃×C₄	C₂×C₃×C₁₁₃	1	3	yes	—
44	C₃×C₄²	C₂×C₃×C₁₁₃	1	4	yes	—
45	C₂²×C₉	C₂×C₄×C₁₁₃	1	6	yes	—
46	C₂²×C₉	C₂³×C₁₁₃	1	1	yes	characteristic
47	C₄×C₉	C₂×C₄×C₁₁₃	1	12	yes	direct product factor
48	C₂×C₄²	C₉×C₁₁₃	1	1	yes	characteristic, direct product factor, Sylow
49	C₂³×C₃	C₂²×C₃×C₁₁₃	1	1	yes	characteristic
50	C₂×C₃×C₄	C₃×C₄×C₁₁₃	1	12	yes	—
51	C₂×C₃×C₄	C₂²×C₃×C₁₁₃	1	6	yes	—
52	C₂×C₉	C₄²×C₁₁₃	1	4	yes	direct product factor
53	C₂×C₉	C₂²×C₄×C₁₁₃	1	3	yes	—
54	C₂²×C₄	C₂×C₉×C₁₁₃	1	3	yes	—
55	C₄²	C₂×C₉×C₁₁₃	1	4	yes	direct product factor
56	C₂²×C₃	C₂×C₃×C₄×C₁₁₃	1	6	yes	—
57	C₂²×C₃	C₂³×C₃×C₁₁₃	1	1	yes	characteristic
58	C₃×C₄	C₂×C₃×C₄×C₁₁₃	1	12	yes	—
59	C₉	C₂×C₄²×C₁₁₃	1	1	yes	characteristic, direct product factor, Sylow
60	C₂³	C₂²×C₉×C₁₁₃	1	1	yes	characteristic
61	C₂×C₄	C₄×C₉×C₁₁₃	1	12	yes	direct product factor
62	C₂×C₄	C₂²×C₉×C₁₁₃	1	6	yes	—
63	C₂×C₃	C₃×C₄²×C₁₁₃	1	4	yes	—
64	C₂×C₃	C₂²×C₃×C₄×C₁₁₃	1	3	yes	—
65	C₂²	C₂×C₄×C₉×C₁₁₃	1	6	yes	—
66	C₂²	C₂³×C₉×C₁₁₃	1	1	yes	characteristic
67	C₄	C₂×C₄×C₉×C₁₁₃	1	12	yes	direct product factor
68	C₃	C₂×C₃×C₄²×C₁₁₃	1	1	yes	characteristic
69	C₂	C₄²×C₉×C₁₁₃	1	4	yes	direct product factor
70	C₂	C₂²×C₄×C₉×C₁₁₃	1	3	yes	—

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂²×C₃	C₂³×C₃×C₁₁₃
socle, prime radical	C₂³×C₃×C₁₁₃	C₂²×C₃
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄²×C₉×C₁₁₃	C₂	direct (N × H)
5. C₂×C₄×C₉×C₁₁₃	C₄	direct (N × H)
14. C₄×C₉×C₁₁₃	C₂×C₄	direct (N × H)
15. C₂×C₄²×C₁₁₃	C₉	direct (N × H)
16. C₂×C₉×C₁₁₃	C₄²	direct (N × H)
19. C₄²×C₁₁₃	C₂×C₉	direct (N × H)
23. C₉×C₁₁₃	C₂×C₄²	direct (N × H)
25. C₂×C₄×C₁₁₃	C₄×C₉	direct (N × H)
31. C₄×C₁₁₃	C₂×C₄×C₉	direct (N × H)
33. C₂×C₄²×C₉	C₁₁₃	direct (N × H)
34. C₂×C₁₁₃	C₄²×C₉	direct (N × H)
37. C₄²×C₉	C₂×C₁₁₃	direct (N × H)
38. C₁₁₃	C₂×C₄²×C₉	direct (N × H)
41. C₂×C₄×C₉	C₄×C₁₁₃	direct (N × H)
47. C₄×C₉	C₂×C₄×C₁₁₃	direct (N × H)
48. C₂×C₄²	C₉×C₁₁₃	direct (N × H)
52. C₂×C₉	C₄²×C₁₁₃	direct (N × H)
55. C₄²	C₂×C₉×C₁₁₃	direct (N × H)
59. C₉	C₂×C₄²×C₁₁₃	direct (N × H)
61. C₂×C₄	C₄×C₉×C₁₁₃	direct (N × H)
67. C₄	C₂×C₄×C₉×C₁₁₃	direct (N × H)
69. C₂	C₄²×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×C₄×C₉×C₁₁₃	2
C₄²×C₉×C₁₁₃	2
C₂×C₃×C₄²×C₁₁₃	3
C₂×C₄²×C₉	113

Subgroups — C2×C42×C9×C113 #72209253

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄²×C₉×C₁₁₃ #72209253