Subgroups — C₂²×C₃×C₄²×C₁₀₉ #72198523

Order 20928.

996 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃×C₄×C₁₀₉	C₂	1	3	yes	maximal
2	C₂×C₃×C₄²×C₁₀₉	C₂	1	12	yes	maximal, direct product factor
3	C₂⁴×C₃×C₁₀₉	C₂²	1	1	yes	characteristic
4	C₂²×C₃×C₄×C₁₀₉	C₄	1	24	yes	direct product factor
5	C₂²×C₃×C₄×C₁₀₉	C₂²	1	18	yes	—
6	C₃×C₄²×C₁₀₉	C₂²	1	16	yes	direct product factor
7	C₂³×C₃×C₁₀₉	C₂×C₄	1	12	yes	—
8	C₂³×C₃×C₁₀₉	C₂³	1	3	yes	—
9	C₂×C₃×C₄×C₁₀₉	C₂×C₄	1	72	yes	direct product factor
10	C₂×C₃×C₄×C₁₀₉	C₂³	1	12	yes	—
11	C₂²×C₄²×C₁₀₉	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
12	C₂³×C₄×C₁₀₉	C₂×C₃	1	3	yes	—
13	C₂×C₄²×C₁₀₉	C₂×C₃	1	12	yes	direct product factor
14	C₂⁴×C₁₀₉	C₂²×C₃	1	1	yes	characteristic
15	C₂²×C₄×C₁₀₉	C₃×C₄	1	24	yes	direct product factor
16	C₂²×C₄×C₁₀₉	C₂²×C₃	1	18	yes	—
17	C₄²×C₁₀₉	C₂²×C₃	1	16	yes	direct product factor
18	C₂²×C₃×C₁₀₉	C₄²	1	16	yes	direct product factor
19	C₂²×C₃×C₁₀₉	C₂²×C₄	1	18	yes	—
20	C₂²×C₃×C₁₀₉	C₂⁴	1	1	yes	characteristic
21	C₃×C₄×C₁₀₉	C₂²×C₄	1	24	yes	direct product factor
22	C₂³×C₁₀₉	C₂×C₃×C₄	1	12	yes	—
23	C₂³×C₁₀₉	C₂³×C₃	1	3	yes	—
24	C₂×C₄×C₁₀₉	C₂×C₃×C₄	1	72	yes	direct product factor
25	C₂×C₄×C₁₀₉	C₂³×C₃	1	12	yes	—
26	C₂×C₃×C₁₀₉	C₂×C₄²	1	12	yes	direct product factor
27	C₂×C₃×C₁₀₉	C₂³×C₄	1	3	yes	—
28	C₂²×C₁₀₉	C₃×C₄²	1	16	yes	direct product factor
29	C₂²×C₁₀₉	C₂²×C₃×C₄	1	18	yes	—
30	C₂²×C₁₀₉	C₂⁴×C₃	1	1	yes	characteristic
31	C₄×C₁₀₉	C₂²×C₃×C₄	1	24	yes	direct product factor
32	C₃×C₁₀₉	C₂²×C₄²	1	1	yes	characteristic, direct product factor, Hall
33	C₂×C₁₀₉	C₂×C₃×C₄²	1	12	yes	direct product factor
34	C₂×C₁₀₉	C₂³×C₃×C₄	1	3	yes	—
35	C₂²×C₃×C₄²	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Hall
36	C₁₀₉	C₂²×C₃×C₄²	1	1	yes	characteristic, direct product factor, Sylow
37	C₂³×C₃×C₄	C₂×C₁₀₉	1	3	yes	—
38	C₂×C₃×C₄²	C₂×C₁₀₉	1	12	yes	direct product factor
39	C₂²×C₄²	C₃×C₁₀₉	1	1	yes	characteristic, direct product factor, Sylow
40	C₂⁴×C₃	C₂²×C₁₀₉	1	1	yes	characteristic
41	C₂²×C₃×C₄	C₄×C₁₀₉	1	24	yes	direct product factor
42	C₂²×C₃×C₄	C₂²×C₁₀₉	1	18	yes	—
43	C₃×C₄²	C₂²×C₁₀₉	1	16	yes	direct product factor
44	C₂³×C₄	C₂×C₃×C₁₀₉	1	3	yes	—
45	C₂×C₄²	C₂×C₃×C₁₀₉	1	12	yes	direct product factor
46	C₂³×C₃	C₂×C₄×C₁₀₉	1	12	yes	—
47	C₂³×C₃	C₂³×C₁₀₉	1	3	yes	—
48	C₂×C₃×C₄	C₂×C₄×C₁₀₉	1	72	yes	direct product factor
49	C₂×C₃×C₄	C₂³×C₁₀₉	1	12	yes	—
50	C₂⁴	C₂²×C₃×C₁₀₉	1	1	yes	characteristic
51	C₂²×C₄	C₃×C₄×C₁₀₉	1	24	yes	direct product factor
52	C₂²×C₄	C₂²×C₃×C₁₀₉	1	18	yes	—
53	C₄²	C₂²×C₃×C₁₀₉	1	16	yes	direct product factor
54	C₂²×C₃	C₄²×C₁₀₉	1	16	yes	direct product factor
55	C₂²×C₃	C₂²×C₄×C₁₀₉	1	18	yes	—
56	C₂²×C₃	C₂⁴×C₁₀₉	1	1	yes	characteristic
57	C₃×C₄	C₂²×C₄×C₁₀₉	1	24	yes	direct product factor
58	C₂³	C₂×C₃×C₄×C₁₀₉	1	12	yes	—
59	C₂³	C₂³×C₃×C₁₀₉	1	3	yes	—
60	C₂×C₄	C₂×C₃×C₄×C₁₀₉	1	72	yes	direct product factor
61	C₂×C₄	C₂³×C₃×C₁₀₉	1	12	yes	—
62	C₂×C₃	C₂×C₄²×C₁₀₉	1	12	yes	direct product factor
63	C₂×C₃	C₂³×C₄×C₁₀₉	1	3	yes	—
64	C₂²	C₃×C₄²×C₁₀₉	1	16	yes	direct product factor
65	C₂²	C₂²×C₃×C₄×C₁₀₉	1	18	yes	—
66	C₂²	C₂⁴×C₃×C₁₀₉	1	1	yes	characteristic
67	C₄	C₂²×C₃×C₄×C₁₀₉	1	24	yes	direct product factor
68	C₃	C₂²×C₄²×C₁₀₉	1	1	yes	characteristic, direct product factor, Sylow
69	C₂	C₂×C₃×C₄²×C₁₀₉	1	12	yes	direct product factor
70	C₂	C₂³×C₃×C₄×C₁₀₉	1	3	yes	—

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂²	C₂⁴×C₃×C₁₀₉
socle, prime radical	C₂⁴×C₃×C₁₀₉	C₂²
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×C₄²×C₁₀₉	C₂	direct (N × H)
4. C₂²×C₃×C₄×C₁₀₉	C₄	direct (N × H)
6. C₃×C₄²×C₁₀₉	C₂²	direct (N × H)
9. C₂×C₃×C₄×C₁₀₉	C₂×C₄	direct (N × H)
11. C₂²×C₄²×C₁₀₉	C₃	direct (N × H)
13. C₂×C₄²×C₁₀₉	C₂×C₃	direct (N × H)
15. C₂²×C₄×C₁₀₉	C₃×C₄	direct (N × H)
17. C₄²×C₁₀₉	C₂²×C₃	direct (N × H)
18. C₂²×C₃×C₁₀₉	C₄²	direct (N × H)
21. C₃×C₄×C₁₀₉	C₂²×C₄	direct (N × H)
24. C₂×C₄×C₁₀₉	C₂×C₃×C₄	direct (N × H)
26. C₂×C₃×C₁₀₉	C₂×C₄²	direct (N × H)
28. C₂²×C₁₀₉	C₃×C₄²	direct (N × H)
31. C₄×C₁₀₉	C₂²×C₃×C₄	direct (N × H)
32. C₃×C₁₀₉	C₂²×C₄²	direct (N × H)
33. C₂×C₁₀₉	C₂×C₃×C₄²	direct (N × H)
35. C₂²×C₃×C₄²	C₁₀₉	direct (N × H)
36. C₁₀₉	C₂²×C₃×C₄²	direct (N × H)
38. C₂×C₃×C₄²	C₂×C₁₀₉	direct (N × H)
39. C₂²×C₄²	C₃×C₁₀₉	direct (N × H)
41. C₂²×C₃×C₄	C₄×C₁₀₉	direct (N × H)
43. C₃×C₄²	C₂²×C₁₀₉	direct (N × H)
45. C₂×C₄²	C₂×C₃×C₁₀₉	direct (N × H)
48. C₂×C₃×C₄	C₂×C₄×C₁₀₉	direct (N × H)
51. C₂²×C₄	C₃×C₄×C₁₀₉	direct (N × H)
53. C₄²	C₂²×C₃×C₁₀₉	direct (N × H)
54. C₂²×C₃	C₄²×C₁₀₉	direct (N × H)
57. C₃×C₄	C₂²×C₄×C₁₀₉	direct (N × H)
60. C₂×C₄	C₂×C₃×C₄×C₁₀₉	direct (N × H)
62. C₂×C₃	C₂×C₄²×C₁₀₉	direct (N × H)
64. C₂²	C₃×C₄²×C₁₀₉	direct (N × H)
67. C₄	C₂²×C₃×C₄×C₁₀₉	direct (N × H)
68. C₃	C₂²×C₄²×C₁₀₉	direct (N × H)
69. C₂	C₂×C₃×C₄²×C₁₀₉	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₃×C₄×C₁₀₉	2
C₂×C₃×C₄²×C₁₀₉	2
C₂²×C₄²×C₁₀₉	3
C₂²×C₃×C₄²	109

Subgroups — C22×C3×C42×C109 #72198523

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃×C₄²×C₁₀₉ #72198523