Subgroups — C₂×C₄²×C₇×C₁₀₉ #72201708

Order 24416.

216 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×C₇×C₁₀₉	C₂	1	3	yes	maximal
2	C₄²×C₇×C₁₀₉	C₂	1	4	yes	maximal, direct product factor
3	C₂³×C₇×C₁₀₉	C₂²	1	1	yes	characteristic
4	C₂×C₄×C₇×C₁₀₉	C₄	1	12	yes	direct product factor
5	C₂×C₄×C₇×C₁₀₉	C₂²	1	6	yes	—
6	C₂×C₄²×C₁₀₉	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂²×C₇×C₁₀₉	C₂×C₄	1	6	yes	—
8	C₂²×C₇×C₁₀₉	C₂³	1	1	yes	characteristic
9	C₄×C₇×C₁₀₉	C₂×C₄	1	12	yes	direct product factor
10	C₂²×C₄×C₁₀₉	C₂×C₇	1	3	yes	—
11	C₄²×C₁₀₉	C₂×C₇	1	4	yes	direct product factor
12	C₂×C₇×C₁₀₉	C₄²	1	4	yes	direct product factor
13	C₂×C₇×C₁₀₉	C₂²×C₄	1	3	yes	—
14	C₂³×C₁₀₉	C₂²×C₇	1	1	yes	characteristic
15	C₂×C₄×C₁₀₉	C₄×C₇	1	12	yes	direct product factor
16	C₂×C₄×C₁₀₉	C₂²×C₇	1	6	yes	—
17	C₇×C₁₀₉	C₂×C₄²	1	1	yes	characteristic, direct product factor, Hall
18	C₂²×C₁₀₉	C₂×C₄×C₇	1	6	yes	—
19	C₂²×C₁₀₉	C₂³×C₇	1	1	yes	characteristic
20	C₄×C₁₀₉	C₂×C₄×C₇	1	12	yes	direct product factor
21	C₂×C₄²×C₇	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Hall
22	C₂×C₁₀₉	C₄²×C₇	1	4	yes	direct product factor
23	C₂×C₁₀₉	C₂²×C₄×C₇	1	3	yes	—
24	C₂²×C₄×C₇	C₂×C₁₀₉	1	3	yes	—
25	C₄²×C₇	C₂×C₁₀₉	1	4	yes	direct product factor
26	C₁₀₉	C₂×C₄²×C₇	1	1	yes	characteristic, direct product factor, Sylow
27	C₂³×C₇	C₂²×C₁₀₉	1	1	yes	characteristic
28	C₂×C₄×C₇	C₄×C₁₀₉	1	12	yes	direct product factor
29	C₂×C₄×C₇	C₂²×C₁₀₉	1	6	yes	—
30	C₂×C₄²	C₇×C₁₀₉	1	1	yes	characteristic, direct product factor, Sylow
31	C₂²×C₇	C₂×C₄×C₁₀₉	1	6	yes	—
32	C₂²×C₇	C₂³×C₁₀₉	1	1	yes	characteristic
33	C₄×C₇	C₂×C₄×C₁₀₉	1	12	yes	direct product factor
34	C₂²×C₄	C₂×C₇×C₁₀₉	1	3	yes	—
35	C₄²	C₂×C₇×C₁₀₉	1	4	yes	direct product factor
36	C₂×C₇	C₄²×C₁₀₉	1	4	yes	direct product factor
37	C₂×C₇	C₂²×C₄×C₁₀₉	1	3	yes	—
38	C₂³	C₂²×C₇×C₁₀₉	1	1	yes	characteristic
39	C₂×C₄	C₄×C₇×C₁₀₉	1	12	yes	direct product factor
40	C₂×C₄	C₂²×C₇×C₁₀₉	1	6	yes	—
41	C₇	C₂×C₄²×C₁₀₉	1	1	yes	characteristic, direct product factor, Sylow
42	C₂²	C₂×C₄×C₇×C₁₀₉	1	6	yes	—
43	C₂²	C₂³×C₇×C₁₀₉	1	1	yes	characteristic
44	C₄	C₂×C₄×C₇×C₁₀₉	1	12	yes	direct product factor
45	C₂	C₄²×C₇×C₁₀₉	1	4	yes	direct product factor
46	C₂	C₂²×C₄×C₇×C₁₀₉	1	3	yes	—

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂²	C₂³×C₇×C₁₀₉
socle, prime radical	C₂³×C₇×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. C₄²×C₇×C₁₀₉	C₂	direct (N × H)
4. C₂×C₄×C₇×C₁₀₉	C₄	direct (N × H)
6. C₂×C₄²×C₁₀₉	C₇	direct (N × H)
9. C₄×C₇×C₁₀₉	C₂×C₄	direct (N × H)
11. C₄²×C₁₀₉	C₂×C₇	direct (N × H)
12. C₂×C₇×C₁₀₉	C₄²	direct (N × H)
15. C₂×C₄×C₁₀₉	C₄×C₇	direct (N × H)
17. C₇×C₁₀₉	C₂×C₄²	direct (N × H)
20. C₄×C₁₀₉	C₂×C₄×C₇	direct (N × H)
21. C₂×C₄²×C₇	C₁₀₉	direct (N × H)
22. C₂×C₁₀₉	C₄²×C₇	direct (N × H)
25. C₄²×C₇	C₂×C₁₀₉	direct (N × H)
26. C₁₀₉	C₂×C₄²×C₇	direct (N × H)
28. C₂×C₄×C₇	C₄×C₁₀₉	direct (N × H)
30. C₂×C₄²	C₇×C₁₀₉	direct (N × H)
33. C₄×C₇	C₂×C₄×C₁₀₉	direct (N × H)
35. C₄²	C₂×C₇×C₁₀₉	direct (N × H)
36. C₂×C₇	C₄²×C₁₀₉	direct (N × H)
39. C₂×C₄	C₄×C₇×C₁₀₉	direct (N × H)
41. C₇	C₂×C₄²×C₁₀₉	direct (N × H)
44. C₄	C₂×C₄×C₇×C₁₀₉	direct (N × H)
45. C₂	C₄²×C₇×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
C₂²×C₄×C₇×C₁₀₉	2
C₄²×C₇×C₁₀₉	2
C₂×C₄²×C₁₀₉	7
C₂×C₄²×C₇	109

Subgroups — C2×C42×C7×C109 #72201708

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄²×C₇×C₁₀₉ #72201708