Subgroups — C₂³×C₁₆×C₁₀₉ #28360658

Order 13952.

440 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₁₆×C₁₀₉	C₂	1	14	yes	maximal, direct product factor
3	C₂³×C₄×C₁₀₉	C₄	1	1	yes	characteristic
4	C₂²×C₈×C₁₀₉	C₄	1	7	yes	—
5	C₂²×C₈×C₁₀₉	C₂²	1	7	yes	—
6	C₂×C₁₆×C₁₀₉	C₂²	1	28	yes	direct product factor
7	C₂⁴×C₁₀₉	C₈	1	1	yes	characteristic
8	C₂²×C₄×C₁₀₉	C₈	1	7	yes	—
9	C₂²×C₄×C₁₀₉	C₂×C₄	1	7	yes	—
10	C₂×C₈×C₁₀₉	C₂×C₄	1	21	yes	—
11	C₂×C₈×C₁₀₉	C₂³	1	7	yes	—
12	C₁₆×C₁₀₉	C₂³	1	8	yes	direct product factor
13	C₂³×C₁₀₉	C₁₆	1	8	yes	direct product factor
14	C₂³×C₁₀₉	C₂×C₈	1	7	yes	—
15	C₂×C₄×C₁₀₉	C₂×C₈	1	21	yes	—
16	C₂×C₄×C₁₀₉	C₂²×C₄	1	7	yes	—
17	C₈×C₁₀₉	C₂²×C₄	1	7	yes	—
18	C₈×C₁₀₉	C₂⁴	1	1	yes	characteristic
19	C₂²×C₁₀₉	C₂×C₁₆	1	28	yes	direct product factor
20	C₂²×C₁₀₉	C₂²×C₈	1	7	yes	—
21	C₄×C₁₀₉	C₂²×C₈	1	7	yes	—
22	C₄×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
23	C₂×C₁₀₉	C₂²×C₁₆	1	14	yes	direct product factor
24	C₂×C₁₀₉	C₂³×C₈	1	1	yes	characteristic
25	C₂³×C₁₆	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
26	C₁₀₉	C₂³×C₁₆	1	1	yes	characteristic, direct product factor, Sylow
27	C₂³×C₈	C₂×C₁₀₉	1	1	yes	characteristic
28	C₂²×C₁₆	C₂×C₁₀₉	1	14	yes	direct product factor
29	C₂³×C₄	C₄×C₁₀₉	1	1	yes	characteristic
30	C₂²×C₈	C₄×C₁₀₉	1	7	yes	—
31	C₂²×C₈	C₂²×C₁₀₉	1	7	yes	—
32	C₂×C₁₆	C₂²×C₁₀₉	1	28	yes	direct product factor
33	C₂⁴	C₈×C₁₀₉	1	1	yes	characteristic
34	C₂²×C₄	C₈×C₁₀₉	1	7	yes	—
35	C₂²×C₄	C₂×C₄×C₁₀₉	1	7	yes	—
36	C₂×C₈	C₂×C₄×C₁₀₉	1	21	yes	—
37	C₂×C₈	C₂³×C₁₀₉	1	7	yes	—
38	C₁₆	C₂³×C₁₀₉	1	8	yes	direct product factor
39	C₂³	C₁₆×C₁₀₉	1	8	yes	direct product factor
40	C₂³	C₂×C₈×C₁₀₉	1	7	yes	—
41	C₂×C₄	C₂×C₈×C₁₀₉	1	21	yes	—
42	C₂×C₄	C₂²×C₄×C₁₀₉	1	7	yes	—
43	C₈	C₂²×C₄×C₁₀₉	1	7	yes	—
44	C₈	C₂⁴×C₁₀₉	1	1	yes	characteristic
45	C₂²	C₂×C₁₆×C₁₀₉	1	28	yes	direct product factor
46	C₂²	C₂²×C₈×C₁₀₉	1	7	yes	—
47	C₄	C₂²×C₈×C₁₀₉	1	7	yes	—
48	C₄	C₂³×C₄×C₁₀₉	1	1	yes	characteristic
49	C₂	C₂²×C₁₆×C₁₀₉	1	14	yes	direct product factor
50	C₂	C₂³×C₈×C₁₀₉	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₈	C₂⁴×C₁₀₉
socle, prime radical	C₂⁴×C₁₀₉	C₈
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₁₆×C₁₀₉	C₂	direct (N × H)
6. C₂×C₁₆×C₁₀₉	C₂²	direct (N × H)
12. C₁₆×C₁₀₉	C₂³	direct (N × H)
13. C₂³×C₁₀₉	C₁₆	direct (N × H)
19. C₂²×C₁₀₉	C₂×C₁₆	direct (N × H)
23. C₂×C₁₀₉	C₂²×C₁₆	direct (N × H)
25. C₂³×C₁₆	C₁₀₉	direct (N × H)
26. C₁₀₉	C₂³×C₁₆	direct (N × H)
28. C₂²×C₁₆	C₂×C₁₀₉	direct (N × H)
32. C₂×C₁₆	C₂²×C₁₀₉	direct (N × H)
38. C₁₆	C₂³×C₁₀₉	direct (N × H)
39. C₂³	C₁₆×C₁₀₉	direct (N × H)
45. C₂²	C₂×C₁₆×C₁₀₉	direct (N × H)
49. C₂	C₂²×C₁₆×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×C₈×C₁₀₉	2
C₂²×C₁₆×C₁₀₉	2
C₂³×C₁₆	109

Subgroups — C23×C16×C109 #28360658

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₁₆×C₁₀₉ #28360658