Subgroups — C₂×C₇×C₁₆×C₁₀₉ #72201706

Order 24416.

56 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×C₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic
2	C₇×C₁₆×C₁₀₉	C₂	1	2	yes	maximal, direct product factor
3	C₂×C₄×C₇×C₁₀₉	C₄	1	1	yes	characteristic
4	C₇×C₈×C₁₀₉	C₄	1	1	yes	characteristic
5	C₇×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
6	C₂×C₁₆×C₁₀₉	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂²×C₇×C₁₀₉	C₈	1	1	yes	characteristic
8	C₄×C₇×C₁₀₉	C₈	1	1	yes	characteristic
9	C₄×C₇×C₁₀₉	C₂×C₄	1	1	yes	characteristic
10	C₂×C₈×C₁₀₉	C₂×C₇	1	1	yes	characteristic
11	C₁₆×C₁₀₉	C₂×C₇	1	2	yes	direct product factor
12	C₂×C₇×C₁₀₉	C₁₆	1	2	yes	direct product factor
13	C₂×C₇×C₁₀₉	C₂×C₈	1	1	yes	characteristic
14	C₂×C₄×C₁₀₉	C₄×C₇	1	1	yes	characteristic
15	C₈×C₁₀₉	C₄×C₇	1	1	yes	characteristic
16	C₈×C₁₀₉	C₂²×C₇	1	1	yes	characteristic
17	C₇×C₁₀₉	C₂×C₁₆	1	1	yes	characteristic, direct product factor, Hall
18	C₂²×C₁₀₉	C₇×C₈	1	1	yes	characteristic
19	C₄×C₁₀₉	C₇×C₈	1	1	yes	characteristic
20	C₄×C₁₀₉	C₂×C₄×C₇	1	1	yes	characteristic
21	C₂×C₇×C₁₆	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Hall
22	C₂×C₁₀₉	C₇×C₁₆	1	2	yes	direct product factor
23	C₂×C₁₀₉	C₂×C₇×C₈	1	1	yes	characteristic
24	C₂×C₇×C₈	C₂×C₁₀₉	1	1	yes	characteristic
25	C₇×C₁₆	C₂×C₁₀₉	1	2	yes	direct product factor
26	C₁₀₉	C₂×C₇×C₁₆	1	1	yes	characteristic, direct product factor, Sylow
27	C₂×C₄×C₇	C₄×C₁₀₉	1	1	yes	characteristic
28	C₇×C₈	C₄×C₁₀₉	1	1	yes	characteristic
29	C₇×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
30	C₂×C₁₆	C₇×C₁₀₉	1	1	yes	characteristic, direct product factor, Sylow
31	C₂²×C₇	C₈×C₁₀₉	1	1	yes	characteristic
32	C₄×C₇	C₈×C₁₀₉	1	1	yes	characteristic
33	C₄×C₇	C₂×C₄×C₁₀₉	1	1	yes	characteristic
34	C₂×C₈	C₂×C₇×C₁₀₉	1	1	yes	characteristic
35	C₁₆	C₂×C₇×C₁₀₉	1	2	yes	direct product factor
36	C₂×C₇	C₁₆×C₁₀₉	1	2	yes	direct product factor
37	C₂×C₇	C₂×C₈×C₁₀₉	1	1	yes	characteristic
38	C₂×C₄	C₄×C₇×C₁₀₉	1	1	yes	characteristic
39	C₈	C₄×C₇×C₁₀₉	1	1	yes	characteristic
40	C₈	C₂²×C₇×C₁₀₉	1	1	yes	characteristic
41	C₇	C₂×C₁₆×C₁₀₉	1	1	yes	characteristic, direct product factor, Sylow
42	C₂²	C₇×C₈×C₁₀₉	1	1	yes	characteristic
43	C₄	C₇×C₈×C₁₀₉	1	1	yes	characteristic
44	C₄	C₂×C₄×C₇×C₁₀₉	1	1	yes	characteristic
45	C₂	C₇×C₁₆×C₁₀₉	1	2	yes	direct product factor
46	C₂	C₂×C₇×C₈×C₁₀₉	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₈	C₂²×C₇×C₁₀₉
socle, prime radical	C₂²×C₇×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. C₇×C₁₆×C₁₀₉	C₂	direct (N × H)
6. C₂×C₁₆×C₁₀₉	C₇	direct (N × H)
11. C₁₆×C₁₀₉	C₂×C₇	direct (N × H)
12. C₂×C₇×C₁₀₉	C₁₆	direct (N × H)
17. C₇×C₁₀₉	C₂×C₁₆	direct (N × H)
21. C₂×C₇×C₁₆	C₁₀₉	direct (N × H)
22. C₂×C₁₀₉	C₇×C₁₆	direct (N × H)
25. C₇×C₁₆	C₂×C₁₀₉	direct (N × H)
26. C₁₀₉	C₂×C₇×C₁₆	direct (N × H)
30. C₂×C₁₆	C₇×C₁₀₉	direct (N × H)
35. C₁₆	C₂×C₇×C₁₀₉	direct (N × H)
36. C₂×C₇	C₁₆×C₁₀₉	direct (N × H)
41. C₇	C₂×C₁₆×C₁₀₉	direct (N × H)
45. C₂	C₇×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
C₂×C₇×C₈×C₁₀₉	2
C₇×C₁₆×C₁₀₉	2
C₂×C₁₆×C₁₀₉	7
C₂×C₇×C₁₆	109

Subgroups — C2×C7×C16×C109 #72201706

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇×C₁₆×C₁₀₉ #72201706