Subgroups — C₅×C₇×C₈×C₁₀₉ #72207293

Order 30520.

32 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×C₈×C₁₀₉	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
2	C₅×C₈×C₁₀₉	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
3	C₄×C₅×C₇×C₁₀₉	C₂	1	1	yes	maximal, characteristic
4	C₂×C₅×C₇×C₁₀₉	C₄	1	1	yes	characteristic
5	C₅×C₇×C₁₀₉	C₈	1	1	yes	characteristic, direct product factor, Hall
6	C₄×C₇×C₁₀₉	C₂×C₅	1	1	yes	characteristic
7	C₄×C₅×C₁₀₉	C₂×C₇	1	1	yes	characteristic
8	C₂×C₇×C₁₀₉	C₄×C₅	1	1	yes	characteristic
9	C₂×C₅×C₁₀₉	C₄×C₇	1	1	yes	characteristic
10	C₈×C₁₀₉	C₅×C₇	1	1	yes	characteristic, direct product factor, Hall
11	C₇×C₁₀₉	C₅×C₈	1	1	yes	characteristic, direct product factor, Hall
12	C₅×C₁₀₉	C₇×C₈	1	1	yes	characteristic, direct product factor, Hall
13	C₄×C₁₀₉	C₂×C₅×C₇	1	1	yes	characteristic
14	C₅×C₇×C₈	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Hall
15	C₂×C₁₀₉	C₄×C₅×C₇	1	1	yes	characteristic
16	C₄×C₅×C₇	C₂×C₁₀₉	1	1	yes	characteristic
17	C₁₀₉	C₅×C₇×C₈	1	1	yes	characteristic, direct product factor, Sylow
18	C₂×C₅×C₇	C₄×C₁₀₉	1	1	yes	characteristic
19	C₇×C₈	C₅×C₁₀₉	1	1	yes	characteristic, direct product factor, Hall
20	C₅×C₈	C₇×C₁₀₉	1	1	yes	characteristic, direct product factor, Hall
21	C₅×C₇	C₈×C₁₀₉	1	1	yes	characteristic, direct product factor, Hall
22	C₄×C₇	C₂×C₅×C₁₀₉	1	1	yes	characteristic
23	C₄×C₅	C₂×C₇×C₁₀₉	1	1	yes	characteristic
24	C₂×C₇	C₄×C₅×C₁₀₉	1	1	yes	characteristic
25	C₂×C₅	C₄×C₇×C₁₀₉	1	1	yes	characteristic
26	C₈	C₅×C₇×C₁₀₉	1	1	yes	characteristic, direct product factor, Sylow
27	C₇	C₅×C₈×C₁₀₉	1	1	yes	characteristic, direct product factor, Sylow
28	C₅	C₇×C₈×C₁₀₉	1	1	yes	characteristic, direct product factor, Sylow
29	C₄	C₂×C₅×C₇×C₁₀₉	1	1	yes	characteristic
30	C₂	C₄×C₅×C₇×C₁₀₉	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₄	C₂×C₅×C₇×C₁₀₉
socle, prime radical	C₂×C₅×C₇×C₁₀₉	C₄
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₇×C₈×C₁₀₉	C₅	direct (N × H)
2. C₅×C₈×C₁₀₉	C₇	direct (N × H)
5. C₅×C₇×C₁₀₉	C₈	direct (N × H)
10. C₈×C₁₀₉	C₅×C₇	direct (N × H)
11. C₇×C₁₀₉	C₅×C₈	direct (N × H)
12. C₅×C₁₀₉	C₇×C₈	direct (N × H)
14. C₅×C₇×C₈	C₁₀₉	direct (N × H)
17. C₁₀₉	C₅×C₇×C₈	direct (N × H)
19. C₇×C₈	C₅×C₁₀₉	direct (N × H)
20. C₅×C₈	C₇×C₁₀₉	direct (N × H)
21. C₅×C₇	C₈×C₁₀₉	direct (N × H)
26. C₈	C₅×C₇×C₁₀₉	direct (N × H)
27. C₇	C₅×C₈×C₁₀₉	direct (N × H)
28. C₅	C₇×C₈×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
C₇×C₈×C₁₀₉	5
C₅×C₈×C₁₀₉	7
C₄×C₅×C₇×C₁₀₉	2
C₅×C₇×C₈	109

Subgroups — C5×C7×C8×C109 #72207293

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×C₇×C₈×C₁₀₉ #72207293