Subgroups — C₅×C₃⋊Dic₃×C₁₃₇ #37590387

Order 24660.

72 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×C₅×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	C₅×Dic₃×C₁₃₇	—	3	4	no	maximal
3	C₃²×C₅×C₁₃₇	C₄	1	1	yes	characteristic, complemented, Hall
4	C₃⋊Dic₃×C₁₃₇	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₂×C₃×C₅×C₁₃₇	D₃	1	4	yes	—
6	C₄×C₅×C₁₃₇	—	9	1	no	Hall
7	C₂×C₃²×C₁₃₇	C₂×C₅	1	1	yes	characteristic
8	C₃×C₅×C₁₃₇	Dic₃	1	4	yes	complemented
9	Dic₃×C₁₃₇	—	3	4	no	—
10	C₂×C₅×C₁₃₇	C₃⋊D₃	1	1	yes	characteristic
11	C₃²×C₁₃₇	C₄×C₅	1	1	yes	characteristic, complemented, Hall
12	C₂×C₃×C₁₃₇	C₅×D₃	1	4	yes	—
13	C₅×C₁₃₇	C₃⋊Dic₃	1	1	yes	characteristic, direct product factor, Hall
14	C₄×C₁₃₇	—	9	1	no	Hall
15	C₃×C₁₃₇	C₅×Dic₃	1	4	yes	complemented
16	C₂×C₁₃₇	C₅×C₃⋊D₃	1	1	yes	characteristic
17	C₅×C₃⋊Dic₃	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Hall
18	C₁₃₇	C₅×C₃⋊Dic₃	1	1	yes	characteristic, direct product factor, Sylow
19	C₂×C₃²×C₅	C₂×C₁₃₇	1	1	yes	characteristic
20	C₅×Dic₃	—	3	4	no	—
21	C₃²×C₅	C₄×C₁₃₇	1	1	yes	characteristic, complemented, Hall
22	C₃⋊Dic₃	C₅×C₁₃₇	1	1	yes	characteristic, direct product factor, Hall
23	C₂×C₃×C₅	D₃×C₁₃₇	1	4	yes	—
24	C₄×C₅	—	9	1	no	Hall
25	C₂×C₃²	C₂×C₅×C₁₃₇	1	1	yes	characteristic
26	C₃×C₅	Dic₃×C₁₃₇	1	4	yes	complemented
27	Dic₃	—	3	4	no	—
28	C₂×C₅	C₃⋊D₃×C₁₃₇	1	1	yes	characteristic
29	C₃²	C₄×C₅×C₁₃₇	1	1	yes	characteristic, complemented, Sylow
30	C₂×C₃	C₅×D₃×C₁₃₇	1	4	yes	—
31	C₅	C₃⋊Dic₃×C₁₃₇	1	1	yes	characteristic, direct product factor, Sylow
32	C₄	—	9	1	no	Sylow
33	C₃	C₅×Dic₃×C₁₃₇	1	4	yes	complemented
34	C₂	C₅×C₃⋊D₃×C₁₃₇	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂×C₅×C₁₃₇	C₃⋊D₃
commutator, hypocenter	C₃²	C₄×C₅×C₁₃₇
Frattini	C₂	C₅×C₃⋊D₃×C₁₃₇
Fitting, socle, prime radical	C₂×C₃²×C₅×C₁₃₇	C₂
cosocle	C₂×C₃²	C₂×C₅×C₁₃₇
LCS	G ▷ C₃²
UCS	C₁ ◁ C₂×C₅×C₁₃₇

Complements

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

N	H ≅ G/N	type
3. C₃²×C₅×C₁₃₇	C₄	semidirect
4. C₃⋊Dic₃×C₁₃₇	C₅	direct (N × H)
8. C₃×C₅×C₁₃₇	Dic₃	semidirect
11. C₃²×C₁₃₇	C₄×C₅	semidirect
13. C₅×C₁₃₇	C₃⋊Dic₃	direct (N × H)
15. C₃×C₁₃₇	C₅×Dic₃	semidirect
17. C₅×C₃⋊Dic₃	C₁₃₇	direct (N × H)
18. C₁₃₇	C₅×C₃⋊Dic₃	direct (N × H)
21. C₃²×C₅	C₄×C₁₃₇	semidirect
22. C₃⋊Dic₃	C₅×C₁₃₇	direct (N × H)
26. C₃×C₅	Dic₃×C₁₃₇	semidirect
29. C₃²	C₄×C₅×C₁₃₇	semidirect
31. C₅	C₃⋊Dic₃×C₁₃₇	direct (N × H)
33. C₃	C₅×Dic₃×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃²×C₅×C₁₃₇	2
C₅×Dic₃×C₁₃₇	3
C₃⋊Dic₃×C₁₃₇	5
C₅×C₃⋊Dic₃	137

Subgroups — C5×C3⋊Dic3×C137 #37590387

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×C₃⋊Dic₃×C₁₃₇ #37590387