Subgroups — C₂×C₃⋊(C₃⋊(D₁₃₁⋊C₅)) #37556094

Order 23580.

120 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×C₁₃₁⋊C₅	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃⋊(C₃⋊(D₁₃₁⋊C₅))	C₂	1	2	yes	maximal, direct product factor
3	C₂×C₃⋊(D₁₃₁⋊C₅)	—	3	4	no	maximal
4	C₃²×C₁₃₁⋊C₅	C₂²	1	1	yes	characteristic, complemented, Hall
5	C₂×C₃⋊D₃₉₃	C₅	1	1	yes	maximal, characteristic, complemented, Hall
6	C₃⋊(D₁₃₁⋊C₅)	—	3	8	no	—
7	C₂×C₃×C₁₃₁⋊C₅	D₃	1	4	yes	complemented
8	C₂×D₁₃₁⋊C₅	—	9	1	no	Hall
9	C₃⋊D₃₉₃	C₂×C₅	1	2	yes	complemented
10	C₂×C₃²×C₁₃₁	C₂×C₅	1	1	yes	characteristic, complemented
11	C₃×C₁₃₁⋊C₅	C₂×D₃	1	4	yes	complemented
12	C₂×D₃₉₃	—	3	4	no	—
13	C₂×C₁₃₁⋊C₅	C₃⋊D₃	1	1	yes	characteristic, complemented
14	D₁₃₁⋊C₅	—	9	2	no	—
15	C₃²×C₁₃₁	C₂²×C₅	1	1	yes	characteristic, complemented, Hall
16	C₂×C₃×C₁₃₁	C₅×D₃	1	4	yes	complemented
17	D₃₉₃	—	3	8	no	—
18	C₁₃₁⋊C₅	C₂×C₃⋊D₃	1	1	yes	characteristic, complemented, Hall
19	C₂×D₁₃₁	—	9	1	no	Hall
20	C₃×C₁₃₁	C₂×C₅×D₃	1	4	yes	complemented
21	C₂×C₁₃₁	C₅×C₃⋊D₃	1	1	yes	characteristic, complemented
22	D₁₃₁	—	9	2	no	—
23	C₂×C₅×C₃⋊D₃	—	131	1	no	maximal, Hall
24	C₁₃₁	C₂×C₅×C₃⋊D₃	1	1	yes	characteristic, complemented, Sylow
25	C₂×C₃²×C₅	—	131	1	no	—
26	C₅×C₃⋊D₃	—	131	2	no	—
27	C₂×C₅×D₃	—	393	4	no	—
28	C₃²×C₅	—	131	1	no	Hall
29	C₂×C₃⋊D₃	—	131	1	no	Hall
30	C₂×C₃×C₅	—	131	4	no	—
31	C₅×D₃	—	393	8	no	—
32	C₂²×C₅	—	1179	1	no	Hall
33	C₂×C₃²	D₁₃₁⋊C₅	1	1	yes	characteristic, complemented
34	C₃⋊D₃	—	131	2	no	—
35	C₃×C₅	—	131	4	no	—
36	C₂×D₃	—	393	4	no	—
37	C₂×C₅	—	131	1	no	—
38	C₂×C₅	—	1179	2	no	—
39	C₃²	C₂×D₁₃₁⋊C₅	1	1	yes	characteristic, complemented, Sylow
40	C₂×C₃	C₃⋊(D₁₃₁⋊C₅)	1	4	yes	complemented
41	D₃	—	393	8	no	—
42	C₅	—	131	1	no	Sylow
43	C₂²	—	1179	1	no	Sylow
44	C₃	C₂×C₃⋊(D₁₃₁⋊C₅)	1	4	yes	complemented
45	C₂	—	1179	2	no	—
46	C₂	C₃⋊(C₃⋊(D₁₃₁⋊C₅))	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂	C₃⋊(C₃⋊(D₁₃₁⋊C₅))
commutator, cosocle, hypocenter	C₃²×C₁₃₁	C₂²×C₅
Frattini	C₁	G
Fitting, socle	C₂×C₃²×C₁₃₁	C₂×C₅
LCS	G ▷ C₃²×C₁₃₁
UCS	C₁ ◁ C₂

Complements

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

N	H ≅ G/N	type
1. C₂×C₃²×C₁₃₁⋊C₅	C₂	semidirect
2. C₃⋊(C₃⋊(D₁₃₁⋊C₅))	C₂	direct (N × H)
4. C₃²×C₁₃₁⋊C₅	C₂²	semidirect
5. C₂×C₃⋊D₃₉₃	C₅	semidirect
7. C₂×C₃×C₁₃₁⋊C₅	D₃	semidirect
9. C₃⋊D₃₉₃	C₂×C₅	semidirect
10. C₂×C₃²×C₁₃₁	C₂×C₅	semidirect
11. C₃×C₁₃₁⋊C₅	C₂×D₃	semidirect
13. C₂×C₁₃₁⋊C₅	C₃⋊D₃	semidirect
15. C₃²×C₁₃₁	C₂²×C₅	semidirect
16. C₂×C₃×C₁₃₁	C₅×D₃	semidirect
18. C₁₃₁⋊C₅	C₂×C₃⋊D₃	semidirect
20. C₃×C₁₃₁	C₂×C₅×D₃	semidirect
21. C₂×C₁₃₁	C₅×C₃⋊D₃	semidirect
24. C₁₃₁	C₂×C₅×C₃⋊D₃	semidirect
33. C₂×C₃²	D₁₃₁⋊C₅	semidirect
39. C₃²	C₂×D₁₃₁⋊C₅	semidirect
40. C₂×C₃	C₃⋊(D₁₃₁⋊C₅)	semidirect
44. C₃	C₂×C₃⋊(D₁₃₁⋊C₅)	semidirect
46. C₂	C₃⋊(C₃⋊(D₁₃₁⋊C₅))	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×C₃²×C₁₃₁⋊C₅	2
C₃⋊(C₃⋊(D₁₃₁⋊C₅))	2
C₂×C₃⋊(D₁₃₁⋊C₅)	3
C₂×C₃⋊D₃₉₃	5
C₂×C₅×C₃⋊D₃	131

Subgroups — C2×C3⋊(C3⋊(D131⋊C5)) #37556094

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃⋊(C₃⋊(D₁₃₁⋊C₅)) #37556094