Subgroups — C₃⋊(C₃⋊(C₃₁⋊#1039)) #42124652

Order 27900.

288 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×C₅²×C₃₁	C₂	1	1	yes	maximal, characteristic
2	C₃⋊(C₃₁⋊#1039)	—	3	4	no	maximal
3	C₃²×C₅²×C₃₁	C₄	1	1	yes	characteristic, complemented, Hall
4	C₃⋊Dic₄₆₅	—	5	6	no	maximal
5	C₂×C₃×C₅²×C₃₁	D₃	1	4	yes	—
6	C₃₁⋊#1039	—	9	1	no	Hall
7	C₂×C₃²×C₅×C₃₁	D₅	1	6	yes	—
8	C₃×C₅²×C₃₁	Dic₃	1	4	yes	complemented
9	Dic₄₆₅	—	15	24	no	—
10	C₂×C₅²×C₃₁	C₃⋊D₃	1	1	yes	characteristic
11	C₃²×C₅×C₃₁	Dic₅	1	6	yes	complemented
12	C₃⋊Dic₉₃	—	25	1	no	Hall
13	C₂×C₃×C₅×C₃₁	D₁₅	1	24	yes	—
14	C₃⋊(C₃⋊#1039)	—	31	1	no	maximal, Hall
15	C₅²×C₃₁	C₃⋊Dic₃	1	1	yes	characteristic, complemented, Hall
16	Dic₁₅₅	—	45	6	no	—
17	C₂×C₃²×C₃₁	C₅⋊D₅	1	1	yes	characteristic
18	C₃×C₅×C₃₁	Dic₁₅	1	24	yes	complemented
19	C₂×C₃²×C₅²	D₃₁	1	1	yes	characteristic
20	Dic₉₃	—	75	4	no	—
21	C₂×C₅×C₃₁	C₃⋊D₁₅	1	6	yes	—
22	C₃⋊#1039	—	93	4	no	—
23	C₃²×C₃₁	#1039	1	1	yes	characteristic, complemented, Hall
24	C₃²×C₅²	Dic₃₁	1	1	yes	characteristic, complemented, Hall
25	C₂×C₃×C₃₁	C₃⋊(C₅⋊D₅)	1	4	yes	—
26	C₃⋊Dic₁₅	—	155	6	no	—
27	C₅×C₃₁	C₃⋊Dic₁₅	1	6	yes	complemented
28	C₂×C₃×C₅²	D₉₃	1	4	yes	—
29	Dic₃₁	—	225	1	no	Hall
30	#1039	—	279	1	no	Hall
31	C₃×C₃₁	C₃⋊#1039	1	4	yes	complemented
32	C₂×C₃²×C₅	D₁₅₅	1	6	yes	—
33	C₃×C₅²	Dic₉₃	1	4	yes	complemented
34	C₂×C₃₁	C₃⋊(C₃⋊(C₅⋊D₅))	1	1	yes	characteristic
35	Dic₁₅	—	465	24	no	—
36	C₂×C₅²	C₃⋊D₉₃	1	1	yes	characteristic
37	C₃²×C₅	Dic₁₅₅	1	6	yes	complemented
38	C₃⋊Dic₃	—	775	1	no	Hall
39	C₃₁	C₃⋊(C₃⋊#1039)	1	1	yes	characteristic, complemented, Sylow
40	C₂×C₃×C₅	D₄₆₅	1	24	yes	—
41	C₅²	C₃⋊Dic₉₃	1	1	yes	characteristic, complemented, Sylow
42	Dic₅	—	1395	6	no	—
43	C₂×C₃²	C₅⋊D₁₅₅	1	1	yes	characteristic
44	C₃×C₅	Dic₄₆₅	1	24	yes	complemented
45	Dic₃	—	2325	4	no	—
46	C₂×C₅	C₃⋊D₄₆₅	1	6	yes	—
47	C₃²	C₃₁⋊#1039	1	1	yes	characteristic, complemented, Sylow
48	C₂×C₃	C₃⋊(C₅⋊D₁₅₅)	1	4	yes	—
49	C₅	C₃⋊Dic₄₆₅	1	6	yes	complemented
50	C₄	—	6975	1	no	Sylow
51	C₃	C₃⋊(C₃₁⋊#1039)	1	4	yes	complemented
52	C₂	C₃⋊(C₃⋊(C₅⋊D₁₅₅))	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₂	C₃⋊(C₃⋊(C₅⋊D₁₅₅))
commutator, hypocenter	C₃²×C₅²×C₃₁	C₄
Fitting, socle, prime radical, cosocle	C₂×C₃²×C₅²×C₃₁	C₂
LCS	G ▷ C₃²×C₅²×C₃₁
UCS	C₁ ◁ C₂

Complements

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

N	H ≅ G/N	type
3. C₃²×C₅²×C₃₁	C₄	semidirect
8. C₃×C₅²×C₃₁	Dic₃	semidirect
11. C₃²×C₅×C₃₁	Dic₅	semidirect
15. C₅²×C₃₁	C₃⋊Dic₃	semidirect
18. C₃×C₅×C₃₁	Dic₁₅	semidirect
23. C₃²×C₃₁	#1039	semidirect
24. C₃²×C₅²	Dic₃₁	semidirect
27. C₅×C₃₁	C₃⋊Dic₁₅	semidirect
31. C₃×C₃₁	C₃⋊#1039	semidirect
33. C₃×C₅²	Dic₉₃	semidirect
37. C₃²×C₅	Dic₁₅₅	semidirect
39. C₃₁	C₃⋊(C₃⋊#1039)	semidirect
41. C₅²	C₃⋊Dic₉₃	semidirect
44. C₃×C₅	Dic₄₆₅	semidirect
47. C₃²	C₃₁⋊#1039	semidirect
49. C₅	C₃⋊Dic₄₆₅	semidirect
51. C₃	C₃⋊(C₃₁⋊#1039)	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃²×C₅²×C₃₁	2
C₃⋊(C₃₁⋊#1039)	3
C₃⋊Dic₄₆₅	5
C₃⋊(C₃⋊#1039)	31

Subgroups — C3⋊(C3⋊(C31⋊#1039)) #42124652

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃⋊(C₃⋊(C₃₁⋊#1039)) #42124652