Subgroups — C₃×C₃⋊(C₅⋊D₃₅₅) #44544089

Order 31950.

144 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₅²×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	C₃×C₅⋊D₃₅₅	—	3	1	no	maximal
3	C₃⋊(C₅⋊D₃₅₅)	C₃	1	1	yes	maximal, characteristic, direct product factor
4	C₃×D₁₀₆₅	—	5	6	no	maximal
5	C₃×C₅²×C₇₁	—	2	1	no	—
6	C₃×C₅²×C₇₁	D₃	1	1	yes	characteristic, complemented
7	C₃×C₅²×C₇₁	C₂×C₃	1	1	yes	characteristic, complemented
8	C₅⋊D₃₅₅	—	3	1	no	Hall
9	C₃²×C₅×C₇₁	D₅	1	6	yes	complemented
10	D₁₀₆₅	—	5	6	no	—
11	C₃×D₃₅₅	—	15	6	no	—
12	C₅²×C₇₁	C₃×D₃	1	1	yes	characteristic, complemented, Hall
13	C₃×D₂₁₃	—	25	1	no	Hall
14	C₃×C₅×C₇₁	—	2	6	no	—
15	C₃×C₅×C₇₁	C₃×D₅	1	6	yes	complemented
16	C₃×C₅×C₇₁	D₁₅	1	6	yes	complemented
17	D₃₅₅	—	15	6	no	—
18	C₃²×C₇₁	C₅⋊D₅	1	1	yes	characteristic, complemented, Hall
19	C₃×C₃⋊(C₅⋊D₅)	—	71	1	no	maximal, Hall
20	D₂₁₃	—	25	1	no	—
21	C₃×D₇₁	—	75	1	no	—
22	C₅×C₇₁	C₃×D₁₅	1	6	yes	complemented
23	C₃²×C₅²	D₇₁	1	1	yes	characteristic, complemented, Hall
24	C₃×C₇₁	—	2	1	no	—
25	C₃×C₇₁	C₃×C₅⋊D₅	1	1	yes	characteristic, complemented
26	C₃×C₇₁	C₃⋊(C₅⋊D₅)	1	1	yes	characteristic, complemented
27	C₃⋊(C₅⋊D₅)	—	71	1	no	—
28	C₃×C₅⋊D₅	—	213	1	no	—
29	D₇₁	—	75	1	no	Hall
30	C₃×D₁₅	—	355	6	no	—
31	C₃×C₅²	—	2	1	no	—
32	C₃×C₅²	C₃×D₇₁	1	1	yes	characteristic, complemented
33	C₃×C₅²	D₂₁₃	1	1	yes	characteristic, complemented
34	C₇₁	C₃×C₃⋊(C₅⋊D₅)	1	1	yes	characteristic, complemented, Sylow
35	C₅⋊D₅	—	213	1	no	Hall
36	C₃²×C₅	D₃₅₅	1	6	yes	complemented
37	D₁₅	—	355	6	no	—
38	C₃×D₅	—	1065	6	no	—
39	C₅²	C₃×D₂₁₃	1	1	yes	characteristic, complemented, Sylow
40	C₃×D₃	—	1775	1	no	Hall
41	C₃×C₅	—	2	6	no	—
42	C₃×C₅	C₃×D₃₅₅	1	6	yes	complemented
43	C₃×C₅	D₁₀₆₅	1	6	yes	complemented
44	D₅	—	1065	6	no	—
45	C₃²	C₅⋊D₃₅₅	1	1	yes	characteristic, complemented, Sylow
46	C₂×C₃	—	5325	1	no	—
47	D₃	—	1775	1	no	—
48	C₅	C₃×D₁₀₆₅	1	6	yes	complemented
49	C₃	—	2	1	no	—
50	C₃	C₃⋊(C₅⋊D₃₅₅)	1	1	yes	characteristic, direct product factor
51	C₃	C₃×C₅⋊D₃₅₅	1	1	yes	characteristic, complemented
52	C₂	—	5325	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₃	C₃⋊(C₅⋊D₃₅₅)
commutator, cosocle, hypocenter	C₃×C₅²×C₇₁	C₂×C₃
Frattini	C₁	G
Fitting, socle	C₃²×C₅²×C₇₁	C₂
LCS	G ▷ C₃×C₅²×C₇₁
UCS	C₁ ◁ C₃

Complements

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

N	H ≅ G/N	type
1. C₃²×C₅²×C₇₁	C₂	semidirect
3. C₃⋊(C₅⋊D₃₅₅)	C₃	direct (N × H)
6. C₃×C₅²×C₇₁	D₃	semidirect
7. C₃×C₅²×C₇₁	C₂×C₃	semidirect
9. C₃²×C₅×C₇₁	D₅	semidirect
12. C₅²×C₇₁	C₃×D₃	semidirect
15. C₃×C₅×C₇₁	C₃×D₅	semidirect
16. C₃×C₅×C₇₁	D₁₅	semidirect
18. C₃²×C₇₁	C₅⋊D₅	semidirect
22. C₅×C₇₁	C₃×D₁₅	semidirect
23. C₃²×C₅²	D₇₁	semidirect
25. C₃×C₇₁	C₃×C₅⋊D₅	semidirect
26. C₃×C₇₁	C₃⋊(C₅⋊D₅)	semidirect
32. C₃×C₅²	C₃×D₇₁	semidirect
33. C₃×C₅²	D₂₁₃	semidirect
34. C₇₁	C₃×C₃⋊(C₅⋊D₅)	semidirect
36. C₃²×C₅	D₃₅₅	semidirect
39. C₅²	C₃×D₂₁₃	semidirect
42. C₃×C₅	C₃×D₃₅₅	semidirect
43. C₃×C₅	D₁₀₆₅	semidirect
45. C₃²	C₅⋊D₃₅₅	semidirect
48. C₅	C₃×D₁₀₆₅	semidirect
50. C₃	C₃⋊(C₅⋊D₃₅₅)	direct (N × H)
51. C₃	C₃×C₅⋊D₃₅₅	semidirect

Maximal subgroups

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

subgroup	index
C₃²×C₅²×C₇₁	2
C₃×C₅⋊D₃₅₅	3
C₃⋊(C₅⋊D₃₅₅)	3
C₃×D₁₀₆₅	5
C₃×C₃⋊(C₅⋊D₅)	71

Subgroups — C3×C3⋊(C5⋊D355) #44544089

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₃⋊(C₅⋊D₃₅₅) #44544089