Subgroups — C₃×C₂²⋊(C₂²⋊(C₂²⋊C₉)) #13180115

Order 1728.

6002 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁶×C₃²	C₃	1	1	yes	maximal, characteristic
2	C₂²⋊(C₂²⋊(C₂²⋊C₉))	C₃	1	3	yes	maximal, direct product factor
3	C₃×C₂²⋊(C₂²⋊C₉)	—	4	21	no	maximal
4	C₂⁵×C₃²	—	3	21	no	—
5	C₂⁶×C₃	C₉	1	3	yes	complemented
6	C₂⁶×C₃	C₃²	1	1	yes	characteristic
7	C₂⁴×C₃²	—	3	210	no	—
8	C₂⁴×C₃²	A₄	1	21	yes	—
9	C₂²⋊(C₂²⋊C₉)	—	4	63	no	—
10	C₃×C₂²⋊C₉	—	16	21	no	—
11	C₂⁵×C₃	—	3	63	no	—
12	C₂⁵×C₃	—	3	21	no	—
13	C₂³×C₃²	—	3	360	no	—
14	C₂³×C₃²	—	3	105	no	—
15	C₂⁶	C₃×C₉	1	1	yes	characteristic, complemented, Sylow
16	C₂⁴×C₃	—	3	210	no	—
17	C₂⁴×C₃	—	3	630	no	—
18	C₂⁴×C₃	C₂²⋊C₉	1	63	yes	complemented
19	C₂⁴×C₃	C₃×A₄	1	21	yes	—
20	C₂²×C₃²	—	3	210	no	—
21	C₂²×C₃²	C₂²⋊A₄	1	21	yes	—
22	C₂²⋊C₉	—	16	63	no	—
23	C₂⁵	—	3	21	no	—
24	C₃×C₉	—	64	1	no	Sylow
25	C₂³×C₃	—	3	1080	no	—
26	C₂³×C₃	—	3	360	no	—
27	C₂³×C₃	—	3	105	no	—
28	C₂³×C₃	—	3	315	no	—
29	C₂×C₃²	—	3	21	no	—
30	C₂⁴	—	3	210	no	—
31	C₂⁴	C₃×C₂²⋊C₉	1	21	yes	complemented
32	C₂²×C₃	—	3	630	no	—
33	C₂²×C₃	—	3	210	no	—
34	C₂²×C₃	C₂²⋊(C₂²⋊C₉)	1	63	yes	complemented
35	C₂²×C₃	C₃×C₂²⋊A₄	1	21	yes	—
36	C₃²	C₂²⋊(C₂²⋊A₄)	1	1	yes	characteristic
37	C₉	—	64	3	no	—
38	C₂³	—	3	360	no	—
39	C₂³	—	3	105	no	—
40	C₂×C₃	—	3	63	no	—
41	C₂×C₃	—	3	21	no	—
42	C₂²	—	3	210	no	—
43	C₂²	C₃×C₂²⋊(C₂²⋊C₉)	1	21	yes	complemented
44	C₃	C₂²⋊(C₂²⋊(C₂²⋊C₉))	1	3	yes	direct product factor
45	C₃	C₃×C₂²⋊(C₂²⋊A₄)	1	1	yes	characteristic
46	C₂	—	3	21	no	—

	subgroup	G/H
center, hypercenter	C₃²	C₂²⋊(C₂²⋊A₄)
commutator, hypocenter	C₂⁶	C₃×C₉
Frattini	C₃	C₃×C₂²⋊(C₂²⋊A₄)
Fitting, socle, prime radical	C₂⁶×C₃²	C₃
cosocle	C₂⁶×C₃	C₃²
LCS	G ▷ C₂⁶
UCS	C₁ ◁ C₃²

Complements

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

N	H ≅ G/N	type
2. C₂²⋊(C₂²⋊(C₂²⋊C₉))	C₃	direct (N × H)
5. C₂⁶×C₃	C₉	semidirect
15. C₂⁶	C₃×C₉	semidirect
18. C₂⁴×C₃	C₂²⋊C₉	semidirect
31. C₂⁴	C₃×C₂²⋊C₉	semidirect
34. C₂²×C₃	C₂²⋊(C₂²⋊C₉)	semidirect
43. C₂²	C₃×C₂²⋊(C₂²⋊C₉)	semidirect
44. C₃	C₂²⋊(C₂²⋊(C₂²⋊C₉))	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁶×C₃²	3
C₂²⋊(C₂²⋊(C₂²⋊C₉))	3
C₃×C₂²⋊(C₂²⋊C₉)	4

Subgroups — C3×C22⋊(C22⋊(C22⋊C9)) #13180115

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₂²⋊(C₂²⋊(C₂²⋊C₉)) #13180115