Subgroups — C₃²⋊(C₂²⋊(C₂²⋊A₄)) #13180117

Order 1728.

6046 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, complemented
2	C₃×C₂²⋊(C₂²⋊A₄)	C₃	1	3	yes	maximal, complemented
3	C₃²⋊(C₂²⋊A₄)	—	4	21	no	maximal
4	C₂⁵×C₃²	—	3	21	no	—
5	C₂⁶×C₃	—	3	1	no	—
6	C₂⁶×C₃	C₃²	1	1	yes	characteristic
7	C₂²⋊(C₂²⋊A₄)	—	3	3	no	—
8	C₂⁴×C₃²	—	3	210	no	—
9	C₂⁴×C₃²	A₄	1	21	yes	complemented
10	C₃×C₂²⋊A₄	—	4	63	no	—
11	C₂²⋊H(3,𝔽₃)	—	16	21	no	—
12	C₂⁵×C₃	—	3	63	no	—
13	C₂⁵×C₃	—	3	21	no	—
14	C₂³×C₃²	—	3	360	no	—
15	C₂³×C₃²	—	3	105	no	—
16	C₂⁶	H(3,𝔽₃)	1	1	yes	characteristic, complemented, Sylow
17	C₂⁴×C₃	—	3	630	no	—
18	C₂⁴×C₃	—	3	210	no	—
19	C₂⁴×C₃	—	3	21	no	—
20	C₂⁴×C₃	C₃×A₄	1	21	yes	—
21	C₂²⋊A₄	—	12	63	no	—
22	C₂²×C₃²	—	3	210	no	—
23	C₂²×C₃²	C₂²⋊A₄	1	21	yes	complemented
24	C₃×A₄	—	16	63	no	—
25	C₂⁵	—	3	21	no	—
26	H(3,𝔽₃)	—	64	1	no	Sylow
27	C₂³×C₃	—	3	1080	no	—
28	C₂³×C₃	—	3	360	no	—
29	C₂³×C₃	—	3	315	no	—
30	C₂³×C₃	—	3	105	no	—
31	C₂×C₃²	—	3	21	no	—
32	C₂⁴	—	3	210	no	—
33	C₂⁴	C₂²⋊H(3,𝔽₃)	1	21	yes	complemented
34	C₂²×C₃	—	3	630	no	—
35	C₂²×C₃	—	3	210	no	—
36	C₂²×C₃	—	3	21	no	—
37	C₂²×C₃	C₃×C₂²⋊A₄	1	21	yes	—
38	A₄	—	48	63	no	—
39	C₃²	—	64	3	no	—
40	C₃²	C₂²⋊(C₂²⋊A₄)	1	1	yes	characteristic, complemented
41	C₂³	—	3	360	no	—
42	C₂³	—	3	105	no	—
43	C₂×C₃	—	3	63	no	—
44	C₂×C₃	—	3	21	no	—
45	C₂²	—	3	210	no	—
46	C₂²	C₃²⋊(C₂²⋊A₄)	1	21	yes	complemented
47	C₃	—	3	1	no	—
48	C₃	—	192	3	no	—
49	C₃	C₃×C₂²⋊(C₂²⋊A₄)	1	1	yes	characteristic
50	C₂	—	3	21	no	—

	subgroup	G/H
center, Frattini	C₃	C₃×C₂²⋊(C₂²⋊A₄)
commutator, socle, cosocle	C₂⁶×C₃	C₃²
Fitting	C₂⁶×C₃²	C₃
hypercenter	C₃²	C₂²⋊(C₂²⋊A₄)
hypocenter	C₂⁶	H(3,𝔽₃)
LCS	G ▷ C₂⁶×C₃ ▷ C₂⁶
UCS	C₁ ◁ C₃ ◁ C₃²

Complements

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

N	H ≅ G/N	type
1. C₂⁶×C₃²	C₃	semidirect
2. C₃×C₂²⋊(C₂²⋊A₄)	C₃	semidirect
9. C₂⁴×C₃²	A₄	semidirect
16. C₂⁶	H(3,𝔽₃)	semidirect
23. C₂²×C₃²	C₂²⋊A₄	semidirect
33. C₂⁴	C₂²⋊H(3,𝔽₃)	semidirect
40. C₃²	C₂²⋊(C₂²⋊A₄)	semidirect
46. C₂²	C₃²⋊(C₂²⋊A₄)	semidirect

Maximal subgroups

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

subgroup	index
C₂⁶×C₃²	3
C₃×C₂²⋊(C₂²⋊A₄)	3
C₃²⋊(C₂²⋊A₄)	4

Subgroups — C32⋊(C22⋊(C22⋊A4)) #13180117

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²⋊(C₂²⋊(C₂²⋊A₄)) #13180117