Subgroups — C₂⁴×C₂₇⋊C₃ #13048207

Order 1296.

804 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₂₇⋊C₃	C₂	1	15	yes	maximal, direct product factor
2	C₂⁴×C₃×C₉	C₃	1	1	yes	maximal, characteristic
3	C₂⁴×C₂₇	C₃	1	3	yes	maximal, complemented
4	C₂²×C₂₇⋊C₃	C₂²	1	35	yes	direct product factor
5	C₂³×C₃×C₉	C₂×C₃	1	15	yes	—
6	C₂³×C₂₇	C₂×C₃	1	45	yes	complemented
7	C₂×C₂₇⋊C₃	C₂³	1	15	yes	direct product factor
8	C₂⁴×C₃²	C₉	1	1	yes	characteristic
9	C₂⁴×C₉	C₉	1	2	yes	—
10	C₂⁴×C₉	C₃²	1	1	yes	characteristic
11	C₂²×C₃×C₉	C₂²×C₃	1	35	yes	—
12	C₂²×C₂₇	C₂²×C₃	1	105	yes	complemented
13	C₂₇⋊C₃	C₂⁴	1	1	yes	characteristic, direct product factor, Sylow
14	C₂³×C₃²	C₂×C₉	1	15	yes	—
15	C₂³×C₉	C₂×C₉	1	30	yes	—
16	C₂³×C₉	C₂×C₃²	1	15	yes	—
17	C₂×C₃×C₉	C₂³×C₃	1	15	yes	—
18	C₂×C₂₇	C₂³×C₃	1	45	yes	complemented
19	C₂⁴×C₃	—	3	1	no	—
20	C₂⁴×C₃	C₃×C₉	1	1	yes	characteristic
21	C₂²×C₃²	C₂²×C₉	1	35	yes	—
22	C₂²×C₉	C₂²×C₉	1	70	yes	—
23	C₂²×C₉	C₂²×C₃²	1	35	yes	—
24	C₃×C₉	C₂⁴×C₃	1	1	yes	characteristic
25	C₂₇	C₂⁴×C₃	1	3	yes	complemented
26	C₂³×C₃	—	3	15	no	—
27	C₂³×C₃	C₂×C₃×C₉	1	15	yes	—
28	C₂×C₃²	C₂³×C₉	1	15	yes	—
29	C₂×C₉	C₂³×C₉	1	30	yes	—
30	C₂×C₉	C₂³×C₃²	1	15	yes	—
31	C₂⁴	C₂₇⋊C₃	1	1	yes	characteristic, direct product factor, Sylow
32	C₂²×C₃	—	3	35	no	—
33	C₂²×C₃	C₂²×C₃×C₉	1	35	yes	—
34	C₃²	C₂⁴×C₉	1	1	yes	characteristic
35	C₉	C₂⁴×C₉	1	2	yes	—
36	C₉	C₂⁴×C₃²	1	1	yes	characteristic
37	C₂³	C₂×C₂₇⋊C₃	1	15	yes	direct product factor
38	C₂×C₃	—	3	15	no	—
39	C₂×C₃	C₂³×C₃×C₉	1	15	yes	—
40	C₂²	C₂²×C₂₇⋊C₃	1	35	yes	direct product factor
41	C₃	—	3	1	no	—
42	C₃	C₂⁴×C₃×C₉	1	1	yes	characteristic
43	C₂	C₂³×C₂₇⋊C₃	1	15	yes	direct product factor

	subgroup	G/H
center	C₂⁴×C₉	C₃²
commutator	C₃	C₂⁴×C₃×C₉
Frattini, cosocle	C₉	C₂⁴×C₃²
socle	C₂⁴×C₃	C₃×C₉
prime radical	C₂⁴×C₃²	C₉
LCS	G ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂³×C₂₇⋊C₃	C₂	direct (N × H)
3. C₂⁴×C₂₇	C₃	semidirect
4. C₂²×C₂₇⋊C₃	C₂²	direct (N × H)
6. C₂³×C₂₇	C₂×C₃	semidirect
7. C₂×C₂₇⋊C₃	C₂³	direct (N × H)
12. C₂²×C₂₇	C₂²×C₃	semidirect
13. C₂₇⋊C₃	C₂⁴	direct (N × H)
18. C₂×C₂₇	C₂³×C₃	semidirect
25. C₂₇	C₂⁴×C₃	semidirect
31. C₂⁴	C₂₇⋊C₃	direct (N × H)
37. C₂³	C₂×C₂₇⋊C₃	direct (N × H)
40. C₂²	C₂²×C₂₇⋊C₃	direct (N × H)
43. C₂	C₂³×C₂₇⋊C₃	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₂₇⋊C₃	2
C₂⁴×C₃×C₉	3
C₂⁴×C₂₇	3

Subgroups — C24×C27⋊C3 #13048207

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂⁴×C₂₇⋊C₃ #13048207