Subgroups — C₂⁴×#733 #13048211

Order 1296.

871 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#733	C₂	1	15	yes	maximal, direct product factor
2	C₂⁴×3₋¹⁺²	C₃	1	3	yes	maximal
3	C₂⁴×C₃×C₉	C₃	1	1	yes	maximal, characteristic
4	C₂²×#733	C₂²	1	35	yes	direct product factor
5	C₂³×3₋¹⁺²	C₂×C₃	1	45	yes	—
6	C₂³×C₃×C₉	C₂×C₃	1	15	yes	—
7	C₂×#733	C₂³	1	15	yes	direct product factor
8	C₂⁴×C₃²	C₃²	1	1	yes	characteristic
9	C₂⁴×C₉	—	3	1	no	—
10	C₂⁴×C₉	—	3	3	no	—
11	C₂²×3₋¹⁺²	C₂²×C₃	1	105	yes	—
12	C₂²×C₃×C₉	C₂²×C₃	1	35	yes	—
13	#733	C₂⁴	1	1	yes	characteristic, direct product factor, Sylow
14	C₂³×C₃²	C₂×C₃²	1	15	yes	—
15	C₂³×C₉	—	3	15	no	—
16	C₂³×C₉	—	3	45	no	—
17	C₂×3₋¹⁺²	C₂³×C₃	1	45	yes	—
18	C₂×C₃×C₉	C₂³×C₃	1	15	yes	—
19	C₂⁴×C₃	—	3	1	no	—
20	C₂⁴×C₃	H(3,𝔽₃)	1	1	yes	characteristic
21	C₂²×C₃²	C₂²×C₃²	1	35	yes	—
22	C₂²×C₉	—	3	35	no	—
23	C₂²×C₉	—	3	105	no	—
24	3₋¹⁺²	C₂⁴×C₃	1	3	yes	—
25	C₃×C₉	C₂⁴×C₃	1	1	yes	characteristic
26	C₂³×C₃	—	3	15	no	—
27	C₂³×C₃	C₂×H(3,𝔽₃)	1	15	yes	—
28	C₂×C₃²	C₂³×C₃²	1	15	yes	—
29	C₂×C₉	—	3	15	no	—
30	C₂×C₉	—	3	45	no	—
31	C₂⁴	#733	1	1	yes	characteristic, direct product factor, Sylow
32	C₂²×C₃	—	3	35	no	—
33	C₂²×C₃	C₂²×H(3,𝔽₃)	1	35	yes	—
34	C₃²	C₂⁴×C₃²	1	1	yes	characteristic
35	C₉	—	3	1	no	—
36	C₉	—	3	3	no	—
37	C₂³	C₂×#733	1	15	yes	direct product factor
38	C₂×C₃	—	3	15	no	—
39	C₂×C₃	C₂³×H(3,𝔽₃)	1	15	yes	—
40	C₂²	C₂²×#733	1	35	yes	direct product factor
41	C₃	—	3	1	no	—
42	C₃	C₂⁴×H(3,𝔽₃)	1	1	yes	characteristic
43	C₂	C₂³×#733	1	15	yes	direct product factor

	subgroup	G/H
center, socle	C₂⁴×C₃	H(3,𝔽₃)
commutator, Frattini, cosocle	C₃²	C₂⁴×C₃²
prime radical	C₂⁴×C₃²	C₃²
LCS	G ▷ C₃² ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₃ ◁ C₂⁴×C₃² ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂³×#733	C₂	direct (N × H)
4. C₂²×#733	C₂²	direct (N × H)
7. C₂×#733	C₂³	direct (N × H)
13. #733	C₂⁴	direct (N × H)
31. C₂⁴	#733	direct (N × H)
37. C₂³	C₂×#733	direct (N × H)
40. C₂²	C₂²×#733	direct (N × H)
43. C₂	C₂³×#733	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×#733	2
C₂⁴×3₋¹⁺²	3
C₂⁴×C₃×C₉	3

Subgroups — C24×#733 #13048211

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂⁴×#733 #13048211