Subgroups — C₂⁴×AGL(1,𝔽₉) #11918429

Order 1152.

2350 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×#157	C₂	1	1	yes	maximal, characteristic
2	C₂³×AGL(1,𝔽₉)	C₂	1	30	yes	maximal, direct product factor
3	C₂⁴×C₃⋊D₃	C₄	1	1	yes	characteristic
4	C₂³×#157	C₄	1	15	yes	—
5	C₂³×#157	C₂²	1	15	yes	—
6	C₂²×AGL(1,𝔽₉)	C₂²	1	140	yes	direct product factor
7	C₂⁴×C₃²	C₈	1	1	yes	characteristic, complemented
8	C₂³×C₃⋊D₃	C₈	1	15	yes	complemented
9	C₂³×C₃⋊D₃	C₂×C₄	1	15	yes	—
10	C₂²×#157	C₂×C₄	1	105	yes	—
11	C₂²×#157	C₂³	1	35	yes	—
12	C₂×AGL(1,𝔽₉)	C₂³	1	120	yes	direct product factor
13	C₂⁴×C₈	—	9	1	no	maximal, Sylow
14	C₂⁴×D₃	—	12	1	no	—
15	C₂³×C₃²	C₂×C₈	1	15	yes	complemented
16	C₂²×C₃⋊D₃	C₂×C₈	1	105	yes	complemented
17	C₂²×C₃⋊D₃	C₂²×C₄	1	35	yes	—
18	C₂×#157	C₂²×C₄	1	105	yes	—
19	C₂×#157	C₂⁴	1	15	yes	—
20	AGL(1,𝔽₉)	C₂⁴	1	16	yes	direct product factor
21	C₂⁴×C₄	—	9	1	no	—
22	C₂³×C₈	—	9	30	no	—
23	C₂⁴×C₃	—	4	1	no	—
24	C₂³×D₃	—	12	15	no	—
25	C₂³×D₃	—	12	15	no	—
26	C₂²×C₃²	C₂²×C₈	1	35	yes	complemented
27	C₂×C₃⋊D₃	C₂²×C₈	1	105	yes	complemented
28	C₂×C₃⋊D₃	C₂³×C₄	1	15	yes	—
29	#157	C₂³×C₄	1	15	yes	—
30	#157	C₂⁵	1	1	yes	characteristic
31	C₂⁵	—	9	1	no	—
32	C₂³×C₄	—	9	15	no	—
33	C₂³×C₄	—	9	15	no	—
34	C₂²×C₈	—	9	140	no	—
35	C₂³×C₃	—	4	15	no	—
36	C₂²×D₃	—	12	105	no	—
37	C₂²×D₃	—	12	35	no	—
38	C₂×C₃²	C₂³×C₈	1	15	yes	complemented
39	C₃⋊D₃	C₂³×C₈	1	15	yes	complemented
40	C₃⋊D₃	C₂⁴×C₄	1	1	yes	characteristic
41	C₂⁴	—	9	15	no	—
42	C₂⁴	—	9	15	no	—
43	C₂⁴	AGL(1,𝔽₉)	1	1	yes	characteristic, direct product factor
44	C₂²×C₄	—	9	105	no	—
45	C₂²×C₄	—	9	35	no	—
46	C₂×C₈	—	9	120	no	—
47	C₂²×C₃	—	4	35	no	—
48	C₂×D₃	—	12	105	no	—
49	C₂×D₃	—	12	15	no	—
50	C₃²	C₂⁴×C₈	1	1	yes	characteristic, complemented, Sylow
51	C₂³	—	9	35	no	—
52	C₂³	—	9	105	no	—
53	C₂³	C₂×AGL(1,𝔽₉)	1	15	yes	direct product factor
54	C₂×C₄	—	9	15	no	—
55	C₂×C₄	—	9	105	no	—
56	C₈	—	9	16	no	—
57	C₂×C₃	—	4	15	no	—
58	D₃	—	12	15	no	—
59	D₃	—	12	1	no	—
60	C₂²	—	9	15	no	—
61	C₂²	—	9	105	no	—
62	C₂²	C₂²×AGL(1,𝔽₉)	1	35	yes	direct product factor
63	C₄	—	9	1	no	—
64	C₄	—	9	15	no	—
65	C₃	—	4	1	no	—
66	C₂	—	9	1	no	—
67	C₂	—	9	15	no	—
68	C₂	C₂³×AGL(1,𝔽₉)	1	15	yes	direct product factor

	subgroup	G/H
center, hypercenter	C₂⁴	AGL(1,𝔽₉)
commutator, hypocenter	C₃²	C₂⁴×C₈
Frattini	C₁	G
Fitting, socle	C₂⁴×C₃²	C₈
prime radical	C₂⁴×C₃⋊D₃	C₄
cosocle	#157	C₂⁵
LCS	G ▷ C₃²
UCS	C₁ ◁ C₂⁴

Complements

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

N	H ≅ G/N	type
2. C₂³×AGL(1,𝔽₉)	C₂	direct (N × H)
6. C₂²×AGL(1,𝔽₉)	C₂²	direct (N × H)
7. C₂⁴×C₃²	C₈	semidirect
8. C₂³×C₃⋊D₃	C₈	semidirect
12. C₂×AGL(1,𝔽₉)	C₂³	direct (N × H)
15. C₂³×C₃²	C₂×C₈	semidirect
16. C₂²×C₃⋊D₃	C₂×C₈	semidirect
20. AGL(1,𝔽₉)	C₂⁴	direct (N × H)
26. C₂²×C₃²	C₂²×C₈	semidirect
27. C₂×C₃⋊D₃	C₂²×C₈	semidirect
38. C₂×C₃²	C₂³×C₈	semidirect
39. C₃⋊D₃	C₂³×C₈	semidirect
43. C₂⁴	AGL(1,𝔽₉)	direct (N × H)
50. C₃²	C₂⁴×C₈	semidirect
53. C₂³	C₂×AGL(1,𝔽₉)	direct (N × H)
62. C₂²	C₂²×AGL(1,𝔽₉)	direct (N × H)
68. C₂	C₂³×AGL(1,𝔽₉)	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×#157	2
C₂³×AGL(1,𝔽₉)	2
C₂⁴×C₈	9

Subgroups — C24×AGL(1,𝔽9) #11918429

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂⁴×AGL(1,𝔽₉) #11918429