Subgroups — C₁₁²×C₃⋊#1044 #49137806

Order 36300.

448 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₁₁²×C₅⋊D₅	C₂	1	1	yes	maximal, characteristic
2	C₁₁²×#1044	—	3	1	no	maximal, Hall
3	C₃×C₅²×C₁₁²	C₄	1	1	yes	characteristic, complemented, Hall
4	C₁₁²×ΓL(1,𝔽₁₆)	—	5	2	no	maximal
5	C₁₁²×C₅⋊D₅	D₃	1	1	yes	characteristic
6	C₃×D₅×C₁₁²	—	5	2	no	—
7	C₃×D₅×C₁₁²	—	10	2	no	—
8	C₁₁×C₃⋊#1044	C₁₁	1	12	yes	maximal, direct product factor
9	C₅²×C₁₁²	Dic₃	1	1	yes	characteristic, complemented, Hall
10	C₁₁²×AGL(1,𝔽₅)	—	15	2	no	—
11	C₃×C₅×C₁₁²	—	2	2	no	—
12	C₃×C₅×C₁₁²	AGL(1,𝔽₅)	1	2	yes	complemented
13	C₃×C₁₁×C₅⋊D₅	C₂×C₁₁	1	12	yes	—
14	C₁₁²×Dic₃	—	25	1	no	Hall
15	D₅×C₁₁²	—	5	2	no	—
16	D₅×C₁₁²	—	10	2	no	—
17	C₁₁×#1044	—	3	12	no	—
18	C₃×C₅²×C₁₁	C₄×C₁₁	1	12	yes	complemented
19	C₂×C₃×C₁₁²	—	25	1	no	—
20	C₁₁×ΓL(1,𝔽₁₆)	—	5	24	no	—
21	C₅×C₁₁²	—	2	2	no	—
22	C₅×C₁₁²	ΓL(1,𝔽₁₆)	1	2	yes	complemented
23	C₁₁×C₅⋊D₅	D₃×C₁₁	1	12	yes	—
24	C₄×C₁₁²	—	75	1	no	Hall
25	C₃×C₁₁²	#1044	1	1	yes	characteristic, complemented, Hall
26	C₃×D₅×C₁₁	—	5	24	no	—
27	C₃×D₅×C₁₁	—	10	24	no	—
28	C₃⋊#1044	C₁₁²	1	1	yes	characteristic, direct product factor, Hall
29	C₅²×C₁₁	C₁₁×Dic₃	1	12	yes	complemented
30	C₂×C₁₁²	—	25	1	no	—
31	C₁₁×AGL(1,𝔽₅)	—	15	24	no	—
32	C₃×C₅×C₁₁	—	2	24	no	—
33	C₃×C₅×C₁₁	C₁₁×AGL(1,𝔽₅)	1	24	yes	complemented
34	C₃×C₅⋊D₅	C₂×C₁₁²	1	1	yes	characteristic
35	C₁₁×Dic₃	—	25	12	no	—
36	C₁₁²	C₃⋊#1044	1	1	yes	characteristic, direct product factor, Sylow
37	D₅×C₁₁	—	5	24	no	—
38	D₅×C₁₁	—	10	24	no	—
39	#1044	—	3	1	no	Hall
40	C₃×C₅²	C₄×C₁₁²	1	1	yes	characteristic, complemented, Hall
41	C₂×C₃×C₁₁	—	25	12	no	—
42	ΓL(1,𝔽₁₆)	—	5	2	no	—
43	C₅×C₁₁	—	2	24	no	—
44	C₅×C₁₁	C₁₁×ΓL(1,𝔽₁₆)	1	24	yes	complemented
45	C₅⋊D₅	D₃×C₁₁²	1	1	yes	characteristic
46	C₄×C₁₁	—	75	12	no	—
47	C₃×C₁₁	C₁₁×#1044	1	12	yes	complemented
48	C₃×D₅	—	5	2	no	—
49	C₃×D₅	—	10	2	no	—
50	C₅²	C₁₁²×Dic₃	1	1	yes	characteristic, complemented, Sylow
51	C₂×C₁₁	—	25	12	no	—
52	AGL(1,𝔽₅)	—	15	2	no	—
53	C₃×C₅	—	2	2	no	—
54	C₃×C₅	C₁₁²×AGL(1,𝔽₅)	1	2	yes	complemented
55	Dic₃	—	25	1	no	Hall
56	C₁₁	C₁₁×C₃⋊#1044	1	12	yes	direct product factor
57	D₅	—	5	2	no	—
58	D₅	—	10	2	no	—
59	C₂×C₃	—	25	1	no	—
60	C₅	—	2	2	no	—
61	C₅	C₁₁²×ΓL(1,𝔽₁₆)	1	2	yes	complemented
62	C₄	—	75	1	no	Sylow
63	C₃	C₁₁²×#1044	1	1	yes	characteristic, complemented, Sylow
64	C₂	—	25	1	no	—

	subgroup	G/H
center, hypercenter	C₁₁²	C₃⋊#1044
commutator, hypocenter	C₃×C₅²	C₄×C₁₁²
Frattini	C₁	G
Fitting, socle	C₃×C₅²×C₁₁²	C₄
prime radical	C₃×C₁₁²×C₅⋊D₅	C₂
cosocle	C₃×C₅⋊D₅	C₂×C₁₁²
LCS	G ▷ C₃×C₅²
UCS	C₁ ◁ C₁₁²

Complements

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

N	H ≅ G/N	type
3. C₃×C₅²×C₁₁²	C₄	semidirect
8. C₁₁×C₃⋊#1044	C₁₁	direct (N × H)
9. C₅²×C₁₁²	Dic₃	semidirect
12. C₃×C₅×C₁₁²	AGL(1,𝔽₅)	semidirect
18. C₃×C₅²×C₁₁	C₄×C₁₁	semidirect
22. C₅×C₁₁²	ΓL(1,𝔽₁₆)	semidirect
25. C₃×C₁₁²	#1044	semidirect
28. C₃⋊#1044	C₁₁²	direct (N × H)
29. C₅²×C₁₁	C₁₁×Dic₃	semidirect
33. C₃×C₅×C₁₁	C₁₁×AGL(1,𝔽₅)	semidirect
36. C₁₁²	C₃⋊#1044	direct (N × H)
40. C₃×C₅²	C₄×C₁₁²	semidirect
44. C₅×C₁₁	C₁₁×ΓL(1,𝔽₁₆)	semidirect
47. C₃×C₁₁	C₁₁×#1044	semidirect
50. C₅²	C₁₁²×Dic₃	semidirect
54. C₃×C₅	C₁₁²×AGL(1,𝔽₅)	semidirect
56. C₁₁	C₁₁×C₃⋊#1044	direct (N × H)
61. C₅	C₁₁²×ΓL(1,𝔽₁₆)	semidirect
63. C₃	C₁₁²×#1044	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₁₁²×C₅⋊D₅	2
C₁₁²×#1044	3
C₁₁²×ΓL(1,𝔽₁₆)	5
C₁₁×C₃⋊#1044	11

Subgroups — C112×C3⋊#1044 #49137806

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₁²×C₃⋊#1044 #49137806