Subgroups — A₄×C₁₃⋊(C₅²⋊C₃) #26043686

Order 11700.

76 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²⋊(C₁₃⋊(C₅²⋊C₃))	C₃	1	2	yes	maximal, complemented
2	C₅²×A₄×C₁₃	C₃	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₁₃⋊(C₅²⋊C₃)	C₃	1	1	yes	maximal, characteristic, complemented
4	C₃×C₁₃⋊(C₅²⋊C₃)	—	4	1	no	maximal, Hall
5	C₂×C₁₃⋊(C₅²⋊C₃)	—	3	1	no	—
6	C₂²×C₅²×C₁₃	C₃²	1	1	yes	characteristic, complemented, Hall
7	C₃×C₅²×C₁₃	—	4	1	no	—
8	C₁₃⋊(C₅²⋊C₃)	—	4	2	no	—
9	C₁₃⋊(C₅²⋊C₃)	A₄	1	1	yes	characteristic, direct product factor
10	A₄×C₅²⋊C₃	—	13	1	no	maximal, Hall
11	C₅×A₄×C₁₃	—	3	2	no	—
12	C₂×C₅²×C₁₃	—	3	1	no	—
13	A₄×C₁₃⋊C₃	—	25	1	no	maximal, Hall
14	C₅²×C₁₃	C₃×A₄	1	1	yes	characteristic, complemented, Hall
15	C₂²⋊(C₅²⋊C₃)	—	13	2	no	—
16	C₅²×A₄	C₁₃⋊C₃	1	1	yes	characteristic, complemented
17	C₂²×C₅²⋊C₃	—	13	1	no	—
18	C₂²×C₅×C₁₃	—	3	2	no	—
19	C₃×C₅²⋊C₃	—	52	1	no	Hall
20	C₃×C₅×C₁₃	—	12	2	no	—
21	C₂²⋊(C₁₃⋊C₃)	—	25	2	no	—
22	A₄×C₁₃	C₅²⋊C₃	1	1	yes	characteristic, complemented
23	C₂²×C₁₃⋊C₃	—	25	1	no	—
24	C₂×C₅²⋊C₃	—	39	1	no	—
25	C₂×C₅×C₁₃	—	9	2	no	—
26	C₃×C₁₃⋊C₃	—	100	1	no	Hall
27	C₂²×C₅²	C₃×C₁₃⋊C₃	1	1	yes	characteristic, complemented, Hall
28	C₂×C₁₃⋊C₃	—	75	1	no	—
29	C₃×C₅²	—	4	1	no	—
30	C₅²⋊C₃	—	13	1	no	—
31	C₅²⋊C₃	—	52	2	no	—
32	C₅×C₁₃	—	3	2	no	—
33	C₅×A₄	—	3	2	no	—
34	C₂²×C₁₃	C₃×C₅²⋊C₃	1	1	yes	characteristic, complemented, Hall
35	C₂×C₅²	—	3	1	no	—
36	C₃×C₁₃	—	4	1	no	—
37	C₁₃⋊C₃	—	25	1	no	—
38	C₁₃⋊C₃	—	100	2	no	—
39	C₃×A₄	—	325	1	no	Hall
40	C₂×C₁₃	—	3	1	no	—
41	C₅²	A₄×C₁₃⋊C₃	1	1	yes	characteristic, complemented, Sylow
42	C₂²×C₅	—	3	2	no	—
43	C₃×C₅	—	12	2	no	—
44	C₁₃	A₄×C₅²⋊C₃	1	1	yes	characteristic, complemented, Sylow
45	C₂²×C₃	—	325	1	no	—
46	A₄	—	325	2	no	—
47	A₄	C₁₃⋊(C₅²⋊C₃)	1	1	yes	characteristic, direct product factor
48	C₂×C₅	—	9	2	no	—
49	C₃²	—	1300	1	no	Sylow
50	C₂×C₃	—	975	1	no	—
51	C₅	—	3	2	no	—
52	C₂²	C₃×C₁₃⋊(C₅²⋊C₃)	1	1	yes	characteristic, complemented, Sylow
53	C₃	—	4	1	no	—
54	C₃	—	1300	2	no	—
55	C₃	—	325	1	no	—
56	C₂	—	3	1	no	—

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₂²×C₅²×C₁₃	C₃²
LCS	G ▷ C₂²×C₅²×C₁₃
UCS	C₁

Complements

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

N	H ≅ G/N	type
1. C₂²⋊(C₁₃⋊(C₅²⋊C₃))	C₃	semidirect
2. C₅²×A₄×C₁₃	C₃	semidirect
3. C₂²×C₁₃⋊(C₅²⋊C₃)	C₃	semidirect
6. C₂²×C₅²×C₁₃	C₃²	semidirect
9. C₁₃⋊(C₅²⋊C₃)	A₄	direct (N × H)
14. C₅²×C₁₃	C₃×A₄	semidirect
16. C₅²×A₄	C₁₃⋊C₃	semidirect
22. A₄×C₁₃	C₅²⋊C₃	semidirect
27. C₂²×C₅²	C₃×C₁₃⋊C₃	semidirect
34. C₂²×C₁₃	C₃×C₅²⋊C₃	semidirect
41. C₅²	A₄×C₁₃⋊C₃	semidirect
44. C₁₃	A₄×C₅²⋊C₃	semidirect
47. A₄	C₁₃⋊(C₅²⋊C₃)	direct (N × H)
52. C₂²	C₃×C₁₃⋊(C₅²⋊C₃)	semidirect

Maximal subgroups

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

subgroup	index
C₂²⋊(C₁₃⋊(C₅²⋊C₃))	3
C₅²×A₄×C₁₃	3
C₂²×C₁₃⋊(C₅²⋊C₃)	3
C₃×C₁₃⋊(C₅²⋊C₃)	4
A₄×C₅²⋊C₃	13
A₄×C₁₃⋊C₃	25

Subgroups — A4×C13⋊(C52⋊C3) #26043686

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — A₄×C₁₃⋊(C₅²⋊C₃) #26043686