Subgroups — A₄×C₃₁×C₅²⋊C₃ #42124834

Order 27900.

76 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃₁×C₅²⋊C₃	C₃	1	1	yes	maximal, characteristic, complemented
2	C₅²×A₄×C₃₁	C₃	1	1	yes	maximal, characteristic, complemented
3	C₃₁×C₂²⋊(C₅²⋊C₃)	C₃	1	2	yes	maximal, complemented
4	C₃×C₃₁×C₅²⋊C₃	—	4	1	no	maximal, Hall
5	C₂×C₃₁×C₅²⋊C₃	—	3	1	no	—
6	C₂²×C₅²×C₃₁	C₃²	1	1	yes	characteristic, complemented, Hall
7	C₃₁×C₅²⋊C₃	—	4	2	no	—
8	C₃₁×C₅²⋊C₃	A₄	1	1	yes	characteristic, direct product factor
9	C₃×C₅²×C₃₁	—	4	1	no	—
10	C₅×A₄×C₃₁	—	3	2	no	—
11	C₂×C₅²×C₃₁	—	3	1	no	—
12	C₃×A₄×C₃₁	—	25	1	no	maximal, Hall
13	A₄×C₅²⋊C₃	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Hall
14	C₅²×C₃₁	C₃×A₄	1	1	yes	characteristic, complemented, Hall
15	C₂²×C₅×C₃₁	—	3	2	no	—
16	C₃×C₅×C₃₁	—	12	2	no	—
17	C₂²×C₃×C₃₁	—	25	1	no	—
18	A₄×C₃₁	—	25	2	no	—
19	A₄×C₃₁	C₅²⋊C₃	1	1	yes	characteristic, direct product factor
20	C₂×C₅×C₃₁	—	9	2	no	—
21	C₂²⋊(C₅²⋊C₃)	C₃×C₃₁	1	2	yes	complemented
22	C₅²×A₄	C₃×C₃₁	1	1	yes	characteristic, complemented
23	C₂²×C₅²⋊C₃	C₃×C₃₁	1	1	yes	characteristic, complemented
24	C₃²×C₃₁	—	100	1	no	Hall
25	C₃×C₅²⋊C₃	—	4	1	no	Hall
26	C₂×C₃×C₃₁	—	75	1	no	—
27	C₅×C₃₁	—	3	2	no	—
28	C₂×C₅²⋊C₃	—	3	1	no	—
29	C₂²×C₃₁	C₃×C₅²⋊C₃	1	1	yes	characteristic, complemented, Hall
30	C₂²×C₅²	C₃²×C₃₁	1	1	yes	characteristic, complemented, Hall
31	C₃×C₃₁	—	4	1	no	—
32	C₃×C₃₁	—	25	1	no	—
33	C₃×C₃₁	—	100	2	no	—
34	C₃×C₅²	—	4	1	no	—
35	C₅²⋊C₃	—	4	2	no	—
36	C₅²⋊C₃	A₄×C₃₁	1	1	yes	characteristic, direct product factor
37	C₂×C₃₁	—	3	1	no	—
38	C₅×A₄	—	3	2	no	—
39	C₂×C₅²	—	3	1	no	—
40	C₃×A₄	—	25	1	no	Hall
41	C₃₁	A₄×C₅²⋊C₃	1	1	yes	characteristic, direct product factor, Sylow
42	C₅²	C₃×A₄×C₃₁	1	1	yes	characteristic, complemented, Sylow
43	C₂²×C₅	—	3	2	no	—
44	C₃×C₅	—	12	2	no	—
45	C₂²×C₃	—	25	1	no	—
46	A₄	—	25	2	no	—
47	A₄	C₃₁×C₅²⋊C₃	1	1	yes	characteristic, direct product factor
48	C₂×C₅	—	9	2	no	—
49	C₃²	—	100	1	no	Sylow
50	C₂×C₃	—	75	1	no	—
51	C₅	—	3	2	no	—
52	C₂²	C₃×C₃₁×C₅²⋊C₃	1	1	yes	characteristic, complemented, Sylow
53	C₃	—	4	1	no	—
54	C₃	—	25	1	no	—
55	C₃	—	100	2	no	—
56	C₂	—	3	1	no	—

	subgroup	G/H
center, hypercenter	C₃₁	A₄×C₅²⋊C₃
commutator, cosocle, hypocenter	C₂²×C₅²	C₃²×C₃₁
Frattini	C₁	G
Fitting, socle	C₂²×C₅²×C₃₁	C₃²
LCS	G ▷ C₂²×C₅²
UCS	C₁ ◁ C₃₁

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃₁×C₅²⋊C₃	C₃	semidirect
2. C₅²×A₄×C₃₁	C₃	semidirect
3. C₃₁×C₂²⋊(C₅²⋊C₃)	C₃	semidirect
6. C₂²×C₅²×C₃₁	C₃²	semidirect
8. C₃₁×C₅²⋊C₃	A₄	direct (N × H)
13. A₄×C₅²⋊C₃	C₃₁	direct (N × H)
14. C₅²×C₃₁	C₃×A₄	semidirect
19. A₄×C₃₁	C₅²⋊C₃	direct (N × H)
21. C₂²⋊(C₅²⋊C₃)	C₃×C₃₁	semidirect
22. C₅²×A₄	C₃×C₃₁	semidirect
23. C₂²×C₅²⋊C₃	C₃×C₃₁	semidirect
29. C₂²×C₃₁	C₃×C₅²⋊C₃	semidirect
30. C₂²×C₅²	C₃²×C₃₁	semidirect
36. C₅²⋊C₃	A₄×C₃₁	direct (N × H)
41. C₃₁	A₄×C₅²⋊C₃	direct (N × H)
42. C₅²	C₃×A₄×C₃₁	semidirect
47. A₄	C₃₁×C₅²⋊C₃	direct (N × H)
52. C₂²	C₃×C₃₁×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×C₃₁×C₅²⋊C₃	3
C₅²×A₄×C₃₁	3
C₃₁×C₂²⋊(C₅²⋊C₃)	3
C₃×C₃₁×C₅²⋊C₃	4
C₃×A₄×C₃₁	25
A₄×C₅²⋊C₃	31

Subgroups — A4×C31×C52⋊C3 #42124834

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — A₄×C₃₁×C₅²⋊C₃ #42124834