Subgroups — C₅²×A₄×C₃₁⋊C₃ #42124838

Order 27900.

224 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅²×C₃₁⋊C₃	C₃	1	1	yes	maximal, characteristic, complemented
2	C₅²×A₄×C₃₁	C₃	1	1	yes	maximal, characteristic, complemented
3	C₅²×C₂²⋊(C₃₁⋊C₃)	C₃	1	2	yes	maximal, complemented
4	C₃×C₅²×C₃₁⋊C₃	—	4	1	no	maximal, Hall
5	C₅×A₄×C₃₁⋊C₃	C₅	1	6	yes	maximal, direct product factor
6	C₂×C₅²×C₃₁⋊C₃	—	3	1	no	—
7	C₂²×C₅²×C₃₁	C₃²	1	1	yes	characteristic, complemented, Hall
8	C₅²×C₃₁⋊C₃	—	4	2	no	—
9	C₅²×C₃₁⋊C₃	A₄	1	1	yes	characteristic, direct product factor
10	C₃×C₅²×C₃₁	—	4	1	no	—
11	C₅×C₂²⋊(C₃₁⋊C₃)	C₃×C₅	1	12	yes	complemented
12	C₅×A₄×C₃₁	C₃×C₅	1	6	yes	complemented
13	C₂²×C₅×C₃₁⋊C₃	C₃×C₅	1	6	yes	complemented
14	C₂×C₅²×C₃₁	—	3	1	no	—
15	C₃×C₅×C₃₁⋊C₃	—	4	6	no	—
16	A₄×C₃₁⋊C₃	C₅²	1	1	yes	characteristic, direct product factor, Hall
17	C₂×C₅×C₃₁⋊C₃	—	3	6	no	—
18	C₃×C₅²×A₄	—	31	1	no	maximal, Hall
19	C₅²×C₃₁	C₃×A₄	1	1	yes	characteristic, complemented, Hall
20	C₂²×C₅×C₃₁	C₃²×C₅	1	6	yes	complemented
21	C₃×C₅×C₃₁	—	4	6	no	—
22	C₅×C₃₁⋊C₃	—	4	12	no	—
23	C₅×C₃₁⋊C₃	C₅×A₄	1	6	yes	direct product factor
24	C₂²⋊(C₃₁⋊C₃)	C₃×C₅²	1	2	yes	complemented
25	A₄×C₃₁	C₃×C₅²	1	1	yes	characteristic, complemented
26	C₂²×C₃₁⋊C₃	C₃×C₅²	1	1	yes	characteristic, complemented
27	C₂×C₅×C₃₁	—	3	6	no	—
28	C₂²×C₃×C₅²	—	31	1	no	—
29	C₅²×A₄	—	31	2	no	—
30	C₅²×A₄	C₃₁⋊C₃	1	1	yes	characteristic, direct product factor
31	C₃×C₃₁⋊C₃	—	4	1	no	Hall
32	C₃²×C₅²	—	124	1	no	Hall
33	C₂×C₃₁⋊C₃	—	3	1	no	—
34	C₃×C₅×A₄	—	31	6	no	—
35	C₅×C₃₁	C₃×C₅×A₄	1	6	yes	complemented
36	C₂×C₃×C₅²	—	93	1	no	—
37	C₂²×C₃₁	C₃²×C₅²	1	1	yes	characteristic, complemented, Hall
38	C₂²×C₅²	C₃×C₃₁⋊C₃	1	1	yes	characteristic, complemented, Hall
39	C₃×C₃₁	—	4	1	no	—
40	C₃₁⋊C₃	—	4	2	no	—
41	C₃₁⋊C₃	C₅²×A₄	1	1	yes	characteristic, direct product factor
42	C₃×C₅²	—	4	1	no	—
43	C₃×C₅²	—	31	1	no	—
44	C₃×C₅²	—	124	2	no	—
45	C₂×C₃₁	—	3	1	no	—
46	C₂²×C₃×C₅	—	31	6	no	—
47	C₅×A₄	—	31	12	no	—
48	C₅×A₄	C₅×C₃₁⋊C₃	1	6	yes	direct product factor
49	C₂×C₅²	—	3	1	no	—
50	C₃²×C₅	—	124	6	no	—
51	C₃×A₄	—	31	1	no	Hall
52	C₃₁	C₃×C₅²×A₄	1	1	yes	characteristic, complemented, Sylow
53	C₂×C₃×C₅	—	93	6	no	—
54	C₅²	A₄×C₃₁⋊C₃	1	1	yes	characteristic, direct product factor, Sylow
55	C₂²×C₅	C₃×C₅×C₃₁⋊C₃	1	6	yes	complemented
56	C₃×C₅	—	4	6	no	—
57	C₃×C₅	—	31	6	no	—
58	C₃×C₅	—	124	12	no	—
59	C₂²×C₃	—	31	1	no	—
60	A₄	—	31	2	no	—
61	A₄	C₅²×C₃₁⋊C₃	1	1	yes	characteristic, direct product factor
62	C₂×C₅	—	3	6	no	—
63	C₃²	—	124	1	no	Sylow
64	C₂×C₃	—	93	1	no	—
65	C₅	C₅×A₄×C₃₁⋊C₃	1	6	yes	direct product factor
66	C₂²	C₃×C₅²×C₃₁⋊C₃	1	1	yes	characteristic, complemented, Sylow
67	C₃	—	4	1	no	—
68	C₃	—	31	1	no	—
69	C₃	—	124	2	no	—
70	C₂	—	3	1	no	—

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

Complements

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

N	H ≅ G/N	type
1. C₂²×C₅²×C₃₁⋊C₃	C₃	semidirect
2. C₅²×A₄×C₃₁	C₃	semidirect
3. C₅²×C₂²⋊(C₃₁⋊C₃)	C₃	semidirect
5. C₅×A₄×C₃₁⋊C₃	C₅	direct (N × H)
7. C₂²×C₅²×C₃₁	C₃²	semidirect
9. C₅²×C₃₁⋊C₃	A₄	direct (N × H)
11. C₅×C₂²⋊(C₃₁⋊C₃)	C₃×C₅	semidirect
12. C₅×A₄×C₃₁	C₃×C₅	semidirect
13. C₂²×C₅×C₃₁⋊C₃	C₃×C₅	semidirect
16. A₄×C₃₁⋊C₃	C₅²	direct (N × H)
19. C₅²×C₃₁	C₃×A₄	semidirect
20. C₂²×C₅×C₃₁	C₃²×C₅	semidirect
23. C₅×C₃₁⋊C₃	C₅×A₄	direct (N × H)
24. C₂²⋊(C₃₁⋊C₃)	C₃×C₅²	semidirect
25. A₄×C₃₁	C₃×C₅²	semidirect
26. C₂²×C₃₁⋊C₃	C₃×C₅²	semidirect
30. C₅²×A₄	C₃₁⋊C₃	direct (N × H)
35. C₅×C₃₁	C₃×C₅×A₄	semidirect
37. C₂²×C₃₁	C₃²×C₅²	semidirect
38. C₂²×C₅²	C₃×C₃₁⋊C₃	semidirect
41. C₃₁⋊C₃	C₅²×A₄	direct (N × H)
48. C₅×A₄	C₅×C₃₁⋊C₃	direct (N × H)
52. C₃₁	C₃×C₅²×A₄	semidirect
54. C₅²	A₄×C₃₁⋊C₃	direct (N × H)
55. C₂²×C₅	C₃×C₅×C₃₁⋊C₃	semidirect
61. A₄	C₅²×C₃₁⋊C₃	direct (N × H)
65. C₅	C₅×A₄×C₃₁⋊C₃	direct (N × H)
66. C₂²	C₃×C₅²×C₃₁⋊C₃	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₅²×C₃₁⋊C₃	3
C₅²×A₄×C₃₁	3
C₅²×C₂²⋊(C₃₁⋊C₃)	3
C₃×C₅²×C₃₁⋊C₃	4
C₅×A₄×C₃₁⋊C₃	5
C₃×C₅²×A₄	31

Subgroups — C52×A4×C31⋊C3 #42124838

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅²×A₄×C₃₁⋊C₃ #42124838