Subgroups — C₅²×C₃⋊D₃×D₃₁ #42124947

Order 27900.

480 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₅²×D₃₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅²×C₃⋊D₃×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅²×C₃⋊D₉₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅²×D₃×D₃₁	—	3	4	no	maximal
5	C₃²×C₅²×C₃₁	C₂²	1	1	yes	characteristic, complemented, Hall
6	C₅×C₃⋊D₃×D₃₁	C₅	1	6	yes	maximal, direct product factor
7	C₅²×D₉₃	—	3	4	no	—
8	C₅²×D₃×C₃₁	—	3	4	no	—
9	C₃×C₅²×D₃₁	D₃	1	4	yes	complemented
10	C₂×C₅²×D₃₁	—	9	1	no	Hall
11	C₅×C₃⋊D₉₃	C₂×C₅	1	6	yes	complemented
12	C₅×C₃⋊D₃×C₃₁	C₂×C₅	1	6	yes	complemented
13	C₃²×C₅×D₃₁	C₂×C₅	1	6	yes	complemented
14	C₃×C₅²×C₃₁	C₂×D₃	1	4	yes	complemented
15	C₅×D₃×D₃₁	—	3	24	no	—
16	C₂×C₅²×C₃₁	—	9	1	no	—
17	C₅²×D₃₁	—	9	1	no	—
18	C₅²×D₃₁	C₃⋊D₃	1	1	yes	characteristic, direct product factor
19	C₃²×C₅×C₃₁	C₂²×C₅	1	6	yes	complemented
20	C₃⋊D₃×D₃₁	C₅²	1	1	yes	characteristic, direct product factor, Hall
21	C₅×D₉₃	—	3	24	no	—
22	C₅×D₃×C₃₁	—	3	24	no	—
23	C₃×C₅×D₃₁	C₅×D₃	1	24	yes	complemented
24	C₂×C₅²×C₃⋊D₃	—	31	1	no	maximal, Hall
25	C₅²×C₃₁	C₂×C₃⋊D₃	1	1	yes	characteristic, complemented, Hall
26	C₂×C₅×D₃₁	—	9	6	no	—
27	C₃⋊D₉₃	C₂×C₅²	1	1	yes	characteristic, complemented
28	C₃⋊D₃×C₃₁	C₂×C₅²	1	1	yes	characteristic, complemented
29	C₃²×D₃₁	C₂×C₅²	1	1	yes	characteristic, complemented
30	C₃×C₅×C₃₁	C₂×C₅×D₃	1	24	yes	complemented
31	C₂×C₃²×C₅²	—	31	1	no	—
32	C₅²×C₃⋊D₃	—	31	1	no	—
33	C₅²×C₃⋊D₃	D₃₁	1	1	yes	characteristic, direct product factor
34	D₃×D₃₁	—	3	4	no	—
35	C₂×C₅×C₃₁	—	9	6	no	—
36	C₅×D₃₁	—	9	6	no	—
37	C₅×D₃₁	C₅×C₃⋊D₃	1	6	yes	direct product factor
38	C₂×C₅²×D₃	—	93	4	no	—
39	C₃²×C₃₁	C₂²×C₅²	1	1	yes	characteristic, complemented, Hall
40	C₃²×C₅²	C₂×D₃₁	1	1	yes	characteristic, complemented, Hall
41	D₉₃	—	3	4	no	—
42	C₃×D₃₁	C₅²×D₃	1	4	yes	complemented
43	D₃×C₃₁	—	3	4	no	—
44	C₂×C₅×C₃⋊D₃	—	31	6	no	—
45	C₅×C₃₁	C₂×C₅×C₃⋊D₃	1	6	yes	complemented
46	C₂×C₃×C₅²	—	31	4	no	—
47	C₅²×D₃	—	3	4	no	—
48	C₅²×D₃	—	93	4	no	—
49	C₂×D₃₁	—	9	1	no	Hall
50	C₂²×C₅²	—	279	1	no	Hall
51	C₃×C₃₁	C₂×C₅²×D₃	1	4	yes	complemented
52	C₂×C₃²×C₅	—	31	6	no	—
53	C₅×C₃⋊D₃	—	31	6	no	—
54	C₅×C₃⋊D₃	C₅×D₃₁	1	6	yes	direct product factor
55	C₃×C₅²	D₃×D₃₁	1	4	yes	complemented
56	C₂×C₃₁	—	9	1	no	—
57	D₃₁	—	9	1	no	—
58	D₃₁	C₅²×C₃⋊D₃	1	1	yes	characteristic, direct product factor
59	C₂×C₅×D₃	—	93	24	no	—
60	C₂×C₅²	—	9	1	no	—
61	C₂×C₅²	—	279	1	no	—
62	C₂×C₅²	—	31	1	no	—
63	C₃²×C₅	C₂×C₅×D₃₁	1	6	yes	complemented
64	C₂×C₃⋊D₃	—	31	1	no	Hall
65	C₃₁	C₂×C₅²×C₃⋊D₃	1	1	yes	characteristic, complemented, Sylow
66	C₂×C₃×C₅	—	31	24	no	—
67	C₅×D₃	—	3	24	no	—
68	C₅×D₃	—	93	24	no	—
69	C₅²	C₃⋊D₃×D₃₁	1	1	yes	characteristic, direct product factor, Sylow
70	C₂²×C₅	—	279	6	no	—
71	C₂×C₃²	—	31	1	no	—
72	C₃⋊D₃	—	31	1	no	—
73	C₃⋊D₃	C₅²×D₃₁	1	1	yes	characteristic, direct product factor
74	C₃×C₅	C₅×D₃×D₃₁	1	24	yes	complemented
75	C₂×D₃	—	93	4	no	—
76	C₂×C₅	—	9	6	no	—
77	C₂×C₅	—	279	6	no	—
78	C₂×C₅	—	31	6	no	—
79	C₃²	C₂×C₅²×D₃₁	1	1	yes	characteristic, complemented, Sylow
80	C₂×C₃	—	31	4	no	—
81	D₃	—	3	4	no	—
82	D₃	—	93	4	no	—
83	C₅	C₅×C₃⋊D₃×D₃₁	1	6	yes	direct product factor
84	C₂²	—	279	1	no	Sylow
85	C₃	C₅²×D₃×D₃₁	1	4	yes	complemented
86	C₂	—	9	1	no	—
87	C₂	—	279	1	no	—
88	C₂	—	31	1	no	—

	subgroup	G/H
center, hypercenter	C₅²	C₃⋊D₃×D₃₁
commutator, cosocle, hypocenter	C₃²×C₃₁	C₂²×C₅²
Frattini	C₁	G
Fitting, socle	C₃²×C₅²×C₃₁	C₂²
LCS	G ▷ C₃²×C₃₁
UCS	C₁ ◁ C₅²

Complements

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

N	H ≅ G/N	type
1. C₃²×C₅²×D₃₁	C₂	semidirect
2. C₅²×C₃⋊D₃×C₃₁	C₂	semidirect
3. C₅²×C₃⋊D₉₃	C₂	semidirect
5. C₃²×C₅²×C₃₁	C₂²	semidirect
6. C₅×C₃⋊D₃×D₃₁	C₅	direct (N × H)
9. C₃×C₅²×D₃₁	D₃	semidirect
11. C₅×C₃⋊D₉₃	C₂×C₅	semidirect
12. C₅×C₃⋊D₃×C₃₁	C₂×C₅	semidirect
13. C₃²×C₅×D₃₁	C₂×C₅	semidirect
14. C₃×C₅²×C₃₁	C₂×D₃	semidirect
18. C₅²×D₃₁	C₃⋊D₃	direct (N × H)
19. C₃²×C₅×C₃₁	C₂²×C₅	semidirect
20. C₃⋊D₃×D₃₁	C₅²	direct (N × H)
23. C₃×C₅×D₃₁	C₅×D₃	semidirect
25. C₅²×C₃₁	C₂×C₃⋊D₃	semidirect
27. C₃⋊D₉₃	C₂×C₅²	semidirect
28. C₃⋊D₃×C₃₁	C₂×C₅²	semidirect
29. C₃²×D₃₁	C₂×C₅²	semidirect
30. C₃×C₅×C₃₁	C₂×C₅×D₃	semidirect
33. C₅²×C₃⋊D₃	D₃₁	direct (N × H)
37. C₅×D₃₁	C₅×C₃⋊D₃	direct (N × H)
39. C₃²×C₃₁	C₂²×C₅²	semidirect
40. C₃²×C₅²	C₂×D₃₁	semidirect
42. C₃×D₃₁	C₅²×D₃	semidirect
45. C₅×C₃₁	C₂×C₅×C₃⋊D₃	semidirect
51. C₃×C₃₁	C₂×C₅²×D₃	semidirect
54. C₅×C₃⋊D₃	C₅×D₃₁	direct (N × H)
55. C₃×C₅²	D₃×D₃₁	semidirect
58. D₃₁	C₅²×C₃⋊D₃	direct (N × H)
63. C₃²×C₅	C₂×C₅×D₃₁	semidirect
65. C₃₁	C₂×C₅²×C₃⋊D₃	semidirect
69. C₅²	C₃⋊D₃×D₃₁	direct (N × H)
73. C₃⋊D₃	C₅²×D₃₁	direct (N × H)
74. C₃×C₅	C₅×D₃×D₃₁	semidirect
79. C₃²	C₂×C₅²×D₃₁	semidirect
83. C₅	C₅×C₃⋊D₃×D₃₁	direct (N × H)
85. C₃	C₅²×D₃×D₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₃²×C₅²×D₃₁	2
C₅²×C₃⋊D₃×C₃₁	2
C₅²×C₃⋊D₉₃	2
C₅²×D₃×D₃₁	3
C₅×C₃⋊D₃×D₃₁	5
C₂×C₅²×C₃⋊D₃	31

Subgroups — C52×C3⋊D3×D31 #42124947

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅²×C₃⋊D₃×D₃₁ #42124947