Subgroups — C₃×Dic₃×C₂₅×C₃₁ #42124331

Order 27900.

84 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×C₂₅×C₃₁	C₂	1	1	yes	maximal, characteristic
2	Dic₃×C₂₅×C₃₁	C₃	1	1	yes	maximal, characteristic, direct product factor
3	C₃×C₄×C₂₅×C₃₁	—	3	1	no	maximal
4	C₃²×C₂₅×C₃₁	C₄	1	1	yes	characteristic, complemented, Hall
5	C₃×C₅×Dic₃×C₃₁	C₅	1	1	yes	maximal, characteristic
6	C₂×C₃×C₂₅×C₃₁	—	2	1	no	—
7	C₂×C₃×C₂₅×C₃₁	D₃	1	1	yes	characteristic
8	C₂×C₃×C₂₅×C₃₁	C₂×C₃	1	1	yes	characteristic
9	C₄×C₂₅×C₃₁	—	3	1	no	Hall
10	C₂×C₃²×C₅×C₃₁	C₂×C₅	1	1	yes	characteristic
11	C₃×C₂₅×C₃₁	—	2	1	no	—
12	C₃×C₂₅×C₃₁	Dic₃	1	1	yes	characteristic, direct product factor
13	C₃×C₂₅×C₃₁	C₃×C₄	1	1	yes	characteristic, complemented
14	C₃×C₄×C₅×C₃₁	—	3	1	no	—
15	C₅×Dic₃×C₃₁	C₃×C₅	1	1	yes	characteristic
16	C₂×C₂₅×C₃₁	C₃×D₃	1	1	yes	characteristic
17	C₃²×C₅×C₃₁	C₄×C₅	1	1	yes	characteristic
18	C₃×Dic₃×C₃₁	C₂₅	1	1	yes	characteristic, direct product factor, Hall
19	C₂×C₃×C₅×C₃₁	—	2	1	no	—
20	C₂×C₃×C₅×C₃₁	C₅×D₃	1	1	yes	characteristic
21	C₂×C₃×C₅×C₃₁	C₂×C₃×C₅	1	1	yes	characteristic
22	C₃×Dic₃×C₂₅	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Hall
23	C₂₅×C₃₁	C₃×Dic₃	1	1	yes	characteristic, direct product factor, Hall
24	C₄×C₅×C₃₁	—	3	1	no	—
25	C₂×C₃²×C₃₁	C₂×C₂₅	1	1	yes	characteristic
26	C₃×C₅×C₃₁	—	2	1	no	—
27	C₃×C₅×C₃₁	C₅×Dic₃	1	1	yes	characteristic
28	C₃×C₅×C₃₁	C₃×C₄×C₅	1	1	yes	characteristic
29	C₂×C₃²×C₂₅	C₂×C₃₁	1	1	yes	characteristic
30	C₃×C₄×C₃₁	—	3	1	no	—
31	Dic₃×C₃₁	C₃×C₂₅	1	1	yes	characteristic, direct product factor
32	C₂×C₅×C₃₁	C₃×C₅×D₃	1	1	yes	characteristic
33	C₃×C₄×C₂₅	—	3	1	no	—
34	Dic₃×C₂₅	C₃×C₃₁	1	1	yes	characteristic, direct product factor
35	C₃²×C₃₁	C₄×C₂₅	1	1	yes	characteristic, complemented, Hall
36	C₃²×C₂₅	C₄×C₃₁	1	1	yes	characteristic, complemented, Hall
37	C₂×C₃×C₃₁	—	2	1	no	—
38	C₂×C₃×C₃₁	D₃×C₂₅	1	1	yes	characteristic
39	C₂×C₃×C₃₁	C₂×C₃×C₂₅	1	1	yes	characteristic
40	C₃×C₅×Dic₃	C₅×C₃₁	1	1	yes	characteristic
41	C₅×C₃₁	C₃×C₅×Dic₃	1	1	yes	characteristic
42	C₂×C₃×C₂₅	—	2	1	no	—
43	C₂×C₃×C₂₅	D₃×C₃₁	1	1	yes	characteristic
44	C₂×C₃×C₂₅	C₂×C₃×C₃₁	1	1	yes	characteristic
45	C₄×C₃₁	—	3	1	no	Hall
46	C₄×C₂₅	—	3	1	no	Hall
47	C₃×C₃₁	—	2	1	no	—
48	C₃×C₃₁	Dic₃×C₂₅	1	1	yes	characteristic, direct product factor
49	C₃×C₃₁	C₃×C₄×C₂₅	1	1	yes	characteristic, complemented
50	C₂×C₃²×C₅	C₂×C₅×C₃₁	1	1	yes	characteristic
51	C₃×C₂₅	—	2	1	no	—
52	C₃×C₂₅	Dic₃×C₃₁	1	1	yes	characteristic, direct product factor
53	C₃×C₂₅	C₃×C₄×C₃₁	1	1	yes	characteristic, complemented
54	C₂×C₃₁	C₃×D₃×C₂₅	1	1	yes	characteristic
55	C₃×C₄×C₅	—	3	1	no	—
56	C₅×Dic₃	C₃×C₅×C₃₁	1	1	yes	characteristic
57	C₂×C₂₅	C₃×D₃×C₃₁	1	1	yes	characteristic
58	C₃²×C₅	C₄×C₅×C₃₁	1	1	yes	characteristic
59	C₃×Dic₃	C₂₅×C₃₁	1	1	yes	characteristic, direct product factor, Hall
60	C₃₁	C₃×Dic₃×C₂₅	1	1	yes	characteristic, direct product factor, Sylow
61	C₂×C₃×C₅	—	2	1	no	—
62	C₂×C₃×C₅	C₅×D₃×C₃₁	1	1	yes	characteristic
63	C₂×C₃×C₅	C₂×C₃×C₅×C₃₁	1	1	yes	characteristic
64	C₂₅	C₃×Dic₃×C₃₁	1	1	yes	characteristic, direct product factor, Sylow
65	C₄×C₅	—	3	1	no	—
66	C₂×C₃²	C₂×C₂₅×C₃₁	1	1	yes	characteristic
67	C₃×C₅	—	2	1	no	—
68	C₃×C₅	C₅×Dic₃×C₃₁	1	1	yes	characteristic
69	C₃×C₅	C₃×C₄×C₅×C₃₁	1	1	yes	characteristic
70	C₃×C₄	—	3	1	no	—
71	Dic₃	C₃×C₂₅×C₃₁	1	1	yes	characteristic, direct product factor
72	C₂×C₅	C₃×C₅×D₃×C₃₁	1	1	yes	characteristic
73	C₃²	C₄×C₂₅×C₃₁	1	1	yes	characteristic, complemented, Sylow
74	C₂×C₃	—	2	1	no	—
75	C₂×C₃	C₂×C₃×C₂₅×C₃₁	1	1	yes	characteristic
76	C₂×C₃	D₃×C₂₅×C₃₁	1	1	yes	characteristic
77	C₅	C₃×C₅×Dic₃×C₃₁	1	1	yes	characteristic
78	C₄	—	3	1	no	Sylow
79	C₃	—	2	1	no	—
80	C₃	C₃×C₄×C₂₅×C₃₁	1	1	yes	characteristic, complemented
81	C₃	Dic₃×C₂₅×C₃₁	1	1	yes	characteristic, direct product factor
82	C₂	C₃×D₃×C₂₅×C₃₁	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂×C₃×C₂₅×C₃₁	D₃
commutator, hypocenter	C₃	C₃×C₄×C₂₅×C₃₁
Frattini	C₂×C₅	C₃×C₅×D₃×C₃₁
Fitting	C₂×C₃²×C₂₅×C₃₁	C₂
socle, prime radical	C₂×C₃²×C₅×C₃₁	C₂×C₅
cosocle	C₂×C₃×C₅	C₂×C₃×C₅×C₃₁
LCS	G ▷ C₃
UCS	C₁ ◁ C₂×C₃×C₂₅×C₃₁

Complements

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

N	H ≅ G/N	type
2. Dic₃×C₂₅×C₃₁	C₃	direct (N × H)
4. C₃²×C₂₅×C₃₁	C₄	semidirect
12. C₃×C₂₅×C₃₁	Dic₃	direct (N × H)
13. C₃×C₂₅×C₃₁	C₃×C₄	semidirect
18. C₃×Dic₃×C₃₁	C₂₅	direct (N × H)
22. C₃×Dic₃×C₂₅	C₃₁	direct (N × H)
23. C₂₅×C₃₁	C₃×Dic₃	direct (N × H)
31. Dic₃×C₃₁	C₃×C₂₅	direct (N × H)
34. Dic₃×C₂₅	C₃×C₃₁	direct (N × H)
35. C₃²×C₃₁	C₄×C₂₅	semidirect
36. C₃²×C₂₅	C₄×C₃₁	semidirect
48. C₃×C₃₁	Dic₃×C₂₅	direct (N × H)
49. C₃×C₃₁	C₃×C₄×C₂₅	semidirect
52. C₃×C₂₅	Dic₃×C₃₁	direct (N × H)
53. C₃×C₂₅	C₃×C₄×C₃₁	semidirect
59. C₃×Dic₃	C₂₅×C₃₁	direct (N × H)
60. C₃₁	C₃×Dic₃×C₂₅	direct (N × H)
64. C₂₅	C₃×Dic₃×C₃₁	direct (N × H)
71. Dic₃	C₃×C₂₅×C₃₁	direct (N × H)
73. C₃²	C₄×C₂₅×C₃₁	semidirect
80. C₃	C₃×C₄×C₂₅×C₃₁	semidirect
81. C₃	Dic₃×C₂₅×C₃₁	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃²×C₂₅×C₃₁	2
Dic₃×C₂₅×C₃₁	3
C₃×C₄×C₂₅×C₃₁	3
C₃×C₅×Dic₃×C₃₁	5
C₃×Dic₃×C₂₅	31

Subgroups — C3×Dic3×C25×C31 #42124331

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×Dic₃×C₂₅×C₃₁ #42124331