Subgroups — C₂×C₃²×C₂₅×D₃₁ #42124468

Order 27900.

180 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₂₅×D₃₁	C₂	1	2	yes	maximal, direct product factor
2	C₂×C₃²×C₂₅×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×C₂₅×D₃₁	C₃	1	4	yes	maximal, direct product factor
4	C₃²×C₂₅×C₃₁	C₂²	1	1	yes	characteristic, complemented, Hall
5	C₂×C₃²×C₅×D₃₁	C₅	1	1	yes	maximal, characteristic
6	C₃×C₂₅×D₃₁	C₂×C₃	1	8	yes	direct product factor
7	C₂×C₃×C₂₅×C₃₁	C₂×C₃	1	4	yes	complemented
8	C₂×C₂₅×D₃₁	C₃²	1	1	yes	characteristic, direct product factor, Hall
9	C₃²×C₅×D₃₁	C₂×C₅	1	2	yes	—
10	C₂×C₃²×C₅×C₃₁	C₂×C₅	1	1	yes	characteristic
11	C₃×C₂₅×C₃₁	C₂²×C₃	1	4	yes	complemented
12	C₂×C₃×C₅×D₃₁	C₃×C₅	1	4	yes	—
13	C₂×C₂₅×C₃₁	C₂×C₃²	1	1	yes	characteristic, complemented
14	C₂₅×D₃₁	C₂×C₃²	1	2	yes	direct product factor
15	C₃²×C₅×C₃₁	C₂²×C₅	1	1	yes	characteristic
16	C₂×C₃²×D₃₁	C₂₅	1	1	yes	characteristic, direct product factor, Hall
17	C₂×C₃×C₅×C₃₁	C₂×C₃×C₅	1	4	yes	—
18	C₃×C₅×D₃₁	C₂×C₃×C₅	1	8	yes	—
19	C₂²×C₃²×C₂₅	—	31	1	no	maximal, Hall
20	C₂₅×C₃₁	C₂²×C₃²	1	1	yes	characteristic, complemented, Hall
21	C₂×C₅×D₃₁	C₃²×C₅	1	1	yes	characteristic
22	C₂×C₃²×C₃₁	C₂×C₂₅	1	1	yes	characteristic, complemented
23	C₃²×D₃₁	C₂×C₂₅	1	2	yes	direct product factor
24	C₃×C₅×C₃₁	C₂²×C₃×C₅	1	4	yes	—
25	C₂×C₃²×C₂₅	—	31	2	no	—
26	C₂×C₃²×C₂₅	D₃₁	1	1	yes	characteristic, direct product factor
27	C₂×C₃×D₃₁	C₃×C₂₅	1	4	yes	direct product factor
28	C₂×C₅×C₃₁	C₂×C₃²×C₅	1	1	yes	characteristic
29	C₅×D₃₁	C₂×C₃²×C₅	1	2	yes	—
30	C₂²×C₃×C₂₅	—	31	4	no	—
31	C₃²×C₃₁	C₂²×C₂₅	1	1	yes	characteristic, complemented, Hall
32	C₃²×C₂₅	C₂×D₃₁	1	1	yes	characteristic, direct product factor, Hall
33	C₂×C₃×C₃₁	C₂×C₃×C₂₅	1	4	yes	complemented
34	C₃×D₃₁	C₂×C₃×C₂₅	1	8	yes	direct product factor
35	C₂²×C₃²×C₅	—	31	1	no	—
36	C₅×C₃₁	C₂²×C₃²×C₅	1	1	yes	characteristic
37	C₂×C₃×C₂₅	—	31	8	no	—
38	C₂×C₃×C₂₅	C₃×D₃₁	1	4	yes	direct product factor
39	C₂×D₃₁	C₃²×C₂₅	1	1	yes	characteristic, direct product factor, Hall
40	C₂²×C₂₅	—	31	1	no	Hall
41	C₃×C₃₁	C₂²×C₃×C₂₅	1	4	yes	complemented
42	C₂×C₃²×C₅	—	31	2	no	—
43	C₂×C₃²×C₅	C₅×D₃₁	1	1	yes	characteristic
44	C₃×C₂₅	C₂×C₃×D₃₁	1	4	yes	direct product factor
45	C₂×C₃₁	C₂×C₃²×C₂₅	1	1	yes	characteristic, complemented
46	D₃₁	C₂×C₃²×C₂₅	1	2	yes	direct product factor
47	C₂²×C₃×C₅	—	31	4	no	—
48	C₂×C₂₅	—	31	2	no	—
49	C₂×C₂₅	C₃²×D₃₁	1	1	yes	characteristic, direct product factor
50	C₃²×C₅	C₂×C₅×D₃₁	1	1	yes	characteristic
51	C₂²×C₃²	—	31	1	no	Hall
52	C₃₁	C₂²×C₃²×C₂₅	1	1	yes	characteristic, complemented, Sylow
53	C₂×C₃×C₅	—	31	8	no	—
54	C₂×C₃×C₅	C₃×C₅×D₃₁	1	4	yes	—
55	C₂₅	C₂×C₃²×D₃₁	1	1	yes	characteristic, direct product factor, Sylow
56	C₂²×C₅	—	31	1	no	—
57	C₂×C₃²	—	31	2	no	—
58	C₂×C₃²	C₂₅×D₃₁	1	1	yes	characteristic, direct product factor
59	C₃×C₅	C₂×C₃×C₅×D₃₁	1	4	yes	—
60	C₂²×C₃	—	31	4	no	—
61	C₂×C₅	—	31	2	no	—
62	C₂×C₅	C₃²×C₅×D₃₁	1	1	yes	characteristic
63	C₃²	C₂×C₂₅×D₃₁	1	1	yes	characteristic, direct product factor, Sylow
64	C₂×C₃	—	31	8	no	—
65	C₂×C₃	C₃×C₂₅×D₃₁	1	4	yes	direct product factor
66	C₅	C₂×C₃²×C₅×D₃₁	1	1	yes	characteristic
67	C₂²	—	31	1	no	Sylow
68	C₃	C₂×C₃×C₂₅×D₃₁	1	4	yes	direct product factor
69	C₂	—	31	2	no	—
70	C₂	C₃²×C₂₅×D₃₁	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂×C₃²×C₂₅	D₃₁
commutator, hypocenter	C₃₁	C₂²×C₃²×C₂₅
Frattini	C₅	C₂×C₃²×C₅×D₃₁
Fitting	C₂×C₃²×C₂₅×C₃₁	C₂
socle	C₂×C₃²×C₅×C₃₁	C₂×C₅
prime radical	C₂×C₃²×C₅×D₃₁	C₅
cosocle	C₅×C₃₁	C₂²×C₃²×C₅
LCS	G ▷ C₃₁
UCS	C₁ ◁ C₂×C₃²×C₂₅

Complements

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

N	H ≅ G/N	type
1. C₃²×C₂₅×D₃₁	C₂	direct (N × H)
2. C₂×C₃²×C₂₅×C₃₁	C₂	semidirect
3. C₂×C₃×C₂₅×D₃₁	C₃	direct (N × H)
4. C₃²×C₂₅×C₃₁	C₂²	semidirect
6. C₃×C₂₅×D₃₁	C₂×C₃	direct (N × H)
7. C₂×C₃×C₂₅×C₃₁	C₂×C₃	semidirect
8. C₂×C₂₅×D₃₁	C₃²	direct (N × H)
11. C₃×C₂₅×C₃₁	C₂²×C₃	semidirect
13. C₂×C₂₅×C₃₁	C₂×C₃²	semidirect
14. C₂₅×D₃₁	C₂×C₃²	direct (N × H)
16. C₂×C₃²×D₃₁	C₂₅	direct (N × H)
20. C₂₅×C₃₁	C₂²×C₃²	semidirect
22. C₂×C₃²×C₃₁	C₂×C₂₅	semidirect
23. C₃²×D₃₁	C₂×C₂₅	direct (N × H)
26. C₂×C₃²×C₂₅	D₃₁	direct (N × H)
27. C₂×C₃×D₃₁	C₃×C₂₅	direct (N × H)
31. C₃²×C₃₁	C₂²×C₂₅	semidirect
32. C₃²×C₂₅	C₂×D₃₁	direct (N × H)
33. C₂×C₃×C₃₁	C₂×C₃×C₂₅	semidirect
34. C₃×D₃₁	C₂×C₃×C₂₅	direct (N × H)
38. C₂×C₃×C₂₅	C₃×D₃₁	direct (N × H)
39. C₂×D₃₁	C₃²×C₂₅	direct (N × H)
41. C₃×C₃₁	C₂²×C₃×C₂₅	semidirect
44. C₃×C₂₅	C₂×C₃×D₃₁	direct (N × H)
45. C₂×C₃₁	C₂×C₃²×C₂₅	semidirect
46. D₃₁	C₂×C₃²×C₂₅	direct (N × H)
49. C₂×C₂₅	C₃²×D₃₁	direct (N × H)
52. C₃₁	C₂²×C₃²×C₂₅	semidirect
55. C₂₅	C₂×C₃²×D₃₁	direct (N × H)
58. C₂×C₃²	C₂₅×D₃₁	direct (N × H)
63. C₃²	C₂×C₂₅×D₃₁	direct (N × H)
65. C₂×C₃	C₃×C₂₅×D₃₁	direct (N × H)
68. C₃	C₂×C₃×C₂₅×D₃₁	direct (N × H)
70. C₂	C₃²×C₂₅×D₃₁	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃²×C₂₅×D₃₁	2
C₂×C₃²×C₂₅×C₃₁	2
C₂×C₃×C₂₅×D₃₁	3
C₂×C₃²×C₅×D₃₁	5
C₂²×C₃²×C₂₅	31

Subgroups — C2×C32×C25×D31 #42124468

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃²×C₂₅×D₃₁ #42124468