Subgroups — C₂³×C₅×C₂₇×C₂₉ #72208113

Order 31320.

256 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×C₂₇×C₂₉	C₂	1	7	yes	maximal, direct product factor
2	C₂³×C₅×C₉×C₂₉	C₃	1	1	yes	maximal, characteristic
3	C₂×C₅×C₂₇×C₂₉	C₂²	1	7	yes	direct product factor
4	C₂³×C₂₇×C₂₉	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₅×C₂₇×C₂₉	C₂³	1	1	yes	characteristic, direct product factor, Hall
6	C₂³×C₃×C₅×C₂₉	C₉	1	1	yes	characteristic
7	C₂²×C₂₇×C₂₉	C₂×C₅	1	7	yes	direct product factor
8	C₂³×C₉×C₂₉	C₃×C₅	1	1	yes	characteristic
9	C₂²×C₅×C₉×C₂₉	C₂×C₃	1	7	yes	—
10	C₂×C₅×C₉×C₂₉	C₂²×C₃	1	7	yes	—
11	C₂²×C₃×C₅×C₂₉	C₂×C₉	1	7	yes	—
12	C₂×C₂₇×C₂₉	C₂²×C₅	1	7	yes	direct product factor
13	C₅×C₉×C₂₉	C₂³×C₃	1	1	yes	characteristic
14	C₂³×C₅×C₂₉	C₂₇	1	1	yes	characteristic, direct product factor, Hall
15	C₂³×C₅×C₂₇	C₂₉	1	1	yes	maximal, characteristic, direct product factor, Hall
16	C₂²×C₉×C₂₉	C₂×C₃×C₅	1	7	yes	—
17	C₂×C₃×C₅×C₂₉	C₂²×C₉	1	7	yes	—
18	C₂₇×C₂₉	C₂³×C₅	1	1	yes	characteristic, direct product factor, Hall
19	C₂³×C₃×C₂₉	C₅×C₉	1	1	yes	characteristic
20	C₂²×C₅×C₂₉	C₂×C₂₇	1	7	yes	direct product factor
21	C₂²×C₅×C₂₇	C₂×C₂₉	1	7	yes	direct product factor
22	C₂×C₉×C₂₉	C₂²×C₃×C₅	1	7	yes	—
23	C₃×C₅×C₂₉	C₂³×C₉	1	1	yes	characteristic
24	C₂³×C₅×C₉	C₃×C₂₉	1	1	yes	characteristic
25	C₂²×C₃×C₂₉	C₂×C₅×C₉	1	7	yes	—
26	C₂×C₅×C₂₉	C₂²×C₂₇	1	7	yes	direct product factor
27	C₂×C₅×C₂₇	C₂²×C₂₉	1	7	yes	direct product factor
28	C₉×C₂₉	C₂³×C₃×C₅	1	1	yes	characteristic
29	C₂³×C₂₉	C₅×C₂₇	1	1	yes	characteristic, direct product factor, Hall
30	C₂³×C₂₇	C₅×C₂₉	1	1	yes	characteristic, direct product factor, Hall
31	C₂²×C₅×C₉	C₂×C₃×C₂₉	1	7	yes	—
32	C₂×C₃×C₂₉	C₂²×C₅×C₉	1	7	yes	—
33	C₅×C₂₉	C₂³×C₂₇	1	1	yes	characteristic, direct product factor, Hall
34	C₅×C₂₇	C₂³×C₂₉	1	1	yes	characteristic, direct product factor, Hall
35	C₂³×C₃×C₅	C₉×C₂₉	1	1	yes	characteristic
36	C₂²×C₂₉	C₂×C₅×C₂₇	1	7	yes	direct product factor
37	C₂²×C₂₇	C₂×C₅×C₂₉	1	7	yes	direct product factor
38	C₂×C₅×C₉	C₂²×C₃×C₂₉	1	7	yes	—
39	C₃×C₂₉	C₂³×C₅×C₉	1	1	yes	characteristic
40	C₂³×C₉	C₃×C₅×C₂₉	1	1	yes	characteristic
41	C₂²×C₃×C₅	C₂×C₉×C₂₉	1	7	yes	—
42	C₂×C₂₉	C₂²×C₅×C₂₇	1	7	yes	direct product factor
43	C₂×C₂₇	C₂²×C₅×C₂₉	1	7	yes	direct product factor
44	C₅×C₉	C₂³×C₃×C₂₉	1	1	yes	characteristic
45	C₂³×C₅	C₂₇×C₂₉	1	1	yes	characteristic, direct product factor, Hall
46	C₂²×C₉	C₂×C₃×C₅×C₂₉	1	7	yes	—
47	C₂×C₃×C₅	C₂²×C₉×C₂₉	1	7	yes	—
48	C₂₉	C₂³×C₅×C₂₇	1	1	yes	characteristic, direct product factor, Sylow
49	C₂₇	C₂³×C₅×C₂₉	1	1	yes	characteristic, direct product factor, Sylow
50	C₂³×C₃	C₅×C₉×C₂₉	1	1	yes	characteristic
51	C₂²×C₅	C₂×C₂₇×C₂₉	1	7	yes	direct product factor
52	C₂×C₉	C₂²×C₃×C₅×C₂₉	1	7	yes	—
53	C₃×C₅	C₂³×C₉×C₂₉	1	1	yes	characteristic
54	C₂²×C₃	C₂×C₅×C₉×C₂₉	1	7	yes	—
55	C₂×C₅	C₂²×C₂₇×C₂₉	1	7	yes	direct product factor
56	C₉	C₂³×C₃×C₅×C₂₉	1	1	yes	characteristic
57	C₂³	C₅×C₂₇×C₂₉	1	1	yes	characteristic, direct product factor, Sylow
58	C₂×C₃	C₂²×C₅×C₉×C₂₉	1	7	yes	—
59	C₅	C₂³×C₂₇×C₂₉	1	1	yes	characteristic, direct product factor, Sylow
60	C₂²	C₂×C₅×C₂₇×C₂₉	1	7	yes	direct product factor
61	C₃	C₂³×C₅×C₉×C₂₉	1	1	yes	characteristic
62	C₂	C₂²×C₅×C₂₇×C₂₉	1	7	yes	direct product factor

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₉	C₂³×C₃×C₅×C₂₉
socle, prime radical	C₂³×C₃×C₅×C₂₉	C₉
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₅×C₂₇×C₂₉	C₂	direct (N × H)
3. C₂×C₅×C₂₇×C₂₉	C₂²	direct (N × H)
4. C₂³×C₂₇×C₂₉	C₅	direct (N × H)
5. C₅×C₂₇×C₂₉	C₂³	direct (N × H)
7. C₂²×C₂₇×C₂₉	C₂×C₅	direct (N × H)
12. C₂×C₂₇×C₂₉	C₂²×C₅	direct (N × H)
14. C₂³×C₅×C₂₉	C₂₇	direct (N × H)
15. C₂³×C₅×C₂₇	C₂₉	direct (N × H)
18. C₂₇×C₂₉	C₂³×C₅	direct (N × H)
20. C₂²×C₅×C₂₉	C₂×C₂₇	direct (N × H)
21. C₂²×C₅×C₂₇	C₂×C₂₉	direct (N × H)
26. C₂×C₅×C₂₉	C₂²×C₂₇	direct (N × H)
27. C₂×C₅×C₂₇	C₂²×C₂₉	direct (N × H)
29. C₂³×C₂₉	C₅×C₂₇	direct (N × H)
30. C₂³×C₂₇	C₅×C₂₉	direct (N × H)
33. C₅×C₂₉	C₂³×C₂₇	direct (N × H)
34. C₅×C₂₇	C₂³×C₂₉	direct (N × H)
36. C₂²×C₂₉	C₂×C₅×C₂₇	direct (N × H)
37. C₂²×C₂₇	C₂×C₅×C₂₉	direct (N × H)
42. C₂×C₂₉	C₂²×C₅×C₂₇	direct (N × H)
43. C₂×C₂₇	C₂²×C₅×C₂₉	direct (N × H)
45. C₂³×C₅	C₂₇×C₂₉	direct (N × H)
48. C₂₉	C₂³×C₅×C₂₇	direct (N × H)
49. C₂₇	C₂³×C₅×C₂₉	direct (N × H)
51. C₂²×C₅	C₂×C₂₇×C₂₉	direct (N × H)
55. C₂×C₅	C₂²×C₂₇×C₂₉	direct (N × H)
57. C₂³	C₅×C₂₇×C₂₉	direct (N × H)
59. C₅	C₂³×C₂₇×C₂₉	direct (N × H)
60. C₂²	C₂×C₅×C₂₇×C₂₉	direct (N × H)
62. C₂	C₂²×C₅×C₂₇×C₂₉	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₅×C₂₇×C₂₉	2
C₂³×C₅×C₉×C₂₉	3
C₂³×C₂₇×C₂₉	5
C₂³×C₅×C₂₇	29

Subgroups — C23×C5×C27×C29 #72208113

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₅×C₂₇×C₂₉ #72208113