Subgroups — C₂²×C₃×C₂₇×C₇₃ #72201005

Order 23652.

140 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₂₇×C₇₃	C₂	1	3	yes	maximal, direct product factor
2	C₂²×C₃×C₉×C₇₃	C₃	1	1	yes	maximal, characteristic
3	C₂²×C₂₇×C₇₃	C₃	1	3	yes	maximal, direct product factor
4	C₂×C₃×C₉×C₇₃	C₂×C₃	1	3	yes	—
5	C₂×C₂₇×C₇₃	C₂×C₃	1	9	yes	direct product factor
6	C₃×C₂₇×C₇₃	C₂²	1	1	yes	characteristic, direct product factor, Hall
7	C₂²×C₃²×C₇₃	C₉	1	1	yes	characteristic
8	C₂²×C₉×C₇₃	C₉	1	2	yes	—
9	C₂²×C₉×C₇₃	C₃²	1	1	yes	characteristic
10	C₃×C₉×C₇₃	C₂²×C₃	1	1	yes	characteristic
11	C₂₇×C₇₃	C₂²×C₃	1	3	yes	direct product factor
12	C₂×C₃²×C₇₃	C₂×C₉	1	3	yes	—
13	C₂×C₉×C₇₃	C₂×C₉	1	6	yes	—
14	C₂×C₉×C₇₃	C₂×C₃²	1	3	yes	—
15	C₂²×C₃×C₇₃	C₂₇	1	3	yes	direct product factor
16	C₂²×C₃×C₇₃	C₃×C₉	1	1	yes	characteristic
17	C₃²×C₇₃	C₂²×C₉	1	1	yes	characteristic
18	C₉×C₇₃	C₂²×C₉	1	2	yes	—
19	C₉×C₇₃	C₂²×C₃²	1	1	yes	characteristic
20	C₂×C₃×C₇₃	C₂×C₂₇	1	9	yes	direct product factor
21	C₂×C₃×C₇₃	C₂×C₃×C₉	1	3	yes	—
22	C₂²×C₃×C₂₇	C₇₃	1	1	yes	maximal, characteristic, direct product factor, Hall
23	C₂²×C₇₃	C₃×C₂₇	1	1	yes	characteristic, direct product factor, Hall
24	C₃×C₇₃	C₂²×C₂₇	1	3	yes	direct product factor
25	C₃×C₇₃	C₂²×C₃×C₉	1	1	yes	characteristic
26	C₂×C₃×C₂₇	C₂×C₇₃	1	3	yes	direct product factor
27	C₂×C₇₃	C₂×C₃×C₂₇	1	3	yes	direct product factor
28	C₂²×C₃×C₉	C₃×C₇₃	1	1	yes	characteristic
29	C₂²×C₂₇	C₃×C₇₃	1	3	yes	direct product factor
30	C₃×C₂₇	C₂²×C₇₃	1	1	yes	characteristic, direct product factor, Sylow
31	C₇₃	C₂²×C₃×C₂₇	1	1	yes	characteristic, direct product factor, Sylow
32	C₂×C₃×C₉	C₂×C₃×C₇₃	1	3	yes	—
33	C₂×C₂₇	C₂×C₃×C₇₃	1	9	yes	direct product factor
34	C₂²×C₃²	C₉×C₇₃	1	1	yes	characteristic
35	C₂²×C₉	C₉×C₇₃	1	2	yes	—
36	C₂²×C₉	C₃²×C₇₃	1	1	yes	characteristic
37	C₃×C₉	C₂²×C₃×C₇₃	1	1	yes	characteristic
38	C₂₇	C₂²×C₃×C₇₃	1	3	yes	direct product factor
39	C₂×C₃²	C₂×C₉×C₇₃	1	3	yes	—
40	C₂×C₉	C₂×C₉×C₇₃	1	6	yes	—
41	C₂×C₉	C₂×C₃²×C₇₃	1	3	yes	—
42	C₂²×C₃	C₂₇×C₇₃	1	3	yes	direct product factor
43	C₂²×C₃	C₃×C₉×C₇₃	1	1	yes	characteristic
44	C₃²	C₂²×C₉×C₇₃	1	1	yes	characteristic
45	C₉	C₂²×C₉×C₇₃	1	2	yes	—
46	C₉	C₂²×C₃²×C₇₃	1	1	yes	characteristic
47	C₂×C₃	C₂×C₂₇×C₇₃	1	9	yes	direct product factor
48	C₂×C₃	C₂×C₃×C₉×C₇₃	1	3	yes	—
49	C₂²	C₃×C₂₇×C₇₃	1	1	yes	characteristic, direct product factor, Sylow
50	C₃	C₂²×C₂₇×C₇₃	1	3	yes	direct product factor
51	C₃	C₂²×C₃×C₉×C₇₃	1	1	yes	characteristic
52	C₂	C₂×C₃×C₂₇×C₇₃	1	3	yes	direct product factor

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₉	C₂²×C₃²×C₇₃
socle, prime radical	C₂²×C₃²×C₇₃	C₉
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×C₂₇×C₇₃	C₂	direct (N × H)
3. C₂²×C₂₇×C₇₃	C₃	direct (N × H)
5. C₂×C₂₇×C₇₃	C₂×C₃	direct (N × H)
6. C₃×C₂₇×C₇₃	C₂²	direct (N × H)
11. C₂₇×C₇₃	C₂²×C₃	direct (N × H)
15. C₂²×C₃×C₇₃	C₂₇	direct (N × H)
20. C₂×C₃×C₇₃	C₂×C₂₇	direct (N × H)
22. C₂²×C₃×C₂₇	C₇₃	direct (N × H)
23. C₂²×C₇₃	C₃×C₂₇	direct (N × H)
24. C₃×C₇₃	C₂²×C₂₇	direct (N × H)
26. C₂×C₃×C₂₇	C₂×C₇₃	direct (N × H)
27. C₂×C₇₃	C₂×C₃×C₂₇	direct (N × H)
29. C₂²×C₂₇	C₃×C₇₃	direct (N × H)
30. C₃×C₂₇	C₂²×C₇₃	direct (N × H)
31. C₇₃	C₂²×C₃×C₂₇	direct (N × H)
33. C₂×C₂₇	C₂×C₃×C₇₃	direct (N × H)
38. C₂₇	C₂²×C₃×C₇₃	direct (N × H)
42. C₂²×C₃	C₂₇×C₇₃	direct (N × H)
47. C₂×C₃	C₂×C₂₇×C₇₃	direct (N × H)
49. C₂²	C₃×C₂₇×C₇₃	direct (N × H)
50. C₃	C₂²×C₂₇×C₇₃	direct (N × H)
52. C₂	C₂×C₃×C₂₇×C₇₃	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×C₃×C₂₇×C₇₃	2
C₂²×C₃×C₉×C₇₃	3
C₂²×C₂₇×C₇₃	3
C₂²×C₃×C₂₇	73

Subgroups — C22×C3×C27×C73 #72201005

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃×C₂₇×C₇₃ #72201005