Subgroups — C₂²×C₄×C₅×C₂₈₉ #72200484

Order 23120.

162 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₅×C₂₈₉	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₅×C₂₈₉	C₂	1	6	yes	maximal, direct product factor
3	C₂²×C₄×C₂₈₉	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
4	C₂³×C₂₈₉	C₂×C₅	1	1	yes	characteristic
5	C₂×C₄×C₂₈₉	C₂×C₅	1	6	yes	direct product factor
6	C₂²×C₅×C₂₈₉	C₄	1	4	yes	direct product factor
7	C₂²×C₅×C₂₈₉	C₂²	1	3	yes	—
8	C₄×C₅×C₂₈₉	C₂²	1	4	yes	direct product factor
9	C₂×C₅×C₂₈₉	C₂×C₄	1	6	yes	direct product factor
10	C₂×C₅×C₂₈₉	C₂³	1	1	yes	characteristic
11	C₅×C₂₈₉	C₂²×C₄	1	1	yes	characteristic, direct product factor, Hall
12	C₂²×C₄×C₅×C₁₇	C₁₇	1	1	yes	maximal, characteristic
13	C₂²×C₂₈₉	C₄×C₅	1	4	yes	direct product factor
14	C₂²×C₂₈₉	C₂²×C₅	1	3	yes	—
15	C₄×C₂₈₉	C₂²×C₅	1	4	yes	direct product factor
16	C₂³×C₅×C₁₇	C₂×C₁₇	1	1	yes	characteristic
17	C₂×C₄×C₅×C₁₇	C₂×C₁₇	1	6	yes	—
18	C₂×C₂₈₉	C₂×C₄×C₅	1	6	yes	direct product factor
19	C₂×C₂₈₉	C₂³×C₅	1	1	yes	characteristic
20	C₂²×C₅×C₁₇	C₄×C₁₇	1	4	yes	—
21	C₂²×C₅×C₁₇	C₂²×C₁₇	1	3	yes	—
22	C₄×C₅×C₁₇	C₂²×C₁₇	1	4	yes	—
23	C₂₈₉	C₂²×C₄×C₅	1	1	yes	characteristic, direct product factor, Sylow
24	C₂²×C₄×C₁₇	C₅×C₁₇	1	1	yes	characteristic
25	C₂×C₅×C₁₇	C₂×C₄×C₁₇	1	6	yes	—
26	C₂×C₅×C₁₇	C₂³×C₁₇	1	1	yes	characteristic
27	C₂³×C₁₇	C₂×C₅×C₁₇	1	1	yes	characteristic
28	C₂×C₄×C₁₇	C₂×C₅×C₁₇	1	6	yes	—
29	C₅×C₁₇	C₂²×C₄×C₁₇	1	1	yes	characteristic
30	C₂²×C₄×C₅	C₂₈₉	1	1	yes	characteristic, direct product factor, Hall
31	C₂²×C₁₇	C₄×C₅×C₁₇	1	4	yes	—
32	C₂²×C₁₇	C₂²×C₅×C₁₇	1	3	yes	—
33	C₄×C₁₇	C₂²×C₅×C₁₇	1	4	yes	—
34	C₂³×C₅	C₂×C₂₈₉	1	1	yes	characteristic
35	C₂×C₄×C₅	C₂×C₂₈₉	1	6	yes	direct product factor
36	C₂×C₁₇	C₂×C₄×C₅×C₁₇	1	6	yes	—
37	C₂×C₁₇	C₂³×C₅×C₁₇	1	1	yes	characteristic
38	C₂²×C₅	C₄×C₂₈₉	1	4	yes	direct product factor
39	C₂²×C₅	C₂²×C₂₈₉	1	3	yes	—
40	C₄×C₅	C₂²×C₂₈₉	1	4	yes	direct product factor
41	C₁₇	C₂²×C₄×C₅×C₁₇	1	1	yes	characteristic
42	C₂²×C₄	C₅×C₂₈₉	1	1	yes	characteristic, direct product factor, Sylow
43	C₂×C₅	C₂×C₄×C₂₈₉	1	6	yes	direct product factor
44	C₂×C₅	C₂³×C₂₈₉	1	1	yes	characteristic
45	C₂³	C₂×C₅×C₂₈₉	1	1	yes	characteristic
46	C₂×C₄	C₂×C₅×C₂₈₉	1	6	yes	direct product factor
47	C₅	C₂²×C₄×C₂₈₉	1	1	yes	characteristic, direct product factor, Sylow
48	C₂²	C₄×C₅×C₂₈₉	1	4	yes	direct product factor
49	C₂²	C₂²×C₅×C₂₈₉	1	3	yes	—
50	C₄	C₂²×C₅×C₂₈₉	1	4	yes	direct product factor
51	C₂	C₂×C₄×C₅×C₂₈₉	1	6	yes	direct product factor
52	C₂	C₂³×C₅×C₂₈₉	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂×C₁₇	C₂³×C₅×C₁₇
socle, prime radical	C₂³×C₅×C₁₇	C₂×C₁₇
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×C₅×C₂₈₉	C₂	direct (N × H)
3. C₂²×C₄×C₂₈₉	C₅	direct (N × H)
5. C₂×C₄×C₂₈₉	C₂×C₅	direct (N × H)
6. C₂²×C₅×C₂₈₉	C₄	direct (N × H)
8. C₄×C₅×C₂₈₉	C₂²	direct (N × H)
9. C₂×C₅×C₂₈₉	C₂×C₄	direct (N × H)
11. C₅×C₂₈₉	C₂²×C₄	direct (N × H)
13. C₂²×C₂₈₉	C₄×C₅	direct (N × H)
15. C₄×C₂₈₉	C₂²×C₅	direct (N × H)
18. C₂×C₂₈₉	C₂×C₄×C₅	direct (N × H)
23. C₂₈₉	C₂²×C₄×C₅	direct (N × H)
30. C₂²×C₄×C₅	C₂₈₉	direct (N × H)
35. C₂×C₄×C₅	C₂×C₂₈₉	direct (N × H)
38. C₂²×C₅	C₄×C₂₈₉	direct (N × H)
40. C₄×C₅	C₂²×C₂₈₉	direct (N × H)
42. C₂²×C₄	C₅×C₂₈₉	direct (N × H)
43. C₂×C₅	C₂×C₄×C₂₈₉	direct (N × H)
46. C₂×C₄	C₂×C₅×C₂₈₉	direct (N × H)
47. C₅	C₂²×C₄×C₂₈₉	direct (N × H)
48. C₂²	C₄×C₅×C₂₈₉	direct (N × H)
50. C₄	C₂²×C₅×C₂₈₉	direct (N × H)
51. C₂	C₂×C₄×C₅×C₂₈₉	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₅×C₂₈₉	2
C₂×C₄×C₅×C₂₈₉	2
C₂²×C₄×C₂₈₉	5
C₂²×C₄×C₅×C₁₇	17

Subgroups — C22×C4×C5×C289 #72200484

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₄×C₅×C₂₈₉ #72200484