Subgroups — C₂⁴×C₃×C₂₈₉ #72192091

Order 13872.

402 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃×C₂₈₉	C₂	1	15	yes	maximal, direct product factor
2	C₂⁴×C₂₈₉	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
3	C₂³×C₂₈₉	C₂×C₃	1	15	yes	direct product factor
4	C₂²×C₃×C₂₈₉	C₂²	1	35	yes	direct product factor
5	C₂×C₃×C₂₈₉	C₂³	1	15	yes	direct product factor
6	C₂²×C₂₈₉	C₂²×C₃	1	35	yes	direct product factor
7	C₃×C₂₈₉	C₂⁴	1	1	yes	characteristic, direct product factor, Hall
8	C₂⁴×C₃×C₁₇	C₁₇	1	1	yes	maximal, characteristic
9	C₂×C₂₈₉	C₂³×C₃	1	15	yes	direct product factor
10	C₂³×C₃×C₁₇	C₂×C₁₇	1	15	yes	—
11	C₂₈₉	C₂⁴×C₃	1	1	yes	characteristic, direct product factor, Sylow
12	C₂⁴×C₁₇	C₃×C₁₇	1	1	yes	characteristic
13	C₂²×C₃×C₁₇	C₂²×C₁₇	1	35	yes	—
14	C₂³×C₁₇	C₂×C₃×C₁₇	1	15	yes	—
15	C₂×C₃×C₁₇	C₂³×C₁₇	1	15	yes	—
16	C₂²×C₁₇	C₂²×C₃×C₁₇	1	35	yes	—
17	C₃×C₁₇	C₂⁴×C₁₇	1	1	yes	characteristic
18	C₂⁴×C₃	C₂₈₉	1	1	yes	characteristic, direct product factor, Hall
19	C₂×C₁₇	C₂³×C₃×C₁₇	1	15	yes	—
20	C₂³×C₃	C₂×C₂₈₉	1	15	yes	direct product factor
21	C₁₇	C₂⁴×C₃×C₁₇	1	1	yes	characteristic
22	C₂⁴	C₃×C₂₈₉	1	1	yes	characteristic, direct product factor, Sylow
23	C₂²×C₃	C₂²×C₂₈₉	1	35	yes	direct product factor
24	C₂³	C₂×C₃×C₂₈₉	1	15	yes	direct product factor
25	C₂×C₃	C₂³×C₂₈₉	1	15	yes	direct product factor
26	C₂²	C₂²×C₃×C₂₈₉	1	35	yes	direct product factor
27	C₃	C₂⁴×C₂₈₉	1	1	yes	characteristic, direct product factor, Sylow
28	C₂	C₂³×C₃×C₂₈₉	1	15	yes	direct product factor

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₁₇	C₂⁴×C₃×C₁₇
socle, prime radical	C₂⁴×C₃×C₁₇	C₁₇
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂³×C₃×C₂₈₉	C₂	direct (N × H)
2. C₂⁴×C₂₈₉	C₃	direct (N × H)
3. C₂³×C₂₈₉	C₂×C₃	direct (N × H)
4. C₂²×C₃×C₂₈₉	C₂²	direct (N × H)
5. C₂×C₃×C₂₈₉	C₂³	direct (N × H)
6. C₂²×C₂₈₉	C₂²×C₃	direct (N × H)
7. C₃×C₂₈₉	C₂⁴	direct (N × H)
9. C₂×C₂₈₉	C₂³×C₃	direct (N × H)
11. C₂₈₉	C₂⁴×C₃	direct (N × H)
18. C₂⁴×C₃	C₂₈₉	direct (N × H)
20. C₂³×C₃	C₂×C₂₈₉	direct (N × H)
22. C₂⁴	C₃×C₂₈₉	direct (N × H)
23. C₂²×C₃	C₂²×C₂₈₉	direct (N × H)
24. C₂³	C₂×C₃×C₂₈₉	direct (N × H)
25. C₂×C₃	C₂³×C₂₈₉	direct (N × H)
26. C₂²	C₂²×C₃×C₂₈₉	direct (N × H)
27. C₃	C₂⁴×C₂₈₉	direct (N × H)
28. C₂	C₂³×C₃×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×C₃×C₂₈₉	2
C₂⁴×C₂₈₉	3
C₂⁴×C₃×C₁₇	17

Subgroups — C24×C3×C289 #72192091

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂⁴×C₃×C₂₈₉ #72192091