Subgroups — C₂³×C₂₇×C₁₃₁ #72205293

Order 28296.

128 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₂₇×C₁₃₁	C₂	1	7	yes	maximal, direct product factor
2	C₂³×C₉×C₁₃₁	C₃	1	1	yes	maximal, characteristic
3	C₂×C₂₇×C₁₃₁	C₂²	1	7	yes	direct product factor
4	C₂³×C₃×C₁₃₁	C₉	1	1	yes	characteristic
5	C₂²×C₉×C₁₃₁	C₂×C₃	1	7	yes	—
6	C₂₇×C₁₃₁	C₂³	1	1	yes	characteristic, direct product factor, Hall
7	C₂×C₉×C₁₃₁	C₂²×C₃	1	7	yes	—
8	C₂²×C₃×C₁₃₁	C₂×C₉	1	7	yes	—
9	C₉×C₁₃₁	C₂³×C₃	1	1	yes	characteristic
10	C₂³×C₁₃₁	C₂₇	1	1	yes	characteristic, direct product factor, Hall
11	C₂×C₃×C₁₃₁	C₂²×C₉	1	7	yes	—
12	C₂²×C₁₃₁	C₂×C₂₇	1	7	yes	direct product factor
13	C₃×C₁₃₁	C₂³×C₉	1	1	yes	characteristic
14	C₂×C₁₃₁	C₂²×C₂₇	1	7	yes	direct product factor
15	C₂³×C₂₇	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Hall
16	C₁₃₁	C₂³×C₂₇	1	1	yes	characteristic, direct product factor, Sylow
17	C₂²×C₂₇	C₂×C₁₃₁	1	7	yes	direct product factor
18	C₂³×C₉	C₃×C₁₃₁	1	1	yes	characteristic
19	C₂×C₂₇	C₂²×C₁₃₁	1	7	yes	direct product factor
20	C₂²×C₉	C₂×C₃×C₁₃₁	1	7	yes	—
21	C₂₇	C₂³×C₁₃₁	1	1	yes	characteristic, direct product factor, Sylow
22	C₂³×C₃	C₉×C₁₃₁	1	1	yes	characteristic
23	C₂×C₉	C₂²×C₃×C₁₃₁	1	7	yes	—
24	C₂²×C₃	C₂×C₉×C₁₃₁	1	7	yes	—
25	C₉	C₂³×C₃×C₁₃₁	1	1	yes	characteristic
26	C₂³	C₂₇×C₁₃₁	1	1	yes	characteristic, direct product factor, Sylow
27	C₂×C₃	C₂²×C₉×C₁₃₁	1	7	yes	—
28	C₂²	C₂×C₂₇×C₁₃₁	1	7	yes	direct product factor
29	C₃	C₂³×C₉×C₁₃₁	1	1	yes	characteristic
30	C₂	C₂²×C₂₇×C₁₃₁	1	7	yes	direct product factor

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₉	C₂³×C₃×C₁₃₁
socle, prime radical	C₂³×C₃×C₁₃₁	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×C₂₇×C₁₃₁	C₂	direct (N × H)
3. C₂×C₂₇×C₁₃₁	C₂²	direct (N × H)
6. C₂₇×C₁₃₁	C₂³	direct (N × H)
10. C₂³×C₁₃₁	C₂₇	direct (N × H)
12. C₂²×C₁₃₁	C₂×C₂₇	direct (N × H)
14. C₂×C₁₃₁	C₂²×C₂₇	direct (N × H)
15. C₂³×C₂₇	C₁₃₁	direct (N × H)
16. C₁₃₁	C₂³×C₂₇	direct (N × H)
17. C₂²×C₂₇	C₂×C₁₃₁	direct (N × H)
19. C₂×C₂₇	C₂²×C₁₃₁	direct (N × H)
21. C₂₇	C₂³×C₁₃₁	direct (N × H)
26. C₂³	C₂₇×C₁₃₁	direct (N × H)
28. C₂²	C₂×C₂₇×C₁₃₁	direct (N × H)
30. C₂	C₂²×C₂₇×C₁₃₁	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×C₂₇×C₁₃₁	2
C₂³×C₉×C₁₃₁	3
C₂³×C₂₇	131

Subgroups — C23×C27×C131 #72205293

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₂₇×C₁₃₁ #72205293