Subgroups — C₃²×C₄×C₇×C₁₂₇ #44544698

Order 32004.

72 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×C₇×C₁₂₇	C₂	1	1	yes	maximal, characteristic
2	C₃×C₄×C₇×C₁₂₇	C₃	1	4	yes	maximal, direct product factor
3	C₃²×C₇×C₁₂₇	C₄	1	1	yes	characteristic, direct product factor, Hall
4	C₂×C₃×C₇×C₁₂₇	C₂×C₃	1	4	yes	—
5	C₃²×C₄×C₁₂₇	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₄×C₇×C₁₂₇	C₃²	1	1	yes	characteristic, direct product factor, Hall
7	C₃×C₇×C₁₂₇	C₃×C₄	1	4	yes	direct product factor
8	C₂×C₃²×C₁₂₇	C₂×C₇	1	1	yes	characteristic
9	C₂×C₇×C₁₂₇	C₂×C₃²	1	1	yes	characteristic
10	C₃×C₄×C₁₂₇	C₃×C₇	1	4	yes	direct product factor
11	C₃²×C₁₂₇	C₄×C₇	1	1	yes	characteristic, direct product factor, Hall
12	C₇×C₁₂₇	C₃²×C₄	1	1	yes	characteristic, direct product factor, Hall
13	C₂×C₃×C₁₂₇	C₂×C₃×C₇	1	4	yes	—
14	C₄×C₁₂₇	C₃²×C₇	1	1	yes	characteristic, direct product factor, Hall
15	C₃×C₁₂₇	C₃×C₄×C₇	1	4	yes	direct product factor
16	C₂×C₁₂₇	C₂×C₃²×C₇	1	1	yes	characteristic
17	C₃²×C₄×C₇	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Hall
18	C₁₂₇	C₃²×C₄×C₇	1	1	yes	characteristic, direct product factor, Sylow
19	C₂×C₃²×C₇	C₂×C₁₂₇	1	1	yes	characteristic
20	C₃×C₄×C₇	C₃×C₁₂₇	1	4	yes	direct product factor
21	C₃²×C₇	C₄×C₁₂₇	1	1	yes	characteristic, direct product factor, Hall
22	C₂×C₃×C₇	C₂×C₃×C₁₂₇	1	4	yes	—
23	C₃²×C₄	C₇×C₁₂₇	1	1	yes	characteristic, direct product factor, Hall
24	C₄×C₇	C₃²×C₁₂₇	1	1	yes	characteristic, direct product factor, Hall
25	C₃×C₇	C₃×C₄×C₁₂₇	1	4	yes	direct product factor
26	C₂×C₃²	C₂×C₇×C₁₂₇	1	1	yes	characteristic
27	C₂×C₇	C₂×C₃²×C₁₂₇	1	1	yes	characteristic
28	C₃×C₄	C₃×C₇×C₁₂₇	1	4	yes	direct product factor
29	C₃²	C₄×C₇×C₁₂₇	1	1	yes	characteristic, direct product factor, Sylow
30	C₇	C₃²×C₄×C₁₂₇	1	1	yes	characteristic, direct product factor, Sylow
31	C₂×C₃	C₂×C₃×C₇×C₁₂₇	1	4	yes	—
32	C₄	C₃²×C₇×C₁₂₇	1	1	yes	characteristic, direct product factor, Sylow
33	C₃	C₃×C₄×C₇×C₁₂₇	1	4	yes	direct product factor
34	C₂	C₂×C₃²×C₇×C₁₂₇	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂	C₂×C₃²×C₇×C₁₂₇
socle, prime radical	C₂×C₃²×C₇×C₁₂₇	C₂
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃×C₄×C₇×C₁₂₇	C₃	direct (N × H)
3. C₃²×C₇×C₁₂₇	C₄	direct (N × H)
5. C₃²×C₄×C₁₂₇	C₇	direct (N × H)
6. C₄×C₇×C₁₂₇	C₃²	direct (N × H)
7. C₃×C₇×C₁₂₇	C₃×C₄	direct (N × H)
10. C₃×C₄×C₁₂₇	C₃×C₇	direct (N × H)
11. C₃²×C₁₂₇	C₄×C₇	direct (N × H)
12. C₇×C₁₂₇	C₃²×C₄	direct (N × H)
14. C₄×C₁₂₇	C₃²×C₇	direct (N × H)
15. C₃×C₁₂₇	C₃×C₄×C₇	direct (N × H)
17. C₃²×C₄×C₇	C₁₂₇	direct (N × H)
18. C₁₂₇	C₃²×C₄×C₇	direct (N × H)
20. C₃×C₄×C₇	C₃×C₁₂₇	direct (N × H)
21. C₃²×C₇	C₄×C₁₂₇	direct (N × H)
23. C₃²×C₄	C₇×C₁₂₇	direct (N × H)
24. C₄×C₇	C₃²×C₁₂₇	direct (N × H)
25. C₃×C₇	C₃×C₄×C₁₂₇	direct (N × H)
28. C₃×C₄	C₃×C₇×C₁₂₇	direct (N × H)
29. C₃²	C₄×C₇×C₁₂₇	direct (N × H)
30. C₇	C₃²×C₄×C₁₂₇	direct (N × H)
32. C₄	C₃²×C₇×C₁₂₇	direct (N × H)
33. C₃	C₃×C₄×C₇×C₁₂₇	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×C₃²×C₇×C₁₂₇	2
C₃×C₄×C₇×C₁₂₇	3
C₃²×C₄×C₁₂₇	7
C₃²×C₄×C₇	127

Subgroups — C32×C4×C7×C127 #44544698

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×C₄×C₇×C₁₂₇ #44544698