Finite Group Databasebeta

← C₂⁷×C₁₃₇ #31778607

Subgroups — C₂⁷×C₁₃₇ #31778607

Order 17536.

58424 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁶×C₁₃₇	C₂	1	127	yes	maximal, direct product factor
2	C₂⁵×C₁₃₇	C₂²	1	2667	yes	direct product factor
3	C₂⁴×C₁₃₇	C₂³	1	11811	yes	direct product factor
4	C₂³×C₁₃₇	C₂⁴	1	11811	yes	direct product factor
5	C₂²×C₁₃₇	C₂⁵	1	2667	yes	direct product factor
6	C₂×C₁₃₇	C₂⁶	1	127	yes	direct product factor
7	C₁₃₇	C₂⁷	1	1	yes	characteristic, direct product factor, Sylow
8	C₂⁷	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
9	C₂⁶	C₂×C₁₃₇	1	127	yes	direct product factor
10	C₂⁵	C₂²×C₁₃₇	1	2667	yes	direct product factor
11	C₂⁴	C₂³×C₁₃₇	1	11811	yes	direct product factor
12	C₂³	C₂⁴×C₁₃₇	1	11811	yes	direct product factor
13	C₂²	C₂⁵×C₁₃₇	1	2667	yes	direct product factor
14	C₂	C₂⁶×C₁₃₇	1	127	yes	direct product factor

	subgroup	G/H
center	G	C₁
commutator, Frattini	C₁	G
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂⁶×C₁₃₇	C₂	direct (N × H)
2. C₂⁵×C₁₃₇	C₂²	direct (N × H)
3. C₂⁴×C₁₃₇	C₂³	direct (N × H)
4. C₂³×C₁₃₇	C₂⁴	direct (N × H)
5. C₂²×C₁₃₇	C₂⁵	direct (N × H)
6. C₂×C₁₃₇	C₂⁶	direct (N × H)
7. C₁₃₇	C₂⁷	direct (N × H)
8. C₂⁷	C₁₃₇	direct (N × H)
9. C₂⁶	C₂×C₁₃₇	direct (N × H)
10. C₂⁵	C₂²×C₁₃₇	direct (N × H)
11. C₂⁴	C₂³×C₁₃₇	direct (N × H)
12. C₂³	C₂⁴×C₁₃₇	direct (N × H)
13. C₂²	C₂⁵×C₁₃₇	direct (N × H)
14. C₂	C₂⁶×C₁₃₇	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁶×C₁₃₇	2
C₂⁷	137