Finite Group Databasebeta

← C₂²×C₄₉×C₁₃₇ #41001639

Subgroups — C₂²×C₄₉×C₁₃₇ #41001639

Order 26852.

30 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₉×C₁₃₇	C₂	1	3	yes	maximal, direct product factor
2	C₄₉×C₁₃₇	C₂²	1	1	yes	characteristic, direct product factor, Hall
3	C₂²×C₇×C₁₃₇	C₇	1	1	yes	maximal, characteristic
4	C₂×C₇×C₁₃₇	C₂×C₇	1	3	yes	—
5	C₇×C₁₃₇	C₂²×C₇	1	1	yes	characteristic
6	C₂²×C₁₃₇	C₄₉	1	1	yes	characteristic, direct product factor, Hall
7	C₂×C₁₃₇	C₂×C₄₉	1	3	yes	direct product factor
8	C₂²×C₄₉	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₁₃₇	C₂²×C₄₉	1	1	yes	characteristic, direct product factor, Sylow
10	C₂×C₄₉	C₂×C₁₃₇	1	3	yes	direct product factor
11	C₄₉	C₂²×C₁₃₇	1	1	yes	characteristic, direct product factor, Sylow
12	C₂²×C₇	C₇×C₁₃₇	1	1	yes	characteristic
13	C₂×C₇	C₂×C₇×C₁₃₇	1	3	yes	—
14	C₇	C₂²×C₇×C₁₃₇	1	1	yes	characteristic
15	C₂²	C₄₉×C₁₃₇	1	1	yes	characteristic, direct product factor, Sylow
16	C₂	C₂×C₄₉×C₁₃₇	1	3	yes	direct product factor

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₇	C₂²×C₇×C₁₃₇
socle, prime radical	C₂²×C₇×C₁₃₇	C₇
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₄₉×C₁₃₇	C₂	direct (N × H)
2. C₄₉×C₁₃₇	C₂²	direct (N × H)
6. C₂²×C₁₃₇	C₄₉	direct (N × H)
7. C₂×C₁₃₇	C₂×C₄₉	direct (N × H)
8. C₂²×C₄₉	C₁₃₇	direct (N × H)
9. C₁₃₇	C₂²×C₄₉	direct (N × H)
10. C₂×C₄₉	C₂×C₁₃₇	direct (N × H)
11. C₄₉	C₂²×C₁₃₇	direct (N × H)
15. C₂²	C₄₉×C₁₃₇	direct (N × H)
16. C₂	C₂×C₄₉×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
C₂×C₄₉×C₁₃₇	2
C₂²×C₇×C₁₃₇	7
C₂²×C₄₉	137