Finite Group Databasebeta

← C₂×C₆₇×D₁₃₉ #49152000

Subgroups — C₂×C₆₇×D₁₃₉ #49152000

Order 37252.

20 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₆₇×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₆₇×D₁₃₉	C₂	1	2	yes	maximal, direct product factor
3	C₆₇×C₁₃₉	C₂²	1	1	yes	characteristic, complemented, Hall
4	C₂×D₁₃₉	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₂×C₁₃₉	C₂×C₆₇	1	1	yes	characteristic, complemented
6	D₁₃₉	C₂×C₆₇	1	2	yes	direct product factor
7	C₂²×C₆₇	—	139	1	no	maximal, Hall
8	C₁₃₉	C₂²×C₆₇	1	1	yes	characteristic, complemented, Sylow
9	C₂×C₆₇	—	139	2	no	—
10	C₂×C₆₇	D₁₃₉	1	1	yes	characteristic, direct product factor
11	C₆₇	C₂×D₁₃₉	1	1	yes	characteristic, direct product factor, Sylow
12	C₂²	—	139	1	no	Sylow
13	C₂	—	139	2	no	—
14	C₂	C₆₇×D₁₃₉	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂×C₆₇	D₁₃₉
commutator, cosocle, hypocenter	C₁₃₉	C₂²×C₆₇
Frattini	C₁	G
Fitting, socle	C₂×C₆₇×C₁₃₉	C₂
LCS	G ▷ C₁₃₉
UCS	C₁ ◁ C₂×C₆₇

Complements

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

N	H ≅ G/N	type
1. C₂×C₆₇×C₁₃₉	C₂	semidirect
2. C₆₇×D₁₃₉	C₂	direct (N × H)
3. C₆₇×C₁₃₉	C₂²	semidirect
4. C₂×D₁₃₉	C₆₇	direct (N × H)
5. C₂×C₁₃₉	C₂×C₆₇	semidirect
6. D₁₃₉	C₂×C₆₇	direct (N × H)
8. C₁₃₉	C₂²×C₆₇	semidirect
10. C₂×C₆₇	D₁₃₉	direct (N × H)
11. C₆₇	C₂×D₁₃₉	direct (N × H)
14. C₂	C₆₇×D₁₃₉	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×C₆₇×C₁₃₉	2
C₆₇×D₁₃₉	2
C₂×D₁₃₉	67
C₂²×C₆₇	139