Finite Group Databasebeta

← C₆₇×D₂₃₇ #44540871

Subgroups — C₆₇×D₂₃₇ #44540871

Order 31758.

16 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₆₇×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	C₆₇×D₇₉	—	3	1	no	maximal, Hall
3	C₆₇×C₇₉	D₃	1	1	yes	characteristic, complemented, Hall
4	D₂₃₇	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Hall
5	D₃×C₆₇	—	79	1	no	maximal, Hall
6	C₃×C₇₉	C₂×C₆₇	1	1	yes	characteristic, complemented, Hall
7	C₃×C₆₇	D₇₉	1	1	yes	characteristic, complemented, Hall
8	D₇₉	—	3	1	no	Hall
9	C₂×C₆₇	—	237	1	no	Hall
10	C₇₉	D₃×C₆₇	1	1	yes	characteristic, complemented, Sylow
11	C₆₇	D₂₃₇	1	1	yes	characteristic, direct product factor, Sylow
12	D₃	—	79	1	no	Hall
13	C₃	C₆₇×D₇₉	1	1	yes	characteristic, complemented, Sylow
14	C₂	—	237	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₆₇	D₂₃₇
commutator, cosocle, hypocenter	C₃×C₇₉	C₂×C₆₇
Frattini	C₁	G
Fitting, socle	C₃×C₆₇×C₇₉	C₂
LCS	G ▷ C₃×C₇₉
UCS	C₁ ◁ C₆₇

Complements

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

N	H ≅ G/N	type
1. C₃×C₆₇×C₇₉	C₂	semidirect
3. C₆₇×C₇₉	D₃	semidirect
4. D₂₃₇	C₆₇	direct (N × H)
6. C₃×C₇₉	C₂×C₆₇	semidirect
7. C₃×C₆₇	D₇₉	semidirect
10. C₇₉	D₃×C₆₇	semidirect
11. C₆₇	D₂₃₇	direct (N × H)
13. C₃	C₆₇×D₇₉	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₆₇×C₇₉	2
C₆₇×D₇₉	3
D₂₃₇	67
D₃×C₆₇	79