Finite Group Databasebeta

← C₂×C₆₇×C₆₇⋊C₃ #41002230

Subgroups — C₂×C₆₇×C₆₇⋊C₃ #41002230

Order 26934.

60 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×C₆₇⋊C₃	C₂	1	1	yes	maximal, characteristic, direct product factor, Hall
2	C₂×C₆₇²	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	C₆₇²	C₂×C₃	1	1	yes	characteristic, complemented, Sylow
4	C₂×C₃×C₆₇	—	67	1	no	maximal
5	C₂×C₆₇⋊C₃	C₆₇	1	1	yes	maximal, characteristic, direct product factor
6	C₃×C₆₇	—	67	1	no	—
7	C₆₇⋊C₃	C₂×C₆₇	1	1	yes	characteristic, direct product factor
8	C₂×C₆₇	—	3	22	no	—
9	C₂×C₆₇	C₆₇⋊C₃	1	1	yes	characteristic, direct product factor
10	C₂×C₆₇	C₃×C₆₇	1	1	yes	characteristic, complemented
11	C₆₇	—	3	22	no	—
12	C₆₇	C₂×C₆₇⋊C₃	1	1	yes	characteristic, direct product factor
13	C₆₇	C₂×C₃×C₆₇	1	1	yes	characteristic, complemented
14	C₂×C₃	—	67	1	no	Hall
15	C₃	—	67	1	no	Sylow
16	C₂	C₆₇×C₆₇⋊C₃	1	1	yes	characteristic, direct product factor, Sylow

	subgroup	G/H
center, hypercenter	C₂×C₆₇	C₆₇⋊C₃
commutator, cosocle, hypocenter	C₆₇	C₂×C₃×C₆₇
Frattini	C₁	G
Fitting, socle	C₂×C₆₇²	C₃
LCS	G ▷ C₆₇
UCS	C₁ ◁ C₂×C₆₇

Complements

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

N	H ≅ G/N	type
1. C₆₇×C₆₇⋊C₃	C₂	direct (N × H)
2. C₂×C₆₇²	C₃	semidirect
3. C₆₇²	C₂×C₃	semidirect
5. C₂×C₆₇⋊C₃	C₆₇	direct (N × H)
7. C₆₇⋊C₃	C₂×C₆₇	direct (N × H)
9. C₂×C₆₇	C₆₇⋊C₃	direct (N × H)
10. C₂×C₆₇	C₃×C₆₇	semidirect
12. C₆₇	C₂×C₆₇⋊C₃	direct (N × H)
13. C₆₇	C₂×C₃×C₆₇	semidirect
16. C₂	C₆₇×C₆₇⋊C₃	direct (N × H)

Maximal subgroups

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

subgroup	index
C₆₇×C₆₇⋊C₃	2
C₂×C₆₇²	3
C₂×C₃×C₆₇	67
C₂×C₆₇⋊C₃	67