Finite Group Databasebeta

← C₇×C₃⋊(C₃⋊(C₃⋊D₃)) #11765822

Subgroups — C₇×C₃⋊(C₃⋊(C₃⋊D₃)) #11765822

Order 1134.

848 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃⁴×C₇	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	C₇×C₃⋊(C₃⋊D₃)	—	3	40	no	maximal
3	C₃³×C₇	D₃	1	40	yes	complemented
4	C₃⋊(C₃⋊(C₃⋊D₃))	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₇×C₃⋊D₃	—	9	130	no	—
6	C₃⁴	C₂×C₇	1	1	yes	characteristic, complemented, Sylow
7	C₃²×C₇	C₃⋊D₃	1	130	yes	complemented
8	C₃⋊(C₃⋊D₃)	—	3	40	no	—
9	D₃×C₇	—	27	40	no	—
10	C₃³	D₃×C₇	1	40	yes	complemented
11	C₃×C₇	C₃⋊(C₃⋊D₃)	1	40	yes	complemented
12	C₃⋊D₃	—	9	130	no	—
13	C₂×C₇	—	81	1	no	Hall
14	C₃²	C₇×C₃⋊D₃	1	130	yes	complemented
15	C₇	C₃⋊(C₃⋊(C₃⋊D₃))	1	1	yes	characteristic, direct product factor, Sylow
16	D₃	—	27	40	no	—
17	C₃	C₇×C₃⋊(C₃⋊D₃)	1	40	yes	complemented
18	C₂	—	81	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₇	C₃⋊(C₃⋊(C₃⋊D₃))
commutator, cosocle, hypocenter	C₃⁴	C₂×C₇
Frattini	C₁	G
Fitting, socle	C₃⁴×C₇	C₂
LCS	G ▷ C₃⁴
UCS	C₁ ◁ C₇

Complements

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

N	H ≅ G/N	type
1. C₃⁴×C₇	C₂	semidirect
3. C₃³×C₇	D₃	semidirect
4. C₃⋊(C₃⋊(C₃⋊D₃))	C₇	direct (N × H)
6. C₃⁴	C₂×C₇	semidirect
7. C₃²×C₇	C₃⋊D₃	semidirect
10. C₃³	D₃×C₇	semidirect
11. C₃×C₇	C₃⋊(C₃⋊D₃)	semidirect
14. C₃²	C₇×C₃⋊D₃	semidirect
15. C₇	C₃⋊(C₃⋊(C₃⋊D₃))	direct (N × H)
17. C₃	C₇×C₃⋊(C₃⋊D₃)	semidirect

Maximal subgroups

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

subgroup	index
C₃⁴×C₇	2
C₇×C₃⋊(C₃⋊D₃)	3
C₃⋊(C₃⋊(C₃⋊D₃))	7