Finite Group Databasebeta

← C₂×C₇×H(3,𝔽₃) #67392

Subgroups — C₂×C₇×H(3,𝔽₃) #67392

Order 378.

44 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×H(3,𝔽₃)	C₂	1	1	yes	maximal, characteristic, direct product factor, Hall
2	C₂×C₃²×C₇	C₃	1	4	yes	maximal, complemented
3	C₃²×C₇	C₂×C₃	1	4	yes	complemented
4	C₂×H(3,𝔽₃)	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₂×C₃×C₇	—	3	4	no	—
6	C₂×C₃×C₇	C₃²	1	1	yes	characteristic
7	H(3,𝔽₃)	C₂×C₇	1	1	yes	characteristic, direct product factor, Sylow
8	C₃×C₇	—	3	4	no	—
9	C₃×C₇	C₂×C₃²	1	1	yes	characteristic
10	C₂×C₃²	C₃×C₇	1	4	yes	complemented
11	C₂×C₇	H(3,𝔽₃)	1	1	yes	characteristic, direct product factor, Hall
12	C₃²	C₂×C₃×C₇	1	4	yes	complemented
13	C₇	C₂×H(3,𝔽₃)	1	1	yes	characteristic, direct product factor, Sylow
14	C₂×C₃	—	3	4	no	—
15	C₂×C₃	C₃²×C₇	1	1	yes	characteristic
16	C₃	—	3	4	no	—
17	C₃	C₂×C₃²×C₇	1	1	yes	characteristic
18	C₂	C₇×H(3,𝔽₃)	1	1	yes	characteristic, direct product factor, Sylow

	subgroup	G/H
center, socle	C₂×C₃×C₇	C₃²
commutator, Frattini, cosocle	C₃	C₂×C₃²×C₇
LCS	G ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₇×H(3,𝔽₃)	C₂	direct (N × H)
2. C₂×C₃²×C₇	C₃	semidirect
3. C₃²×C₇	C₂×C₃	semidirect
4. C₂×H(3,𝔽₃)	C₇	direct (N × H)
7. H(3,𝔽₃)	C₂×C₇	direct (N × H)
10. C₂×C₃²	C₃×C₇	semidirect
11. C₂×C₇	H(3,𝔽₃)	direct (N × H)
12. C₃²	C₂×C₃×C₇	semidirect
13. C₇	C₂×H(3,𝔽₃)	direct (N × H)
18. C₂	C₇×H(3,𝔽₃)	direct (N × H)

Maximal subgroups

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

subgroup	index
C₇×H(3,𝔽₃)	2
C₂×C₃²×C₇	3
C₂×H(3,𝔽₃)	7