Finite Group Databasebeta

← C₇×C₁₃×H(3,𝔽₇) #44530134

Subgroups — C₇×C₁₃×H(3,𝔽₇) #44530134

Order 31213.

472 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇³×C₁₃	C₇	1	8	yes	maximal, complemented
2	C₁₃×H(3,𝔽₇)	C₇	1	49	yes	maximal, direct product factor
3	C₇×H(3,𝔽₇)	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
4	C₇²×C₁₃	—	7	56	no	—
5	C₇²×C₁₃	C₇²	1	1	yes	characteristic
6	C₇²×C₁₃	C₇²	1	56	yes	complemented
7	C₇³	C₇×C₁₃	1	8	yes	complemented
8	H(3,𝔽₇)	C₇×C₁₃	1	49	yes	direct product factor
9	C₇×C₁₃	—	7	56	no	—
10	C₇×C₁₃	H(3,𝔽₇)	1	7	yes	direct product factor
11	C₇×C₁₃	C₇³	1	1	yes	characteristic
12	C₇²	—	7	56	no	—
13	C₇²	C₇²×C₁₃	1	1	yes	characteristic
14	C₇²	C₇²×C₁₃	1	56	yes	complemented
15	C₁₃	C₇×H(3,𝔽₇)	1	1	yes	characteristic, direct product factor, Sylow
16	C₇	—	7	56	no	—
17	C₇	C₁₃×H(3,𝔽₇)	1	7	yes	direct product factor
18	C₇	C₇³×C₁₃	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₇²×C₁₃	C₇²
commutator, Frattini, cosocle	C₇	C₇³×C₁₃
LCS	G ▷ C₇ ▷ C₁
UCS	C₁ ◁ C₇²×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₇³×C₁₃	C₇	semidirect
2. C₁₃×H(3,𝔽₇)	C₇	direct (N × H)
3. C₇×H(3,𝔽₇)	C₁₃	direct (N × H)
6. C₇²×C₁₃	C₇²	semidirect
7. C₇³	C₇×C₁₃	semidirect
8. H(3,𝔽₇)	C₇×C₁₃	direct (N × H)
10. C₇×C₁₃	H(3,𝔽₇)	direct (N × H)
14. C₇²	C₇²×C₁₃	semidirect
15. C₁₃	C₇×H(3,𝔽₇)	direct (N × H)
17. C₇	C₁₃×H(3,𝔽₇)	direct (N × H)

Maximal subgroups

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

subgroup	index
C₇³×C₁₃	7
C₁₃×H(3,𝔽₇)	7
C₇×H(3,𝔽₇)	13