Finite Group Databasebeta

Subgroups — #36340751

Order 21567.

16 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×C₁₃×C₇₉	C₃	1	1	yes	maximal, characteristic, complemented, Hall
2	#15690494	—	7	1	no	maximal, Hall
3	#13110305	—	13	1	no	maximal, Hall
4	C₁₃×C₇₉	ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Hall
5	C₇×C₇₉	C₁₃⋊C₃	1	1	yes	characteristic, complemented, Hall
6	#63271	—	79	1	no	maximal, Hall
7	C₇₉⋊C₃	—	91	1	no	Hall
8	C₇×C₁₃	C₇₉⋊C₃	1	1	yes	characteristic, complemented, Hall
9	C₇₉	#63271	1	1	yes	characteristic, complemented, Sylow
10	C₁₃⋊C₃	—	553	1	no	Hall
11	ΓL(1,𝔽₈)	—	1027	1	no	Hall
12	C₁₃	#13110305	1	1	yes	characteristic, complemented, Sylow
13	C₇	#15690494	1	1	yes	characteristic, complemented, Sylow
14	C₃	—	7189	1	no	Sylow

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₇×C₁₃×C₇₉	C₃
LCS	G ▷ C₇×C₁₃×C₇₉
UCS	C₁

Complements

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

N	H ≅ G/N	type
1. C₇×C₁₃×C₇₉	C₃	semidirect
4. C₁₃×C₇₉	ΓL(1,𝔽₈)	semidirect
5. C₇×C₇₉	C₁₃⋊C₃	semidirect
8. C₇×C₁₃	C₇₉⋊C₃	semidirect
9. C₇₉	#63271	semidirect
12. C₁₃	#13110305	semidirect
13. C₇	#15690494	semidirect

Maximal subgroups

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

subgroup	index
C₇×C₁₃×C₇₉	3
#15690494	7
#13110305	13
#63271	79