Finite Group Databasebeta

Subgroups — #36374983

Order 22336.

14 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26013545	C₂	1	1	yes	maximal, characteristic
2	C₁₆×C₃₄₉	C₄	1	1	yes	characteristic
3	C₈×C₃₄₉	C₈	1	1	yes	characteristic
4	C₄×C₃₄₉	C₁₆	1	1	yes	characteristic
5	C₂×C₃₄₉	C₃₂	1	1	yes	characteristic
6	C₃₄₉	C₆₄	1	1	yes	characteristic, complemented, Sylow
7	C₆₄	—	349	1	no	maximal, Sylow
8	C₃₂	—	349	1	no	—
9	C₁₆	#13063703	1	1	yes	characteristic
10	C₈	#14584102	1	1	yes	characteristic
11	C₄	#19150236	1	1	yes	characteristic
12	C₂	#26013689	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₁₆	#13063703
commutator, hypocenter	C₃₄₉	C₆₄
Fitting	C₁₆×C₃₄₉	C₄
socle, prime radical	C₂×C₃₄₉	C₃₂
cosocle	#26013545	C₂
LCS	G ▷ C₃₄₉
UCS	C₁ ◁ C₁₆

Complements

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

N	H ≅ G/N	type
6. C₃₄₉	C₆₄	semidirect

Maximal subgroups

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

subgroup	index
#26013545	2
C₆₄	349