Subgroups — C₇⋊(Q₈⋊C₉) #92597

Order 504.

22 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₇×Q₈	C₃	1	1	yes	maximal, characteristic
2	C₂×C₇⋊C₉	—	4	1	no	maximal
3	C₃×C₄×C₇	—	3	1	no	—
4	Q₈⋊C₉	—	7	1	no	maximal, Hall
5	C₇⋊C₉	—	4	1	no	Hall
6	C₇×Q₈	C₉	1	1	yes	characteristic, complemented, Hall
7	C₂×C₃×C₇	A₄	1	1	yes	characteristic
8	C₄×C₇	—	3	1	no	—
9	C₃×Q₈	ΓL(1,𝔽₈)	1	1	yes	characteristic
10	C₃×C₇	SL(2,𝔽₃)	1	1	yes	characteristic
11	C₂×C₉	—	28	1	no	—
12	C₂×C₇	C₂²⋊C₉	1	1	yes	characteristic
13	C₃×C₄	—	3	1	no	—
14	C₉	—	28	1	no	Sylow
15	Q₈	C₇⋊C₉	1	1	yes	characteristic, complemented, Sylow
16	C₇	Q₈⋊C₉	1	1	yes	characteristic, complemented, Sylow
17	C₂×C₃	C₂²⋊ΓL(1,𝔽₈)	1	1	yes	characteristic
18	C₄	—	3	1	no	—
19	C₃	C₇⋊SL(2,𝔽₃)	1	1	yes	characteristic
20	C₂	C₂²⋊(C₇⋊C₉)	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₂×C₃	C₂²⋊ΓL(1,𝔽₈)
commutator, hypocenter	C₇×Q₈	C₉
Fitting, cosocle	C₃×C₇×Q₈	C₃
socle, prime radical	C₂×C₃×C₇	A₄
LCS	G ▷ C₇×Q₈
UCS	C₁ ◁ C₂×C₃

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

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