Subgroups — C₃×C₇⋊(Q₈⋊C₉) #13094439

Order 1512.

72 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₇×Q₈	C₃	1	1	yes	maximal, characteristic
2	C₇⋊(Q₈⋊C₉)	C₃	1	3	yes	maximal, direct product factor
3	C₂×C₃×C₇⋊C₉	—	4	1	no	maximal
4	C₃²×C₄×C₇	—	3	1	no	—
5	C₃×Q₈⋊C₉	—	7	1	no	maximal, Hall
6	C₃×C₇⋊C₉	—	4	1	no	Hall
7	C₃×C₇×Q₈	C₉	1	3	yes	complemented
8	C₃×C₇×Q₈	C₃²	1	1	yes	characteristic
9	C₂×C₃²×C₇	A₄	1	1	yes	characteristic
10	C₂×C₇⋊C₉	—	4	3	no	—
11	C₃×C₄×C₇	—	3	3	no	—
12	C₃×C₄×C₇	—	3	1	no	—
13	C₃²×Q₈	ΓL(1,𝔽₈)	1	1	yes	characteristic
14	Q₈⋊C₉	—	7	3	no	—
15	C₃²×C₇	SL(2,𝔽₃)	1	1	yes	characteristic
16	C₇⋊C₉	—	4	3	no	—
17	C₇×Q₈	C₃×C₉	1	1	yes	characteristic, complemented, Hall
18	C₂×C₃×C₉	—	28	1	no	—
19	C₂×C₃×C₇	C₂²⋊C₉	1	3	yes	—
20	C₂×C₃×C₇	C₃×A₄	1	1	yes	characteristic
21	C₃²×C₄	—	3	1	no	—
22	C₄×C₇	—	3	1	no	—
23	C₃×C₉	—	28	1	no	Sylow
24	C₃×Q₈	C₇⋊C₉	1	3	yes	complemented
25	C₃×Q₈	C₃×ΓL(1,𝔽₈)	1	1	yes	characteristic
26	C₃×C₇	Q₈⋊C₉	1	3	yes	complemented
27	C₃×C₇	C₃×SL(2,𝔽₃)	1	1	yes	characteristic
28	C₂×C₃²	C₂²⋊ΓL(1,𝔽₈)	1	1	yes	characteristic
29	C₂×C₉	—	28	3	no	—
30	C₂×C₇	C₃×C₂²⋊C₉	1	1	yes	characteristic
31	C₃×C₄	—	3	3	no	—
32	C₃×C₄	—	3	1	no	—
33	C₃²	C₇⋊SL(2,𝔽₃)	1	1	yes	characteristic
34	C₉	—	28	3	no	—
35	Q₈	C₃×C₇⋊C₉	1	1	yes	characteristic, complemented, Sylow
36	C₇	C₃×Q₈⋊C₉	1	1	yes	characteristic, complemented, Sylow
37	C₂×C₃	C₂²⋊(C₇⋊C₉)	1	3	yes	—
38	C₂×C₃	C₃×C₂²⋊ΓL(1,𝔽₈)	1	1	yes	characteristic
39	C₄	—	3	1	no	—
40	C₃	C₇⋊(Q₈⋊C₉)	1	3	yes	direct product factor
41	C₃	C₃×C₇⋊SL(2,𝔽₃)	1	1	yes	characteristic
42	C₂	C₃×C₂²⋊(C₇⋊C₉)	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂×C₃²	C₂²⋊ΓL(1,𝔽₈)
commutator, hypocenter	C₇×Q₈	C₃×C₉
Frattini	C₂×C₃	C₃×C₂²⋊ΓL(1,𝔽₈)
Fitting	C₃²×C₇×Q₈	C₃
socle, prime radical	C₂×C₃²×C₇	A₄
cosocle	C₃×C₇×Q₈	C₃²
LCS	G ▷ C₇×Q₈
UCS	C₁ ◁ C₂×C₃²

Complements

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

N	H ≅ G/N	type
2. C₇⋊(Q₈⋊C₉)	C₃	direct (N × H)
7. C₃×C₇×Q₈	C₉	semidirect
17. C₇×Q₈	C₃×C₉	semidirect
24. C₃×Q₈	C₇⋊C₉	semidirect
26. C₃×C₇	Q₈⋊C₉	semidirect
35. Q₈	C₃×C₇⋊C₉	semidirect
36. C₇	C₃×Q₈⋊C₉	semidirect
40. C₃	C₇⋊(Q₈⋊C₉)	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃²×C₇×Q₈	3
C₇⋊(Q₈⋊C₉)	3
C₂×C₃×C₇⋊C₉	4
C₃×Q₈⋊C₉	7

Subgroups — C3×C7⋊(Q8⋊C9) #13094439

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₇⋊(Q₈⋊C₉) #13094439