Subgroups — C₃×C₂²⋊#14556233 #42117607

Order 27468.

56 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²⋊#14556233	C₃	1	3	yes	maximal, direct product factor
2	C₂²×C₃×C₇×C₁₀₉	C₃	1	1	yes	maximal, characteristic, complemented
3	C₃×#14556233	—	4	1	no	maximal, Hall
4	C₂×C₃×C₇×C₁₀₉	—	3	1	no	—
5	C₃×C₂²⋊(C₁₀₉⋊C₃)	—	7	1	no	maximal, Hall
6	C₂²×C₇×C₁₀₉	C₃²	1	1	yes	characteristic, complemented, Hall
7	C₃×C₇×C₁₀₉	A₄	1	1	yes	characteristic, complemented
8	#14556233	—	4	3	no	—
9	C₂×C₇×C₁₀₉	—	3	1	no	—
10	C₂²×C₃×C₁₀₉	ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
11	C₂²⋊(C₁₀₉⋊C₃)	—	7	3	no	—
12	C₃×C₁₀₉⋊C₃	—	28	1	no	Hall
13	C₇×C₁₀₉	C₃×A₄	1	1	yes	characteristic, complemented, Hall
14	C₂×C₃×C₁₀₉	—	3	1	no	—
15	C₂²×C₁₀₉	C₃×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Hall
16	C₃×C₁₀₉	C₂²⋊ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
17	C₁₀₉⋊C₃	—	28	3	no	—
18	C₃×C₂²⋊ΓL(1,𝔽₈)	—	109	1	no	maximal, Hall
19	C₂×C₁₀₉	—	3	1	no	—
20	C₁₀₉	C₃×C₂²⋊ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Sylow
21	C₂²×C₃×C₇	C₁₀₉⋊C₃	1	1	yes	characteristic, complemented
22	C₂²⋊ΓL(1,𝔽₈)	—	109	3	no	—
23	C₃×ΓL(1,𝔽₈)	—	436	1	no	Hall
24	C₂×C₃×C₇	—	3	1	no	—
25	C₃×A₄	—	763	1	no	Hall
26	C₂²×C₇	C₃×C₁₀₉⋊C₃	1	1	yes	characteristic, complemented, Hall
27	C₃×C₇	C₂²⋊(C₁₀₉⋊C₃)	1	1	yes	characteristic, complemented
28	ΓL(1,𝔽₈)	—	436	3	no	—
29	C₂×C₇	—	3	1	no	—
30	C₂²×C₃	#14556233	1	1	yes	characteristic, complemented
31	A₄	—	763	3	no	—
32	C₃²	—	3052	1	no	Sylow
33	C₇	C₃×C₂²⋊(C₁₀₉⋊C₃)	1	1	yes	characteristic, complemented, Sylow
34	C₂×C₃	—	3	1	no	—
35	C₂²	C₃×#14556233	1	1	yes	characteristic, complemented, Sylow
36	C₃	—	3052	3	no	—
37	C₃	C₂²⋊#14556233	1	1	yes	characteristic, direct product factor
38	C₂	—	3	1	no	—

	subgroup	G/H
center, hypercenter	C₃	C₂²⋊#14556233
commutator, cosocle, hypocenter	C₂²×C₇×C₁₀₉	C₃²
Frattini	C₁	G
Fitting, socle	C₂²×C₃×C₇×C₁₀₉	C₃
LCS	G ▷ C₂²×C₇×C₁₀₉
UCS	C₁ ◁ C₃

Complements

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

N	H ≅ G/N	type
1. C₂²⋊#14556233	C₃	direct (N × H)
2. C₂²×C₃×C₇×C₁₀₉	C₃	semidirect
6. C₂²×C₇×C₁₀₉	C₃²	semidirect
7. C₃×C₇×C₁₀₉	A₄	semidirect
10. C₂²×C₃×C₁₀₉	ΓL(1,𝔽₈)	semidirect
13. C₇×C₁₀₉	C₃×A₄	semidirect
15. C₂²×C₁₀₉	C₃×ΓL(1,𝔽₈)	semidirect
16. C₃×C₁₀₉	C₂²⋊ΓL(1,𝔽₈)	semidirect
20. C₁₀₉	C₃×C₂²⋊ΓL(1,𝔽₈)	semidirect
21. C₂²×C₃×C₇	C₁₀₉⋊C₃	semidirect
26. C₂²×C₇	C₃×C₁₀₉⋊C₃	semidirect
27. C₃×C₇	C₂²⋊(C₁₀₉⋊C₃)	semidirect
30. C₂²×C₃	#14556233	semidirect
33. C₇	C₃×C₂²⋊(C₁₀₉⋊C₃)	semidirect
35. C₂²	C₃×#14556233	semidirect
37. C₃	C₂²⋊#14556233	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²⋊#14556233	3
C₂²×C₃×C₇×C₁₀₉	3
C₃×#14556233	4
C₃×C₂²⋊(C₁₀₉⋊C₃)	7
C₃×C₂²⋊ΓL(1,𝔽₈)	109

Subgroups — C3×C22⋊#14556233 #42117607

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₂²⋊#14556233 #42117607