Subgroups — C₃×ΓL(1,𝔽₈)⋊#64408 #32913849

Order 18396.

72 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×ΓL(1,𝔽₈)×D₇₃	C₂	1	1	yes	maximal, characteristic
2	C₃×C₇⋊#64408	C₃	1	1	yes	maximal, characteristic, complemented
3	ΓL(1,𝔽₈)⋊#64408	C₃	1	3	yes	maximal, direct product factor
4	C₃×ΓL(1,𝔽₈)×C₇₃	C₄	1	1	yes	characteristic, complemented, Hall
5	ΓL(1,𝔽₈)×D₇₃	C₂×C₃	1	3	yes	—
6	C₃×C₇×D₇₃	C₂×C₃	1	1	yes	characteristic
7	C₃²×#64408	—	7	1	no	maximal, Hall
8	C₇⋊#64408	C₃²	1	1	yes	characteristic, complemented, Hall
9	C₃×C₇×C₇₃	C₃×C₄	1	1	yes	characteristic, complemented
10	ΓL(1,𝔽₈)×C₇₃	C₃×C₄	1	3	yes	complemented
11	C₃²×D₇₃	—	7	1	no	—
12	C₇×D₇₃	C₂×C₃²	1	1	yes	characteristic
13	C₃×#64408	—	7	3	no	—
14	C₃×#64408	—	21	1	no	—
15	C₃²×C₇₃	—	7	1	no	Hall
16	C₇×C₇₃	C₃²×C₄	1	1	yes	characteristic, complemented, Hall
17	C₃×D₇₃	—	7	3	no	—
18	C₃×D₇₃	AGL(1,𝔽₇)	1	1	yes	characteristic
19	#64408	—	21	1	no	Hall
20	C₃×ΓL(1,𝔽₈)⋊C₄	—	73	1	no	maximal, Hall
21	C₃×C₇₃	—	7	3	no	—
22	C₃×C₇₃	ΓL(1,𝔽₈)⋊C₄	1	1	yes	characteristic, complemented
23	D₇₃	C₃×AGL(1,𝔽₇)	1	1	yes	characteristic
24	C₂×C₃×ΓL(1,𝔽₈)	—	73	1	no	—
25	C₃×Dic₇	—	73	1	no	—
26	ΓL(1,𝔽₈)⋊C₄	—	73	3	no	—
27	C₇₃	C₃×ΓL(1,𝔽₈)⋊C₄	1	1	yes	characteristic, complemented, Sylow
28	C₃×ΓL(1,𝔽₈)	#64408	1	1	yes	characteristic, complemented, Hall
29	C₂×C₃×C₇	—	73	1	no	—
30	C₂×ΓL(1,𝔽₈)	—	73	3	no	—
31	C₃²×C₄	—	511	1	no	Hall
32	Dic₇	—	73	1	no	Hall
33	C₃×C₇	C₃×#64408	1	1	yes	characteristic, complemented
34	ΓL(1,𝔽₈)	C₃×#64408	1	3	yes	complemented
35	C₂×C₃²	—	511	1	no	—
36	C₂×C₇	—	73	1	no	—
37	C₃×C₄	—	511	1	no	—
38	C₃×C₄	—	511	3	no	—
39	C₃²	—	7	1	no	Sylow
40	C₇	C₃²×#64408	1	1	yes	characteristic, complemented, Sylow
41	C₂×C₃	—	73	1	no	—
42	C₂×C₃	—	511	3	no	—
43	C₄	—	511	1	no	Sylow
44	C₃	—	7	3	no	—
45	C₃	ΓL(1,𝔽₈)⋊#64408	1	1	yes	characteristic, direct product factor
46	C₂	—	73	1	no	—

	subgroup	G/H
center, hypercenter	C₃	ΓL(1,𝔽₈)⋊#64408
commutator, hypocenter	C₇×C₇₃	C₃²×C₄
Frattini	C₁	G
Fitting, socle	C₃×C₇×C₇₃	C₃×C₄
prime radical	C₃×ΓL(1,𝔽₈)×D₇₃	C₂
cosocle	C₇×D₇₃	C₂×C₃²
LCS	G ▷ C₇×C₇₃
UCS	C₁ ◁ C₃

Complements

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

N	H ≅ G/N	type
2. C₃×C₇⋊#64408	C₃	semidirect
3. ΓL(1,𝔽₈)⋊#64408	C₃	direct (N × H)
4. C₃×ΓL(1,𝔽₈)×C₇₃	C₄	semidirect
8. C₇⋊#64408	C₃²	semidirect
9. C₃×C₇×C₇₃	C₃×C₄	semidirect
10. ΓL(1,𝔽₈)×C₇₃	C₃×C₄	semidirect
16. C₇×C₇₃	C₃²×C₄	semidirect
22. C₃×C₇₃	ΓL(1,𝔽₈)⋊C₄	semidirect
27. C₇₃	C₃×ΓL(1,𝔽₈)⋊C₄	semidirect
28. C₃×ΓL(1,𝔽₈)	#64408	semidirect
33. C₃×C₇	C₃×#64408	semidirect
34. ΓL(1,𝔽₈)	C₃×#64408	semidirect
40. C₇	C₃²×#64408	semidirect
45. C₃	ΓL(1,𝔽₈)⋊#64408	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×ΓL(1,𝔽₈)×D₇₃	2
C₃×C₇⋊#64408	3
ΓL(1,𝔽₈)⋊#64408	3
C₃²×#64408	7
C₃×ΓL(1,𝔽₈)⋊C₄	73

Subgroups — C3×ΓL(1,𝔽8)⋊#64408 #32913849

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×ΓL(1,𝔽₈)⋊#64408 #32913849