Subgroups — C₃⋊(ΓL(1,𝔽₈)⋊Dic₁₀₇) #41004091

Order 26964.

56 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×ΓL(1,𝔽₈)×C₁₀₇	C₂	1	1	yes	maximal, characteristic
2	ΓL(1,𝔽₈)⋊Dic₁₀₇	—	3	1	no	maximal
3	Dic₂₂₄₇	C₃	1	1	yes	maximal, characteristic, complemented
4	C₃×ΓL(1,𝔽₈)×C₁₀₇	C₄	1	1	yes	characteristic, complemented, Hall
5	C₂×ΓL(1,𝔽₈)×C₁₀₇	—	2	1	no	—
6	C₂×ΓL(1,𝔽₈)×C₁₀₇	D₃	1	1	yes	characteristic
7	C₂×C₃×C₇×C₁₀₇	C₂×C₃	1	1	yes	characteristic
8	C₃×Dic₃₂₁	—	7	1	no	maximal, Hall
9	Dic₇₄₉	—	3	1	no	Hall
10	C₃×C₇×C₁₀₇	C₃×C₄	1	1	yes	characteristic, complemented
11	ΓL(1,𝔽₈)×C₁₀₇	—	2	1	no	—
12	ΓL(1,𝔽₈)×C₁₀₇	Dic₃	1	1	yes	characteristic, complemented
13	C₂×C₃²×C₁₀₇	—	14	1	no	—
14	C₂×C₇×C₁₀₇	C₃×D₃	1	1	yes	characteristic
15	Dic₃₂₁	—	7	1	no	—
16	C₃×Dic₁₀₇	—	21	1	no	—
17	C₃²×C₁₀₇	—	14	1	no	Hall
18	C₇×C₁₀₇	C₃×Dic₃	1	1	yes	characteristic, complemented, Hall
19	C₂×C₃×C₁₀₇	—	7	1	no	—
20	C₂×C₃×C₁₀₇	—	14	1	no	—
21	C₂×C₃×C₁₀₇	AGL(1,𝔽₇)	1	1	yes	characteristic
22	Dic₁₀₇	—	21	1	no	Hall
23	C₃×C₁₀₇	—	7	1	no	—
24	C₃×C₁₀₇	—	14	1	no	—
25	C₃×C₁₀₇	ΓL(1,𝔽₈)⋊C₄	1	1	yes	characteristic, complemented
26	C₃⋊(ΓL(1,𝔽₈)⋊C₄)	—	107	1	no	maximal, Hall
27	C₂×C₁₀₇	C₃⋊AGL(1,𝔽₇)	1	1	yes	characteristic
28	C₂×C₃×ΓL(1,𝔽₈)	D₁₀₇	1	1	yes	characteristic
29	C₁₀₇	C₃⋊(ΓL(1,𝔽₈)⋊C₄)	1	1	yes	characteristic, complemented, Sylow
30	Dic₂₁	—	107	1	no	—
31	ΓL(1,𝔽₈)⋊C₄	—	321	1	no	—
32	C₃×ΓL(1,𝔽₈)	Dic₁₀₇	1	1	yes	characteristic, complemented, Hall
33	C₂×C₃×C₇	C₃×D₁₀₇	1	1	yes	characteristic
34	C₂×ΓL(1,𝔽₈)	—	2	1	no	—
35	C₂×ΓL(1,𝔽₈)	D₃₂₁	1	1	yes	characteristic
36	C₃×Dic₃	—	749	1	no	Hall
37	Dic₇	—	321	1	no	Hall
38	C₃×C₇	C₃×Dic₁₀₇	1	1	yes	characteristic, complemented
39	ΓL(1,𝔽₈)	—	2	1	no	—
40	ΓL(1,𝔽₈)	Dic₃₂₁	1	1	yes	characteristic, complemented
41	C₂×C₃²	—	14	1	no	—
42	C₂×C₇	C₃×D₃₂₁	1	1	yes	characteristic
43	C₃×C₄	—	2247	1	no	—
44	Dic₃	—	749	1	no	—
45	C₃²	—	14	1	no	Sylow
46	C₇	C₃×Dic₃₂₁	1	1	yes	characteristic, complemented, Sylow
47	C₂×C₃	—	7	1	no	—
48	C₂×C₃	—	14	1	no	—
49	C₂×C₃	ΓL(1,𝔽₈)⋊D₁₀₇	1	1	yes	characteristic
50	C₄	—	2247	1	no	Sylow
51	C₃	—	7	1	no	—
52	C₃	—	14	1	no	—
53	C₃	ΓL(1,𝔽₈)⋊Dic₁₀₇	1	1	yes	characteristic, complemented
54	C₂	C₃⋊(ΓL(1,𝔽₈)⋊D₁₀₇)	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₂	C₃⋊(ΓL(1,𝔽₈)⋊D₁₀₇)
commutator, hypocenter	C₃×C₇×C₁₀₇	C₃×C₄
Fitting, socle, cosocle	C₂×C₃×C₇×C₁₀₇	C₂×C₃
prime radical	C₂×C₃×ΓL(1,𝔽₈)×C₁₀₇	C₂
LCS	G ▷ C₃×C₇×C₁₀₇
UCS	C₁ ◁ C₂

Complements

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

N	H ≅ G/N	type
3. Dic₂₂₄₇	C₃	semidirect
4. C₃×ΓL(1,𝔽₈)×C₁₀₇	C₄	semidirect
10. C₃×C₇×C₁₀₇	C₃×C₄	semidirect
12. ΓL(1,𝔽₈)×C₁₀₇	Dic₃	semidirect
18. C₇×C₁₀₇	C₃×Dic₃	semidirect
25. C₃×C₁₀₇	ΓL(1,𝔽₈)⋊C₄	semidirect
29. C₁₀₇	C₃⋊(ΓL(1,𝔽₈)⋊C₄)	semidirect
32. C₃×ΓL(1,𝔽₈)	Dic₁₀₇	semidirect
38. C₃×C₇	C₃×Dic₁₀₇	semidirect
40. ΓL(1,𝔽₈)	Dic₃₂₁	semidirect
46. C₇	C₃×Dic₃₂₁	semidirect
53. C₃	ΓL(1,𝔽₈)⋊Dic₁₀₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×ΓL(1,𝔽₈)×C₁₀₇	2
ΓL(1,𝔽₈)⋊Dic₁₀₇	3
Dic₂₂₄₇	3
C₃×Dic₃₂₁	7
C₃⋊(ΓL(1,𝔽₈)⋊C₄)	107

Subgroups — C3⋊(ΓL(1,𝔽8)⋊Dic107) #41004091

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃⋊(ΓL(1,𝔽₈)⋊Dic₁₀₇) #41004091