Subgroups — ΓL(1,𝔽₈)⋊Dic₃₇₁ #44528362

Order 31164.

60 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×ΓL(1,𝔽₈)×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₇⋊Dic₃₇₁	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	C₇×ΓL(1,𝔽₈)×C₅₃	C₄	1	1	yes	characteristic, complemented, Hall
4	C₂×C₇²×C₅₃	C₂×C₃	1	1	yes	characteristic
5	ΓL(1,𝔽₈)⋊Dic₅₃	—	7	1	no	maximal
6	C₃×Dic₃₇₁	—	7	1	no	maximal
7	C₇²×C₅₃	C₃×C₄	1	1	yes	characteristic, complemented, Hall
8	C₂×ΓL(1,𝔽₈)×C₅₃	D₇	1	1	yes	characteristic
9	C₂×C₃×C₇×C₅₃	—	7	1	no	—
10	Dic₃₇₁	—	7	1	no	—
11	Dic₃₇₁	—	7	1	no	—
12	Dic₃₇₁	—	21	2	no	—
13	C₃×C₇×C₅₃	—	7	1	no	—
14	ΓL(1,𝔽₈)×C₅₃	Dic₇	1	1	yes	characteristic, complemented
15	C₂×C₇×C₅₃	—	3	2	no	—
16	C₂×C₇×C₅₃	AGL(1,𝔽₇)	1	1	yes	characteristic
17	C₂×C₇×C₅₃	C₃×D₇	1	1	yes	characteristic
18	C₃×Dic₅₃	—	49	1	no	Hall
19	ΓL(1,𝔽₈)⋊Dic₇	—	53	1	no	maximal, Hall
20	C₇×C₅₃	—	3	2	no	—
21	C₇×C₅₃	ΓL(1,𝔽₈)⋊C₄	1	1	yes	characteristic, complemented
22	C₇×C₅₃	C₃×Dic₇	1	1	yes	characteristic, complemented
23	C₂×C₃×C₅₃	—	7	1	no	—
24	C₂×C₇×ΓL(1,𝔽₈)	D₅₃	1	1	yes	characteristic
25	Dic₅₃	—	49	1	no	Hall
26	C₇⋊Dic₇	—	53	1	no	Hall
27	C₃×C₅₃	—	7	1	no	Hall
28	C₇×ΓL(1,𝔽₈)	Dic₅₃	1	1	yes	characteristic, complemented, Hall
29	C₂×C₅₃	ΓL(1,𝔽₈)⋊D₇	1	1	yes	characteristic
30	C₂×C₇²	C₃×D₅₃	1	1	yes	characteristic
31	C₃×Dic₇	—	371	1	no	—
32	ΓL(1,𝔽₈)⋊C₄	—	371	1	no	—
33	C₅₃	ΓL(1,𝔽₈)⋊Dic₇	1	1	yes	characteristic, complemented, Sylow
34	C₇²	C₃×Dic₅₃	1	1	yes	characteristic, complemented, Sylow
35	C₂×C₃×C₇	—	7	1	no	—
36	C₂×ΓL(1,𝔽₈)	D₃₇₁	1	1	yes	characteristic
37	Dic₇	—	1113	2	no	—
38	Dic₇	—	371	1	no	—
39	Dic₇	—	371	1	no	—
40	C₃×C₇	—	7	1	no	—
41	ΓL(1,𝔽₈)	Dic₃₇₁	1	1	yes	characteristic, complemented
42	C₂×C₇	—	3	2	no	—
43	C₂×C₇	C₃×D₃₇₁	1	1	yes	characteristic
44	C₂×C₇	ΓL(1,𝔽₈)⋊D₅₃	1	1	yes	characteristic
45	C₃×C₄	—	2597	1	no	Hall
46	C₇	—	3	2	no	—
47	C₇	C₃×Dic₃₇₁	1	1	yes	characteristic, complemented
48	C₇	ΓL(1,𝔽₈)⋊Dic₅₃	1	1	yes	characteristic, complemented
49	C₂×C₃	—	7	1	no	—
50	C₄	—	2597	1	no	Sylow
51	C₃	—	7	1	no	Sylow
52	C₂	ΓL(1,𝔽₈)⋊D₃₇₁	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₂	ΓL(1,𝔽₈)⋊D₃₇₁
commutator, hypocenter	C₇²×C₅₃	C₃×C₄
Fitting, socle, cosocle	C₂×C₇²×C₅₃	C₂×C₃
prime radical	C₂×C₇×ΓL(1,𝔽₈)×C₅₃	C₂
LCS	G ▷ C₇²×C₅₃
UCS	C₁ ◁ C₂

Complements

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

N	H ≅ G/N	type
2. C₇⋊Dic₃₇₁	C₃	semidirect
3. C₇×ΓL(1,𝔽₈)×C₅₃	C₄	semidirect
7. C₇²×C₅₃	C₃×C₄	semidirect
14. ΓL(1,𝔽₈)×C₅₃	Dic₇	semidirect
21. C₇×C₅₃	ΓL(1,𝔽₈)⋊C₄	semidirect
22. C₇×C₅₃	C₃×Dic₇	semidirect
28. C₇×ΓL(1,𝔽₈)	Dic₅₃	semidirect
33. C₅₃	ΓL(1,𝔽₈)⋊Dic₇	semidirect
34. C₇²	C₃×Dic₅₃	semidirect
41. ΓL(1,𝔽₈)	Dic₃₇₁	semidirect
47. C₇	C₃×Dic₃₇₁	semidirect
48. C₇	ΓL(1,𝔽₈)⋊Dic₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₇×ΓL(1,𝔽₈)×C₅₃	2
C₇⋊Dic₃₇₁	3
ΓL(1,𝔽₈)⋊Dic₅₃	7
C₃×Dic₃₇₁	7
ΓL(1,𝔽₈)⋊Dic₇	53

Subgroups — ΓL(1,𝔽8)⋊Dic371 #44528362

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — ΓL(1,𝔽₈)⋊Dic₃₇₁ #44528362