Subgroups — ΓL(1,𝔽₈)×Dic₃₇₁ #44528366

Order 31164.

52 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	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
11	C₇×Dic₅₃	—	7	1	no	—
12	C₃×C₇×C₅₃	—	7	1	no	—
13	ΓL(1,𝔽₈)×C₅₃	Dic₇	1	1	yes	characteristic, complemented
14	C₂×C₇×C₅₃	—	6	1	no	—
15	C₂×C₇×C₅₃	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic
16	C₂×C₇×C₅₃	C₃×D₇	1	1	yes	characteristic
17	C₃×Dic₅₃	—	49	1	no	Hall
18	ΓL(1,𝔽₈)×Dic₇	—	53	1	no	maximal, Hall
19	C₇×C₅₃	—	6	1	no	—
20	C₇×C₅₃	C₄×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
21	C₇×C₅₃	C₃×Dic₇	1	1	yes	characteristic, complemented
22	C₂×C₃×C₅₃	—	7	1	no	—
23	C₂×C₇×ΓL(1,𝔽₈)	D₅₃	1	1	yes	characteristic
24	Dic₅₃	—	7	1	no	Hall
25	C₇×Dic₇	—	53	1	no	Hall
26	C₃×C₅₃	—	7	1	no	Hall
27	C₇×ΓL(1,𝔽₈)	Dic₅₃	1	1	yes	characteristic, complemented, Hall
28	C₂×C₅₃	D₇×ΓL(1,𝔽₈)	1	1	yes	characteristic
29	C₂×C₇²	C₃×D₅₃	1	1	yes	characteristic
30	C₃×Dic₇	—	371	1	no	—
31	C₄×ΓL(1,𝔽₈)	—	371	1	no	—
32	C₅₃	ΓL(1,𝔽₈)×Dic₇	1	1	yes	characteristic, complemented, Sylow
33	C₇²	C₃×Dic₅₃	1	1	yes	characteristic, complemented, Sylow
34	C₂×C₃×C₇	—	7	1	no	—
35	C₂×ΓL(1,𝔽₈)	D₃₇₁	1	1	yes	characteristic
36	C₄×C₇	—	371	1	no	—
37	Dic₇	—	53	1	no	—
38	C₃×C₇	—	7	1	no	—
39	ΓL(1,𝔽₈)	Dic₃₇₁	1	1	yes	characteristic, direct product factor
40	C₂×C₇	—	6	1	no	—
41	C₂×C₇	C₃×D₃₇₁	1	1	yes	characteristic
42	C₂×C₇	ΓL(1,𝔽₈)×D₅₃	1	1	yes	characteristic
43	C₃×C₄	—	2597	1	no	Hall
44	C₇	—	6	1	no	—
45	C₇	C₃×Dic₃₇₁	1	1	yes	characteristic, complemented
46	C₇	ΓL(1,𝔽₈)×Dic₅₃	1	1	yes	characteristic, complemented
47	C₂×C₃	—	7	1	no	—
48	C₄	—	371	1	no	Sylow
49	C₃	—	7	1	no	Sylow
50	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
10. Dic₃₇₁	ΓL(1,𝔽₈)	direct (N × H)
13. ΓL(1,𝔽₈)×C₅₃	Dic₇	semidirect
20. C₇×C₅₃	C₄×ΓL(1,𝔽₈)	semidirect
21. C₇×C₅₃	C₃×Dic₇	semidirect
27. C₇×ΓL(1,𝔽₈)	Dic₅₃	semidirect
32. C₅₃	ΓL(1,𝔽₈)×Dic₇	semidirect
33. C₇²	C₃×Dic₅₃	semidirect
39. ΓL(1,𝔽₈)	Dic₃₇₁	direct (N × H)
45. C₇	C₃×Dic₃₇₁	semidirect
46. 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 #44528366

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — ΓL(1,𝔽₈)×Dic₃₇₁ #44528366