Subgroups — ΓL(1,𝔽₈)×Dic₇×C₅₃ #44528365

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×Dic₇×C₅₃	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	C₃×Dic₇×C₅₃	—	7	1	no	maximal
6	C₄×ΓL(1,𝔽₈)×C₅₃	—	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	C₄×C₇×C₅₃	—	7	1	no	—
11	Dic₇×C₅₃	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
12	C₃×C₇×C₅₃	—	7	1	no	—
13	ΓL(1,𝔽₈)×C₅₃	Dic₇	1	1	yes	characteristic, direct product factor
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×C₄×C₅₃	—	49	1	no	Hall
18	ΓL(1,𝔽₈)×Dic₇	C₅₃	1	1	yes	maximal, characteristic, direct product factor, 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)	C₂×C₅₃	1	1	yes	characteristic
24	C₄×C₅₃	—	7	1	no	Hall
25	C₇×Dic₇	C₃×C₅₃	1	1	yes	characteristic, complemented, Hall
26	C₃×C₅₃	—	7	1	no	Hall
27	C₇×ΓL(1,𝔽₈)	C₄×C₅₃	1	1	yes	characteristic, complemented, Hall
28	C₂×C₅₃	D₇×ΓL(1,𝔽₈)	1	1	yes	characteristic
29	C₂×C₇²	C₂×C₃×C₅₃	1	1	yes	characteristic
30	C₃×Dic₇	—	7	1	no	—
31	C₄×ΓL(1,𝔽₈)	—	7	1	no	—
32	C₅₃	ΓL(1,𝔽₈)×Dic₇	1	1	yes	characteristic, direct product factor, Sylow
33	C₇²	C₃×C₄×C₅₃	1	1	yes	characteristic, complemented, Sylow
34	C₂×C₃×C₇	—	7	1	no	—
35	C₂×ΓL(1,𝔽₈)	D₇×C₅₃	1	1	yes	characteristic
36	C₄×C₇	—	7	1	no	—
37	Dic₇	ΓL(1,𝔽₈)×C₅₃	1	1	yes	characteristic, direct product factor
38	C₃×C₇	—	7	1	no	—
39	ΓL(1,𝔽₈)	Dic₇×C₅₃	1	1	yes	characteristic, direct product factor
40	C₂×C₇	—	6	1	no	—
41	C₂×C₇	C₂×ΓL(1,𝔽₈)×C₅₃	1	1	yes	characteristic
42	C₂×C₇	C₃×D₇×C₅₃	1	1	yes	characteristic
43	C₃×C₄	—	49	1	no	Hall
44	C₇	—	6	1	no	—
45	C₇	C₄×ΓL(1,𝔽₈)×C₅₃	1	1	yes	characteristic, complemented
46	C₇	C₃×Dic₇×C₅₃	1	1	yes	characteristic, complemented
47	C₂×C₃	—	7	1	no	—
48	C₄	—	7	1	no	Sylow
49	C₃	—	7	1	no	Sylow
50	C₂	D₇×ΓL(1,𝔽₈)×C₅₃	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂×C₅₃	D₇×ΓL(1,𝔽₈)
commutator, hypocenter	C₇²	C₃×C₄×C₅₃
Frattini	C₂	D₇×ΓL(1,𝔽₈)×C₅₃
Fitting, socle	C₂×C₇²×C₅₃	C₂×C₃
prime radical	C₂×C₇×ΓL(1,𝔽₈)×C₅₃	C₂
cosocle	C₂×C₇²	C₂×C₃×C₅₃
LCS	G ▷ C₇²
UCS	C₁ ◁ C₂×C₅₃

Complements

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

N	H ≅ G/N	type
2. C₇×Dic₇×C₅₃	C₃	semidirect
3. C₇×ΓL(1,𝔽₈)×C₅₃	C₄	semidirect
7. C₇²×C₅₃	C₃×C₄	semidirect
11. Dic₇×C₅₃	ΓL(1,𝔽₈)	direct (N × H)
13. ΓL(1,𝔽₈)×C₅₃	Dic₇	direct (N × H)
18. ΓL(1,𝔽₈)×Dic₇	C₅₃	direct (N × H)
20. C₇×C₅₃	C₄×ΓL(1,𝔽₈)	semidirect
21. C₇×C₅₃	C₃×Dic₇	semidirect
25. C₇×Dic₇	C₃×C₅₃	semidirect
27. C₇×ΓL(1,𝔽₈)	C₄×C₅₃	semidirect
32. C₅₃	ΓL(1,𝔽₈)×Dic₇	direct (N × H)
33. C₇²	C₃×C₄×C₅₃	semidirect
37. Dic₇	ΓL(1,𝔽₈)×C₅₃	direct (N × H)
39. ΓL(1,𝔽₈)	Dic₇×C₅₃	direct (N × H)
45. C₇	C₄×ΓL(1,𝔽₈)×C₅₃	semidirect
46. C₇	C₃×Dic₇×C₅₃	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×Dic₇×C₅₃	3
C₃×Dic₇×C₅₃	7
C₄×ΓL(1,𝔽₈)×C₅₃	7
ΓL(1,𝔽₈)×Dic₇	53

Subgroups — ΓL(1,𝔽8)×Dic7×C53 #44528365

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — ΓL(1,𝔽₈)×Dic₇×C₅₃ #44528365