Subgroups — ΓL(1,𝔽₈)×#13046196 #41002019

Order 26901.

56 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22582226	C₃	1	1	yes	maximal, characteristic, complemented
2	C₇×ΓL(1,𝔽₈)×C₆₁	C₃	1	1	yes	maximal, characteristic, complemented
3	C₇×#13046196	C₃	1	1	yes	maximal, characteristic, complemented
4	C₆₁⋊#3862	C₃	1	1	yes	maximal, characteristic, complemented
5	C₃×#13046196	—	7	1	no	maximal
6	ΓL(1,𝔽₈)×C₆₁⋊C₃	—	7	1	no	maximal
7	C₇²×C₆₁	C₃²	1	1	yes	characteristic, complemented, Hall
8	C₃×C₇×C₆₁	—	7	1	no	—
9	#13046197	—	7	1	no	—
10	#13046196	—	7	1	no	—
11	#13046196	—	7	1	no	—
12	#13046196	—	7	1	no	—
13	#13046196	—	21	2	no	—
14	#13046196	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
15	C₇×C₆₁⋊C₃	—	7	1	no	—
16	ΓL(1,𝔽₈)×C₆₁	ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
17	C₃×C₆₁⋊C₃	—	49	1	no	Hall
18	ΓL(1,𝔽₈)²	—	61	1	no	maximal, Hall
19	C₇×C₆₁	—	3	2	no	—
20	C₇×C₆₁	C₃×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
21	C₇×C₆₁	C₃×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
22	C₃×C₆₁	—	7	1	no	—
23	C₆₁⋊C₃	—	7	1	no	—
24	C₆₁⋊C₃	—	49	1	no	—
25	C₆₁⋊C₃	—	49	1	no	—
26	#3862	—	61	1	no	—
27	#3861	—	61	1	no	—
28	C₇×ΓL(1,𝔽₈)	—	61	1	no	—
29	C₇×ΓL(1,𝔽₈)	C₆₁⋊C₃	1	1	yes	characteristic, complemented
30	C₃×ΓL(1,𝔽₈)	—	427	1	no	—
31	C₃×ΓL(1,𝔽₈)	—	427	1	no	—
32	C₆₁	ΓL(1,𝔽₈)²	1	1	yes	characteristic, complemented, Sylow
33	C₇²	C₃×C₆₁⋊C₃	1	1	yes	characteristic, complemented, Sylow
34	C₃×C₇	—	7	1	no	—
35	C₃×C₇	—	427	1	no	—
36	ΓL(1,𝔽₈)	—	1281	2	no	—
37	ΓL(1,𝔽₈)	—	61	1	no	—
38	ΓL(1,𝔽₈)	—	427	1	no	—
39	ΓL(1,𝔽₈)	—	427	1	no	—
40	ΓL(1,𝔽₈)	—	427	1	no	—
41	ΓL(1,𝔽₈)	—	427	1	no	—
42	ΓL(1,𝔽₈)	#13046196	1	1	yes	characteristic, direct product factor
43	C₃²	—	2989	1	no	Sylow
44	C₇	—	3	2	no	—
45	C₇	ΓL(1,𝔽₈)×C₆₁⋊C₃	1	1	yes	characteristic, complemented
46	C₇	C₃×#13046196	1	1	yes	characteristic, complemented
47	C₃	—	7	1	no	—
48	C₃	—	427	1	no	—
49	C₃	—	2989	1	no	—
50	C₃	—	2989	1	no	—

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₇²×C₆₁	C₃²
LCS	G ▷ C₇²×C₆₁
UCS	C₁

Complements

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

N	H ≅ G/N	type
1. #22582226	C₃	semidirect
2. C₇×ΓL(1,𝔽₈)×C₆₁	C₃	semidirect
3. C₇×#13046196	C₃	semidirect
4. C₆₁⋊#3862	C₃	semidirect
7. C₇²×C₆₁	C₃²	semidirect
14. #13046196	ΓL(1,𝔽₈)	direct (N × H)
16. ΓL(1,𝔽₈)×C₆₁	ΓL(1,𝔽₈)	semidirect
20. C₇×C₆₁	C₃×ΓL(1,𝔽₈)	semidirect
21. C₇×C₆₁	C₃×ΓL(1,𝔽₈)	semidirect
29. C₇×ΓL(1,𝔽₈)	C₆₁⋊C₃	semidirect
32. C₆₁	ΓL(1,𝔽₈)²	semidirect
33. C₇²	C₃×C₆₁⋊C₃	semidirect
42. ΓL(1,𝔽₈)	#13046196	direct (N × H)
45. C₇	ΓL(1,𝔽₈)×C₆₁⋊C₃	semidirect
46. C₇	C₃×#13046196	semidirect

Maximal subgroups

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

subgroup	index
#22582226	3
C₇×ΓL(1,𝔽₈)×C₆₁	3
C₇×#13046196	3
C₆₁⋊#3862	3
C₃×#13046196	7
ΓL(1,𝔽₈)×C₆₁⋊C₃	7
ΓL(1,𝔽₈)²	61

Subgroups — ΓL(1,𝔽8)×#13046196 #41002019

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — ΓL(1,𝔽₈)×#13046196 #41002019