Subgroups — C₄₁×ΓL(1,𝔽₈)⋊AGL(1,𝔽₇) #49116331

Order 36162.

112 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	ΓL(1,𝔽₈)²×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	C₄₁×ΓL(1,𝔽₈)⋊D₇	C₃	1	2	yes	maximal, complemented
3	C₄₁×#64425	C₃	1	1	yes	maximal, characteristic, complemented
4	C₄₁×#64424	C₃	1	1	yes	maximal, characteristic, complemented
5	C₄₁×#3862	C₂×C₃	1	1	yes	characteristic, complemented
6	C₄₁×#3861	C₂×C₃	1	1	yes	characteristic, complemented
7	C₇×ΓL(1,𝔽₈)×C₄₁	C₂×C₃	1	2	yes	complemented
8	C₃×C₄₁×AGL(1,𝔽₇)	—	7	2	no	maximal
9	C₄₁×C₇⋊D₇	C₃²	1	1	yes	characteristic, complemented, Hall
10	C₃×ΓL(1,𝔽₈)×C₄₁	—	7	2	no	—
11	C₇²×C₄₁	C₂×C₃²	1	1	yes	characteristic, complemented, Hall
12	C₃×D₇×C₄₁	—	7	2	no	—
13	C₄₁×AGL(1,𝔽₇)	—	7	2	no	—
14	C₄₁×AGL(1,𝔽₇)	—	7	2	no	—
15	C₄₁×AGL(1,𝔽₇)	—	7	2	no	—
16	C₄₁×AGL(1,𝔽₇)	—	21	2	no	—
17	ΓL(1,𝔽₈)⋊AGL(1,𝔽₇)	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Hall
18	C₃×C₇×C₄₁	—	7	2	no	—
19	ΓL(1,𝔽₈)×C₄₁	—	7	2	no	—
20	ΓL(1,𝔽₈)×C₄₁	—	7	2	no	—
21	ΓL(1,𝔽₈)×C₄₁	—	21	2	no	—
22	ΓL(1,𝔽₈)×C₄₁	AGL(1,𝔽₇)	1	2	yes	complemented
23	C₂×C₃²×C₄₁	—	49	1	no	Hall
24	D₇×C₄₁	—	7	2	no	—
25	D₇×C₄₁	—	21	2	no	—
26	ΓL(1,𝔽₈)²	C₂×C₄₁	1	1	yes	characteristic, complemented, Hall
27	C₃²×C₄₁	—	49	1	no	Hall
28	#64425	C₃×C₄₁	1	1	yes	characteristic, complemented
29	#64424	C₃×C₄₁	1	1	yes	characteristic, complemented
30	ΓL(1,𝔽₈)⋊D₇	C₃×C₄₁	1	2	yes	complemented
31	C₇×C₄₁	—	3	2	no	—
32	C₇×C₄₁	C₃×AGL(1,𝔽₇)	1	2	yes	complemented
33	C₂×C₃×C₄₁	—	49	2	no	—
34	C₂×C₃×C₄₁	—	49	1	no	—
35	C₂×C₃×C₄₁	—	49	1	no	—
36	#3862	C₂×C₃×C₄₁	1	1	yes	characteristic, complemented
37	#3861	C₂×C₃×C₄₁	1	1	yes	characteristic, complemented
38	C₇×ΓL(1,𝔽₈)	C₂×C₃×C₄₁	1	2	yes	complemented
39	C₃×AGL(1,𝔽₇)	—	7	2	no	—
40	C₃×C₄₁	—	7	2	no	—
41	C₃×C₄₁	—	49	1	no	—
42	C₃×C₄₁	—	49	1	no	—
43	C₇⋊D₇	C₃²×C₄₁	1	1	yes	characteristic, complemented, Hall
44	C₂×C₄₁	—	49	1	no	Hall
45	C₃×ΓL(1,𝔽₈)	—	7	2	no	—
46	C₇²	C₂×C₃²×C₄₁	1	1	yes	characteristic, complemented, Sylow
47	C₃×D₇	—	7	2	no	—
48	AGL(1,𝔽₇)	—	7	2	no	—
49	AGL(1,𝔽₇)	—	7	2	no	—
50	AGL(1,𝔽₇)	—	7	2	no	—
51	AGL(1,𝔽₇)	—	21	2	no	—
52	C₄₁	ΓL(1,𝔽₈)⋊AGL(1,𝔽₇)	1	1	yes	characteristic, direct product factor, Sylow
53	C₃×C₇	—	7	2	no	—
54	ΓL(1,𝔽₈)	—	7	2	no	—
55	ΓL(1,𝔽₈)	—	7	2	no	—
56	ΓL(1,𝔽₈)	—	21	2	no	—
57	ΓL(1,𝔽₈)	C₄₁×AGL(1,𝔽₇)	1	2	yes	complemented
58	C₂×C₃²	—	49	1	no	Hall
59	D₇	—	7	2	no	—
60	D₇	—	21	2	no	—
61	C₃²	—	49	1	no	Sylow
62	C₇	—	3	2	no	—
63	C₇	C₃×C₄₁×AGL(1,𝔽₇)	1	2	yes	complemented
64	C₂×C₃	—	49	2	no	—
65	C₂×C₃	—	49	1	no	—
66	C₂×C₃	—	49	1	no	—
67	C₃	—	7	2	no	—
68	C₃	—	49	1	no	—
69	C₃	—	49	1	no	—
70	C₂	—	49	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₄₁	ΓL(1,𝔽₈)⋊AGL(1,𝔽₇)
commutator, cosocle, hypocenter	C₇²	C₂×C₃²×C₄₁
Frattini	C₁	G
Fitting, socle	C₇²×C₄₁	C₂×C₃²
LCS	G ▷ C₇²
UCS	C₁ ◁ C₄₁

Complements

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

N	H ≅ G/N	type
1. ΓL(1,𝔽₈)²×C₄₁	C₂	semidirect
2. C₄₁×ΓL(1,𝔽₈)⋊D₇	C₃	semidirect
3. C₄₁×#64425	C₃	semidirect
4. C₄₁×#64424	C₃	semidirect
5. C₄₁×#3862	C₂×C₃	semidirect
6. C₄₁×#3861	C₂×C₃	semidirect
7. C₇×ΓL(1,𝔽₈)×C₄₁	C₂×C₃	semidirect
9. C₄₁×C₇⋊D₇	C₃²	semidirect
11. C₇²×C₄₁	C₂×C₃²	semidirect
17. ΓL(1,𝔽₈)⋊AGL(1,𝔽₇)	C₄₁	direct (N × H)
22. ΓL(1,𝔽₈)×C₄₁	AGL(1,𝔽₇)	semidirect
26. ΓL(1,𝔽₈)²	C₂×C₄₁	semidirect
28. #64425	C₃×C₄₁	semidirect
29. #64424	C₃×C₄₁	semidirect
30. ΓL(1,𝔽₈)⋊D₇	C₃×C₄₁	semidirect
32. C₇×C₄₁	C₃×AGL(1,𝔽₇)	semidirect
36. #3862	C₂×C₃×C₄₁	semidirect
37. #3861	C₂×C₃×C₄₁	semidirect
38. C₇×ΓL(1,𝔽₈)	C₂×C₃×C₄₁	semidirect
43. C₇⋊D₇	C₃²×C₄₁	semidirect
46. C₇²	C₂×C₃²×C₄₁	semidirect
52. C₄₁	ΓL(1,𝔽₈)⋊AGL(1,𝔽₇)	direct (N × H)
57. ΓL(1,𝔽₈)	C₄₁×AGL(1,𝔽₇)	semidirect
63. C₇	C₃×C₄₁×AGL(1,𝔽₇)	semidirect

Maximal subgroups

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

subgroup	index
ΓL(1,𝔽₈)²×C₄₁	2
C₄₁×ΓL(1,𝔽₈)⋊D₇	3
C₄₁×#64425	3
C₄₁×#64424	3
C₃×C₄₁×AGL(1,𝔽₇)	7
ΓL(1,𝔽₈)⋊AGL(1,𝔽₇)	41

Subgroups — C41×ΓL(1,𝔽8)⋊AGL(1,𝔽7) #49116331

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×ΓL(1,𝔽₈)⋊AGL(1,𝔽₇) #49116331