Subgroups — ΓL(1,𝔽₈)⋊(ΓL(1,𝔽₈)⋊D₃₇) #45657441

Order 32634.

112 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	ΓL(1,𝔽₈)²×C₃₇	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	ΓL(1,𝔽₈)⋊D₂₅₉	C₃	1	2	yes	maximal, complemented
3	#24925955	C₃	1	1	yes	maximal, characteristic, complemented
4	#24925951	C₃	1	1	yes	maximal, characteristic, complemented
5	C₃₇×#3862	C₂×C₃	1	1	yes	characteristic, complemented
6	C₃₇×#3861	C₂×C₃	1	1	yes	characteristic, complemented
7	C₇×ΓL(1,𝔽₈)×C₃₇	C₂×C₃	1	2	yes	complemented
8	C₃×ΓL(1,𝔽₈)⋊D₃₇	—	7	2	no	maximal
9	C₇⋊D₂₅₉	C₃²	1	1	yes	characteristic, complemented, Hall
10	C₃×ΓL(1,𝔽₈)×C₃₇	—	7	2	no	—
11	C₇²×C₃₇	C₂×C₃²	1	1	yes	characteristic, complemented, Hall
12	C₃×D₂₅₉	—	7	2	no	—
13	ΓL(1,𝔽₈)⋊D₃₇	—	7	2	no	—
14	ΓL(1,𝔽₈)⋊D₃₇	—	7	2	no	—
15	ΓL(1,𝔽₈)⋊D₃₇	—	7	2	no	—
16	ΓL(1,𝔽₈)⋊D₃₇	—	21	2	no	—
17	ΓL(1,𝔽₈)⋊AGL(1,𝔽₇)	—	37	1	no	maximal, Hall
18	C₃×C₇×C₃₇	—	7	2	no	—
19	ΓL(1,𝔽₈)×C₃₇	—	7	2	no	—
20	ΓL(1,𝔽₈)×C₃₇	—	7	2	no	—
21	ΓL(1,𝔽₈)×C₃₇	—	21	2	no	—
22	ΓL(1,𝔽₈)×C₃₇	AGL(1,𝔽₇)	1	2	yes	complemented
23	C₃²×D₃₇	—	49	1	no	Hall
24	D₂₅₉	—	7	2	no	—
25	D₂₅₉	—	21	2	no	—
26	ΓL(1,𝔽₈)²	D₃₇	1	1	yes	characteristic, complemented, Hall
27	C₃²×C₃₇	—	49	1	no	Hall
28	#64425	—	37	1	no	—
29	#64424	—	37	1	no	—
30	ΓL(1,𝔽₈)⋊D₇	—	37	2	no	—
31	C₇×C₃₇	—	3	2	no	—
32	C₇×C₃₇	C₃×AGL(1,𝔽₇)	1	2	yes	complemented
33	C₃×D₃₇	—	49	2	no	—
34	C₃×D₃₇	—	49	1	no	—
35	C₃×D₃₇	—	49	1	no	—
36	#3862	C₃×D₃₇	1	1	yes	characteristic, complemented
37	#3861	C₃×D₃₇	1	1	yes	characteristic, complemented
38	C₇×ΓL(1,𝔽₈)	C₃×D₃₇	1	2	yes	complemented
39	C₃×AGL(1,𝔽₇)	—	259	2	no	—
40	C₃×C₃₇	—	7	2	no	—
41	C₃×C₃₇	—	49	1	no	—
42	C₃×C₃₇	—	49	1	no	—
43	C₇⋊D₇	—	37	1	no	Hall
44	D₃₇	—	49	1	no	Hall
45	C₃×ΓL(1,𝔽₈)	—	7	2	no	—
46	C₇²	C₃²×D₃₇	1	1	yes	characteristic, complemented, Sylow
47	C₃×D₇	—	259	2	no	—
48	AGL(1,𝔽₇)	—	259	2	no	—
49	AGL(1,𝔽₇)	—	259	2	no	—
50	AGL(1,𝔽₇)	—	259	2	no	—
51	AGL(1,𝔽₇)	—	777	2	no	—
52	C₃₇	ΓL(1,𝔽₈)⋊AGL(1,𝔽₇)	1	1	yes	characteristic, complemented, 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)	ΓL(1,𝔽₈)⋊D₃₇	1	2	yes	complemented
58	C₂×C₃²	—	1813	1	no	Hall
59	D₇	—	259	2	no	—
60	D₇	—	777	2	no	—
61	C₃²	—	49	1	no	Sylow
62	C₇	—	3	2	no	—
63	C₇	C₃×ΓL(1,𝔽₈)⋊D₃₇	1	2	yes	complemented
64	C₂×C₃	—	1813	2	no	—
65	C₂×C₃	—	1813	1	no	—
66	C₂×C₃	—	1813	1	no	—
67	C₃	—	7	2	no	—
68	C₃	—	49	1	no	—
69	C₃	—	49	1	no	—
70	C₂	—	1813	1	no	Sylow

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₇²×C₃₇	C₂×C₃²
LCS	G ▷ C₇²×C₃₇
UCS	C₁

Complements

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

N	H ≅ G/N	type
1. ΓL(1,𝔽₈)²×C₃₇	C₂	semidirect
2. ΓL(1,𝔽₈)⋊D₂₅₉	C₃	semidirect
3. #24925955	C₃	semidirect
4. #24925951	C₃	semidirect
5. C₃₇×#3862	C₂×C₃	semidirect
6. C₃₇×#3861	C₂×C₃	semidirect
7. C₇×ΓL(1,𝔽₈)×C₃₇	C₂×C₃	semidirect
9. C₇⋊D₂₅₉	C₃²	semidirect
11. C₇²×C₃₇	C₂×C₃²	semidirect
22. ΓL(1,𝔽₈)×C₃₇	AGL(1,𝔽₇)	semidirect
26. ΓL(1,𝔽₈)²	D₃₇	semidirect
32. C₇×C₃₇	C₃×AGL(1,𝔽₇)	semidirect
36. #3862	C₃×D₃₇	semidirect
37. #3861	C₃×D₃₇	semidirect
38. C₇×ΓL(1,𝔽₈)	C₃×D₃₇	semidirect
46. C₇²	C₃²×D₃₇	semidirect
52. C₃₇	ΓL(1,𝔽₈)⋊AGL(1,𝔽₇)	semidirect
57. ΓL(1,𝔽₈)	ΓL(1,𝔽₈)⋊D₃₇	semidirect
63. C₇	C₃×ΓL(1,𝔽₈)⋊D₃₇	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×C₃₇	2
ΓL(1,𝔽₈)⋊D₂₅₉	3
#24925955	3
#24925951	3
C₃×ΓL(1,𝔽₈)⋊D₃₇	7
ΓL(1,𝔽₈)⋊AGL(1,𝔽₇)	37

Subgroups — ΓL(1,𝔽8)⋊(ΓL(1,𝔽8)⋊D37) #45657441

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — ΓL(1,𝔽₈)⋊(ΓL(1,𝔽₈)⋊D₃₇) #45657441