Subgroups — C₂×ΓL(1,𝔽₈)⋊D₃₇₁ #44528428

Order 31164.

100 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, complemented
2	ΓL(1,𝔽₈)⋊D₃₇₁	C₂	1	2	yes	maximal, direct product factor
3	C₂×C₇⋊D₃₇₁	C₃	1	1	yes	maximal, characteristic, complemented, Hall
4	C₇×ΓL(1,𝔽₈)×C₅₃	C₂²	1	1	yes	characteristic, complemented, Hall
5	C₇⋊D₃₇₁	C₂×C₃	1	2	yes	complemented
6	C₂×C₇²×C₅₃	C₂×C₃	1	1	yes	characteristic, complemented
7	C₂×C₃×D₃₇₁	—	7	1	no	maximal
8	C₂×ΓL(1,𝔽₈)⋊D₅₃	—	7	1	no	maximal
9	C₇²×C₅₃	C₂²×C₃	1	1	yes	characteristic, complemented, Hall
10	ΓL(1,𝔽₈)⋊D₅₃	—	7	2	no	—
11	C₃×D₃₇₁	—	7	2	no	—
12	C₂×ΓL(1,𝔽₈)×C₅₃	D₇	1	1	yes	characteristic, complemented
13	C₂×C₃×C₇×C₅₃	—	7	1	no	—
14	C₂×D₃₇₁	—	7	1	no	—
15	C₂×D₃₇₁	—	7	1	no	—
16	C₂×D₃₇₁	—	21	2	no	—
17	C₃×C₇×C₅₃	—	7	1	no	—
18	ΓL(1,𝔽₈)×C₅₃	C₂×D₇	1	1	yes	characteristic, complemented
19	C₂×C₇×C₅₃	—	3	2	no	—
20	C₂×C₇×C₅₃	AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
21	C₂×C₇×C₅₃	C₃×D₇	1	1	yes	characteristic, complemented
22	D₃₇₁	—	7	2	no	—
23	D₃₇₁	—	7	2	no	—
24	D₃₇₁	—	21	4	no	—
25	C₂×C₃×D₅₃	—	49	1	no	Hall
26	C₂×ΓL(1,𝔽₈)⋊D₇	—	53	1	no	maximal, Hall
27	C₇×C₅₃	—	3	2	no	—
28	C₇×C₅₃	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
29	C₇×C₅₃	C₂×C₃×D₇	1	1	yes	characteristic, complemented
30	C₂×C₃×C₅₃	—	7	1	no	—
31	C₃×D₅₃	—	49	2	no	—
32	C₂×C₇×ΓL(1,𝔽₈)	D₅₃	1	1	yes	characteristic, complemented
33	ΓL(1,𝔽₈)⋊D₇	—	53	2	no	—
34	C₂×D₅₃	—	49	1	no	Hall
35	C₂×C₇⋊D₇	—	53	1	no	Hall
36	C₃×C₅₃	—	7	1	no	Hall
37	C₇×ΓL(1,𝔽₈)	C₂×D₅₃	1	1	yes	characteristic, complemented, Hall
38	C₂×C₅₃	ΓL(1,𝔽₈)⋊D₇	1	1	yes	characteristic, complemented
39	D₅₃	—	49	2	no	—
40	C₂×C₇²	C₃×D₅₃	1	1	yes	characteristic, complemented
41	C₇⋊D₇	—	53	2	no	—
42	C₂×C₃×D₇	—	371	1	no	—
43	C₂×AGL(1,𝔽₇)	—	371	1	no	—
44	C₅₃	C₂×ΓL(1,𝔽₈)⋊D₇	1	1	yes	characteristic, complemented, Sylow
45	C₇²	C₂×C₃×D₅₃	1	1	yes	characteristic, complemented, Sylow
46	C₂×C₃×C₇	—	7	1	no	—
47	C₃×D₇	—	371	2	no	—
48	C₂×ΓL(1,𝔽₈)	D₃₇₁	1	1	yes	characteristic, complemented
49	AGL(1,𝔽₇)	—	371	2	no	—
50	C₂×D₇	—	1113	2	no	—
51	C₂×D₇	—	371	1	no	—
52	C₂×D₇	—	371	1	no	—
53	C₃×C₇	—	7	1	no	—
54	ΓL(1,𝔽₈)	C₂×D₃₇₁	1	1	yes	characteristic, complemented
55	C₂×C₇	—	3	2	no	—
56	C₂×C₇	C₃×D₃₇₁	1	1	yes	characteristic, complemented
57	C₂×C₇	ΓL(1,𝔽₈)⋊D₅₃	1	1	yes	characteristic, complemented
58	D₇	—	1113	4	no	—
59	D₇	—	371	2	no	—
60	D₇	—	371	2	no	—
61	C₂²×C₃	—	2597	1	no	Hall
62	C₇	—	3	2	no	—
63	C₇	C₂×ΓL(1,𝔽₈)⋊D₅₃	1	1	yes	characteristic, complemented
64	C₇	C₂×C₃×D₃₇₁	1	1	yes	characteristic, complemented
65	C₂×C₃	—	7	1	no	—
66	C₂×C₃	—	2597	2	no	—
67	C₂²	—	2597	1	no	Sylow
68	C₃	—	7	1	no	Sylow
69	C₂	—	2597	2	no	—
70	C₂	ΓL(1,𝔽₈)⋊D₃₇₁	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂	ΓL(1,𝔽₈)⋊D₃₇₁
commutator, cosocle, hypocenter	C₇²×C₅₃	C₂²×C₃
Frattini	C₁	G
Fitting, socle	C₂×C₇²×C₅₃	C₂×C₃
LCS	G ▷ C₇²×C₅₃
UCS	C₁ ◁ C₂

Complements

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

N	H ≅ G/N	type
1. C₂×C₇×ΓL(1,𝔽₈)×C₅₃	C₂	semidirect
2. ΓL(1,𝔽₈)⋊D₃₇₁	C₂	direct (N × H)
3. C₂×C₇⋊D₃₇₁	C₃	semidirect
4. C₇×ΓL(1,𝔽₈)×C₅₃	C₂²	semidirect
5. C₇⋊D₃₇₁	C₂×C₃	semidirect
6. C₂×C₇²×C₅₃	C₂×C₃	semidirect
9. C₇²×C₅₃	C₂²×C₃	semidirect
12. C₂×ΓL(1,𝔽₈)×C₅₃	D₇	semidirect
18. ΓL(1,𝔽₈)×C₅₃	C₂×D₇	semidirect
20. C₂×C₇×C₅₃	AGL(1,𝔽₇)	semidirect
21. C₂×C₇×C₅₃	C₃×D₇	semidirect
28. C₇×C₅₃	C₂×AGL(1,𝔽₇)	semidirect
29. C₇×C₅₃	C₂×C₃×D₇	semidirect
32. C₂×C₇×ΓL(1,𝔽₈)	D₅₃	semidirect
37. C₇×ΓL(1,𝔽₈)	C₂×D₅₃	semidirect
38. C₂×C₅₃	ΓL(1,𝔽₈)⋊D₇	semidirect
40. C₂×C₇²	C₃×D₅₃	semidirect
44. C₅₃	C₂×ΓL(1,𝔽₈)⋊D₇	semidirect
45. C₇²	C₂×C₃×D₅₃	semidirect
48. C₂×ΓL(1,𝔽₈)	D₃₇₁	semidirect
54. ΓL(1,𝔽₈)	C₂×D₃₇₁	semidirect
56. C₂×C₇	C₃×D₃₇₁	semidirect
57. C₂×C₇	ΓL(1,𝔽₈)⋊D₅₃	semidirect
63. C₇	C₂×ΓL(1,𝔽₈)⋊D₅₃	semidirect
64. C₇	C₂×C₃×D₃₇₁	semidirect
70. C₂	ΓL(1,𝔽₈)⋊D₃₇₁	direct (N × H)

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
ΓL(1,𝔽₈)⋊D₃₇₁	2
C₂×C₇⋊D₃₇₁	3
C₂×C₃×D₃₇₁	7
C₂×ΓL(1,𝔽₈)⋊D₅₃	7
C₂×ΓL(1,𝔽₈)⋊D₇	53

Subgroups — C2×ΓL(1,𝔽8)⋊D371 #44528428

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×ΓL(1,𝔽₈)⋊D₃₇₁ #44528428