Subgroups — ΓL(1,𝔽₈)×Dic₂₁₃ #31822041

Order 17892.

56 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×ΓL(1,𝔽₈)×C₇₁	C₂	1	1	yes	maximal, characteristic
2	ΓL(1,𝔽₈)×Dic₇₁	—	3	1	no	maximal
3	C₇×Dic₂₁₃	C₃	1	1	yes	maximal, characteristic, complemented
4	C₃×ΓL(1,𝔽₈)×C₇₁	C₄	1	1	yes	characteristic, complemented, Hall
5	C₂×ΓL(1,𝔽₈)×C₇₁	—	2	1	no	—
6	C₂×ΓL(1,𝔽₈)×C₇₁	D₃	1	1	yes	characteristic
7	C₂×C₃×C₇×C₇₁	C₂×C₃	1	1	yes	characteristic
8	C₃×Dic₂₁₃	—	7	1	no	maximal, Hall
9	C₇×Dic₇₁	—	3	1	no	Hall
10	C₃×C₇×C₇₁	C₃×C₄	1	1	yes	characteristic, complemented
11	ΓL(1,𝔽₈)×C₇₁	—	2	1	no	—
12	ΓL(1,𝔽₈)×C₇₁	Dic₃	1	1	yes	characteristic, complemented
13	C₂×C₃²×C₇₁	—	7	1	no	—
14	C₂×C₇×C₇₁	C₃×D₃	1	1	yes	characteristic
15	Dic₂₁₃	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
16	C₃×Dic₇₁	—	21	1	no	—
17	C₃²×C₇₁	—	7	1	no	Hall
18	C₇×C₇₁	C₃×Dic₃	1	1	yes	characteristic, complemented, Hall
19	C₂×C₃×C₇₁	—	7	1	no	—
20	C₂×C₃×C₇₁	—	14	1	no	—
21	C₂×C₃×C₇₁	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic
22	Dic₇₁	—	3	1	no	Hall
23	Dic₃×ΓL(1,𝔽₈)	—	71	1	no	maximal, Hall
24	C₃×C₇₁	—	7	1	no	—
25	C₃×C₇₁	—	14	1	no	—
26	C₃×C₇₁	C₄×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
27	C₂×C₇₁	D₃×ΓL(1,𝔽₈)	1	1	yes	characteristic
28	C₂×C₃×ΓL(1,𝔽₈)	D₇₁	1	1	yes	characteristic
29	C₇×Dic₃	—	71	1	no	—
30	C₄×ΓL(1,𝔽₈)	—	213	1	no	—
31	C₇₁	Dic₃×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Sylow
32	C₃×ΓL(1,𝔽₈)	Dic₇₁	1	1	yes	characteristic, complemented, Hall
33	C₂×C₃×C₇	C₃×D₇₁	1	1	yes	characteristic
34	C₂×ΓL(1,𝔽₈)	—	2	1	no	—
35	C₂×ΓL(1,𝔽₈)	D₂₁₃	1	1	yes	characteristic
36	C₃×Dic₃	—	497	1	no	Hall
37	C₄×C₇	—	213	1	no	Hall
38	C₃×C₇	C₃×Dic₇₁	1	1	yes	characteristic, complemented
39	ΓL(1,𝔽₈)	—	2	1	no	—
40	ΓL(1,𝔽₈)	Dic₂₁₃	1	1	yes	characteristic, direct product factor
41	C₂×C₃²	—	7	1	no	—
42	C₂×C₇	C₃×D₂₁₃	1	1	yes	characteristic
43	C₃×C₄	—	1491	1	no	—
44	Dic₃	—	71	1	no	—
45	C₃²	—	7	1	no	Sylow
46	C₇	C₃×Dic₂₁₃	1	1	yes	characteristic, complemented, Sylow
47	C₂×C₃	—	7	1	no	—
48	C₂×C₃	—	14	1	no	—
49	C₂×C₃	ΓL(1,𝔽₈)×D₇₁	1	1	yes	characteristic
50	C₄	—	213	1	no	Sylow
51	C₃	—	7	1	no	—
52	C₃	—	14	1	no	—
53	C₃	ΓL(1,𝔽₈)×Dic₇₁	1	1	yes	characteristic, complemented
54	C₂	ΓL(1,𝔽₈)×D₂₁₃	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₂	ΓL(1,𝔽₈)×D₂₁₃
commutator, hypocenter	C₃×C₇×C₇₁	C₃×C₄
Fitting, socle, cosocle	C₂×C₃×C₇×C₇₁	C₂×C₃
prime radical	C₂×C₃×ΓL(1,𝔽₈)×C₇₁	C₂
LCS	G ▷ C₃×C₇×C₇₁
UCS	C₁ ◁ C₂

Complements

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

N	H ≅ G/N	type
3. C₇×Dic₂₁₃	C₃	semidirect
4. C₃×ΓL(1,𝔽₈)×C₇₁	C₄	semidirect
10. C₃×C₇×C₇₁	C₃×C₄	semidirect
12. ΓL(1,𝔽₈)×C₇₁	Dic₃	semidirect
15. Dic₂₁₃	ΓL(1,𝔽₈)	direct (N × H)
18. C₇×C₇₁	C₃×Dic₃	semidirect
26. C₃×C₇₁	C₄×ΓL(1,𝔽₈)	semidirect
31. C₇₁	Dic₃×ΓL(1,𝔽₈)	semidirect
32. C₃×ΓL(1,𝔽₈)	Dic₇₁	semidirect
38. C₃×C₇	C₃×Dic₇₁	semidirect
40. ΓL(1,𝔽₈)	Dic₂₁₃	direct (N × H)
46. C₇	C₃×Dic₂₁₃	semidirect
53. C₃	ΓL(1,𝔽₈)×Dic₇₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×ΓL(1,𝔽₈)×C₇₁	2
ΓL(1,𝔽₈)×Dic₇₁	3
C₇×Dic₂₁₃	3
C₃×Dic₂₁₃	7
Dic₃×ΓL(1,𝔽₈)	71

Subgroups — ΓL(1,𝔽8)×Dic213 #31822041

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — ΓL(1,𝔽₈)×Dic₂₁₃ #31822041