Subgroups — ΓL(1,𝔽₈)×Dic₃₇₇ #44539744

Order 31668.

48 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×ΓL(1,𝔽₈)×C₂₉	C₂	1	1	yes	maximal, characteristic
2	C₇×Dic₃₇₇	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	C₁₃×ΓL(1,𝔽₈)×C₂₉	C₄	1	1	yes	characteristic, complemented, Hall
4	C₂×C₇×C₁₃×C₂₉	C₂×C₃	1	1	yes	characteristic
5	C₃×Dic₃₇₇	—	7	1	no	maximal, Hall
6	C₇×C₁₃×C₂₉	C₃×C₄	1	1	yes	characteristic, complemented, Hall
7	ΓL(1,𝔽₈)×Dic₂₉	—	13	1	no	maximal, Hall
8	C₂×C₃×C₁₃×C₂₉	—	7	1	no	—
9	Dic₃₇₇	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Hall
10	C₂×ΓL(1,𝔽₈)×C₂₉	D₁₃	1	1	yes	characteristic
11	C₃×C₁₃×C₂₉	—	7	1	no	Hall
12	ΓL(1,𝔽₈)×Dic₁₃	—	29	1	no	maximal, Hall
13	C₇×Dic₂₉	—	13	1	no	Hall
14	C₂×C₁₃×C₂₉	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic
15	ΓL(1,𝔽₈)×C₂₉	Dic₁₃	1	1	yes	characteristic, complemented, Hall
16	C₂×C₁₃×ΓL(1,𝔽₈)	D₂₉	1	1	yes	characteristic
17	C₂×C₇×C₂₉	C₃×D₁₃	1	1	yes	characteristic
18	C₁₃×C₂₉	C₄×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Hall
19	C₇×Dic₁₃	—	29	1	no	Hall
20	C₃×Dic₂₉	—	91	1	no	Hall
21	C₁₃×ΓL(1,𝔽₈)	Dic₂₉	1	1	yes	characteristic, complemented, Hall
22	C₇×C₂₉	C₃×Dic₁₃	1	1	yes	characteristic, complemented, Hall
23	C₂×C₇×C₁₃	C₃×D₂₉	1	1	yes	characteristic
24	C₂×C₃×C₂₉	—	7	1	no	—
25	C₃×Dic₁₃	—	203	1	no	Hall
26	Dic₂₉	—	13	1	no	Hall
27	C₇×C₁₃	C₃×Dic₂₉	1	1	yes	characteristic, complemented, Hall
28	C₃×C₂₉	—	7	1	no	Hall
29	C₄×ΓL(1,𝔽₈)	—	377	1	no	Hall
30	C₂×C₃×C₁₃	—	7	1	no	—
31	C₂×C₂₉	ΓL(1,𝔽₈)×D₁₃	1	1	yes	characteristic
32	Dic₁₃	—	29	1	no	Hall
33	C₂×ΓL(1,𝔽₈)	D₃₇₇	1	1	yes	characteristic
34	C₃×C₁₃	—	7	1	no	Hall
35	C₂₉	ΓL(1,𝔽₈)×Dic₁₃	1	1	yes	characteristic, complemented, Sylow
36	C₄×C₇	—	377	1	no	Hall
37	C₂×C₁₃	ΓL(1,𝔽₈)×D₂₉	1	1	yes	characteristic
38	ΓL(1,𝔽₈)	Dic₃₇₇	1	1	yes	characteristic, direct product factor, Hall
39	C₂×C₇	C₃×D₃₇₇	1	1	yes	characteristic
40	C₁₃	ΓL(1,𝔽₈)×Dic₂₉	1	1	yes	characteristic, complemented, Sylow
41	C₃×C₄	—	2639	1	no	Hall
42	C₇	C₃×Dic₃₇₇	1	1	yes	characteristic, complemented, Sylow
43	C₂×C₃	—	7	1	no	—
44	C₄	—	377	1	no	Sylow
45	C₃	—	7	1	no	Sylow
46	C₂	ΓL(1,𝔽₈)×D₃₇₇	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₂	ΓL(1,𝔽₈)×D₃₇₇
commutator, hypocenter	C₇×C₁₃×C₂₉	C₃×C₄
Fitting, socle, cosocle	C₂×C₇×C₁₃×C₂₉	C₂×C₃
prime radical	C₂×C₁₃×ΓL(1,𝔽₈)×C₂₉	C₂
LCS	G ▷ C₇×C₁₃×C₂₉
UCS	C₁ ◁ C₂

Complements

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

N	H ≅ G/N	type
2. C₇×Dic₃₇₇	C₃	semidirect
3. C₁₃×ΓL(1,𝔽₈)×C₂₉	C₄	semidirect
6. C₇×C₁₃×C₂₉	C₃×C₄	semidirect
9. Dic₃₇₇	ΓL(1,𝔽₈)	direct (N × H)
15. ΓL(1,𝔽₈)×C₂₉	Dic₁₃	semidirect
18. C₁₃×C₂₉	C₄×ΓL(1,𝔽₈)	semidirect
21. C₁₃×ΓL(1,𝔽₈)	Dic₂₉	semidirect
22. C₇×C₂₉	C₃×Dic₁₃	semidirect
27. C₇×C₁₃	C₃×Dic₂₉	semidirect
35. C₂₉	ΓL(1,𝔽₈)×Dic₁₃	semidirect
38. ΓL(1,𝔽₈)	Dic₃₇₇	direct (N × H)
40. C₁₃	ΓL(1,𝔽₈)×Dic₂₉	semidirect
42. C₇	C₃×Dic₃₇₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₁₃×ΓL(1,𝔽₈)×C₂₉	2
C₇×Dic₃₇₇	3
C₃×Dic₃₇₇	7
ΓL(1,𝔽₈)×Dic₂₉	13
ΓL(1,𝔽₈)×Dic₁₃	29

Subgroups — ΓL(1,𝔽8)×Dic377 #44539744

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — ΓL(1,𝔽₈)×Dic₃₇₇ #44539744