Subgroups — C₁₃×ΓL(1,𝔽₈)×#1181 #44539849

Order 31668.

48 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×ΓL(1,𝔽₈)×D₂₉	C₂	1	1	yes	maximal, characteristic
2	C₇×C₁₃×#1181	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	C₁₃×ΓL(1,𝔽₈)×C₂₉	C₄	1	1	yes	characteristic, complemented, Hall
4	C₇×C₁₃×D₂₉	C₂×C₃	1	1	yes	characteristic
5	C₃×C₁₃×#1181	—	7	1	no	maximal, Hall
6	C₇×C₁₃×C₂₉	C₃×C₄	1	1	yes	characteristic, complemented, Hall
7	ΓL(1,𝔽₈)×#1181	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Hall
8	C₃×C₁₃×D₂₉	—	7	1	no	—
9	C₁₃×#1181	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Hall
10	ΓL(1,𝔽₈)×D₂₉	C₂×C₁₃	1	1	yes	characteristic
11	C₃×C₁₃×C₂₉	—	7	1	no	Hall
12	C₄×C₁₃×ΓL(1,𝔽₈)	—	29	1	no	maximal, Hall
13	C₇×#1181	C₃×C₁₃	1	1	yes	characteristic, complemented, Hall
14	C₁₃×D₂₉	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic
15	ΓL(1,𝔽₈)×C₂₉	C₄×C₁₃	1	1	yes	characteristic, complemented, Hall
16	C₂×C₁₃×ΓL(1,𝔽₈)	—	29	1	no	—
17	C₇×D₂₉	C₂×C₃×C₁₃	1	1	yes	characteristic
18	C₁₃×C₂₉	C₄×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Hall
19	C₄×C₇×C₁₃	—	29	1	no	Hall
20	C₃×#1181	—	7	1	no	Hall
21	C₁₃×ΓL(1,𝔽₈)	#1181	1	1	yes	characteristic, direct product factor, Hall
22	C₇×C₂₉	C₃×C₄×C₁₃	1	1	yes	characteristic, complemented, Hall
23	C₂×C₇×C₁₃	—	29	1	no	—
24	C₃×D₂₉	—	7	1	no	—
25	C₃×C₄×C₁₃	—	203	1	no	Hall
26	#1181	C₁₃×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Hall
27	C₇×C₁₃	C₃×#1181	1	1	yes	characteristic, complemented, Hall
28	C₃×C₂₉	—	7	1	no	Hall
29	C₄×ΓL(1,𝔽₈)	—	29	1	no	Hall
30	C₂×C₃×C₁₃	—	203	1	no	—
31	D₂₉	C₂×C₁₃×ΓL(1,𝔽₈)	1	1	yes	characteristic
32	C₄×C₁₃	—	29	1	no	Hall
33	C₂×ΓL(1,𝔽₈)	—	29	1	no	—
34	C₃×C₁₃	—	7	1	no	Hall
35	C₂₉	C₄×C₁₃×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Sylow
36	C₄×C₇	—	29	1	no	Hall
37	C₂×C₁₃	—	29	1	no	—
38	ΓL(1,𝔽₈)	C₁₃×#1181	1	1	yes	characteristic, direct product factor, Hall
39	C₂×C₇	—	29	1	no	—
40	C₁₃	ΓL(1,𝔽₈)×#1181	1	1	yes	characteristic, direct product factor, Sylow
41	C₃×C₄	—	203	1	no	Hall
42	C₇	C₃×C₁₃×#1181	1	1	yes	characteristic, complemented, Sylow
43	C₂×C₃	—	203	1	no	—
44	C₄	—	29	1	no	Sylow
45	C₃	—	7	1	no	Sylow
46	C₂	—	29	1	no	—

	subgroup	G/H
center, hypercenter	C₁₃	ΓL(1,𝔽₈)×#1181
commutator, hypocenter	C₇×C₂₉	C₃×C₄×C₁₃
Frattini	C₁	G
Fitting, socle	C₇×C₁₃×C₂₉	C₃×C₄
prime radical	C₁₃×ΓL(1,𝔽₈)×D₂₉	C₂
cosocle	C₇×D₂₉	C₂×C₃×C₁₃
LCS	G ▷ C₇×C₂₉
UCS	C₁ ◁ C₁₃

Complements

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

N	H ≅ G/N	type
2. C₇×C₁₃×#1181	C₃	semidirect
3. C₁₃×ΓL(1,𝔽₈)×C₂₉	C₄	semidirect
6. C₇×C₁₃×C₂₉	C₃×C₄	semidirect
7. ΓL(1,𝔽₈)×#1181	C₁₃	direct (N × H)
9. C₁₃×#1181	ΓL(1,𝔽₈)	direct (N × H)
13. C₇×#1181	C₃×C₁₃	semidirect
15. ΓL(1,𝔽₈)×C₂₉	C₄×C₁₃	semidirect
18. C₁₃×C₂₉	C₄×ΓL(1,𝔽₈)	semidirect
21. C₁₃×ΓL(1,𝔽₈)	#1181	direct (N × H)
22. C₇×C₂₉	C₃×C₄×C₁₃	semidirect
26. #1181	C₁₃×ΓL(1,𝔽₈)	direct (N × H)
27. C₇×C₁₃	C₃×#1181	semidirect
35. C₂₉	C₄×C₁₃×ΓL(1,𝔽₈)	semidirect
38. ΓL(1,𝔽₈)	C₁₃×#1181	direct (N × H)
40. C₁₃	ΓL(1,𝔽₈)×#1181	direct (N × H)
42. C₇	C₃×C₁₃×#1181	semidirect

Maximal subgroups

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

subgroup	index
C₁₃×ΓL(1,𝔽₈)×D₂₉	2
C₇×C₁₃×#1181	3
C₃×C₁₃×#1181	7
ΓL(1,𝔽₈)×#1181	13
C₄×C₁₃×ΓL(1,𝔽₈)	29

Subgroups — C13×ΓL(1,𝔽8)×#1181 #44539849

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×ΓL(1,𝔽₈)×#1181 #44539849