Subgroups — C₁₃×ΓL(1,𝔽₈)×D₃₃ #31825063

Order 18018.

72 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₁₁×C₁₃×ΓL(1,𝔽₈)	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	C₁₃×ΓL(1,𝔽₈)×D₁₁	—	3	1	no	maximal
3	C₇×C₁₃×D₃₃	C₃	1	1	yes	maximal, characteristic, complemented
4	C₁₁×C₁₃×ΓL(1,𝔽₈)	—	2	1	no	—
5	C₁₁×C₁₃×ΓL(1,𝔽₈)	D₃	1	1	yes	characteristic, complemented
6	C₃×C₇×C₁₁×C₁₃	C₂×C₃	1	1	yes	characteristic, complemented
7	C₃×C₁₃×D₃₃	—	7	1	no	maximal, Hall
8	C₇×C₁₃×D₁₁	—	3	1	no	Hall
9	D₃×C₁₃×ΓL(1,𝔽₈)	—	11	1	no	maximal, Hall
10	ΓL(1,𝔽₈)×D₃₃	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Hall
11	C₃²×C₁₁×C₁₃	—	7	1	no	Hall
12	C₇×C₁₁×C₁₃	C₃×D₃	1	1	yes	characteristic, complemented, Hall
13	C₁₃×D₃₃	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
14	C₃×C₁₃×D₁₁	—	21	1	no	—
15	C₃×C₁₃×ΓL(1,𝔽₈)	D₁₁	1	1	yes	characteristic, complemented, Hall
16	C₃×C₁₁×ΓL(1,𝔽₈)	C₂×C₁₃	1	1	yes	characteristic, complemented, Hall
17	D₃×C₇×C₁₃	—	11	1	no	—
18	C₂×C₁₃×ΓL(1,𝔽₈)	—	33	1	no	—
19	C₇×D₃₃	C₃×C₁₃	1	1	yes	characteristic, complemented
20	ΓL(1,𝔽₈)×D₁₁	—	3	1	no	—
21	C₃×C₁₁×C₁₃	—	7	1	no	—
22	C₃×C₁₁×C₁₃	—	14	1	no	—
23	C₃×C₁₁×C₁₃	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
24	C₁₃×D₁₁	—	3	1	no	Hall
25	C₃×C₇×C₁₃	C₃×D₁₁	1	1	yes	characteristic, complemented
26	C₁₃×ΓL(1,𝔽₈)	—	2	1	no	—
27	C₁₃×ΓL(1,𝔽₈)	D₃₃	1	1	yes	characteristic, direct product factor
28	C₃×D₃×C₁₃	—	77	1	no	Hall
29	C₃×C₇×C₁₁	C₂×C₃×C₁₃	1	1	yes	characteristic, complemented
30	C₁₁×ΓL(1,𝔽₈)	—	2	1	no	—
31	C₁₁×ΓL(1,𝔽₈)	D₃×C₁₃	1	1	yes	characteristic, complemented
32	C₃×D₃₃	—	7	1	no	Hall
33	C₂×C₇×C₁₃	—	33	1	no	Hall
34	C₇×D₁₁	—	3	1	no	Hall
35	C₁₁×C₁₃	D₃×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Hall
36	D₃×ΓL(1,𝔽₈)	—	11	1	no	Hall
37	C₃²×C₁₃	—	7	1	no	Hall
38	C₃²×C₁₁	—	7	1	no	Hall
39	C₇×C₁₃	C₃×D₃₃	1	1	yes	characteristic, complemented, Hall
40	C₂×C₃×C₁₃	—	231	1	no	—
41	D₃×C₁₃	—	11	1	no	—
42	C₇×C₁₁	C₃×D₃×C₁₃	1	1	yes	characteristic, complemented, Hall
43	D₃₃	C₁₃×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
44	C₃×D₁₁	—	21	1	no	—
45	C₃×ΓL(1,𝔽₈)	C₁₃×D₁₁	1	1	yes	characteristic, complemented, Hall
46	D₃×C₇	—	11	1	no	—
47	C₂×ΓL(1,𝔽₈)	—	33	1	no	—
48	C₃×C₁₃	—	7	1	no	—
49	C₃×C₁₃	—	14	1	no	—
50	C₃×C₁₃	ΓL(1,𝔽₈)×D₁₁	1	1	yes	characteristic, complemented
51	C₃×C₁₁	—	7	1	no	—
52	C₃×C₁₁	—	14	1	no	—
53	C₃×C₁₁	C₂×C₁₃×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
54	C₂×C₁₃	—	33	1	no	Hall
55	D₁₁	—	3	1	no	Hall
56	C₃×C₇	C₃×C₁₃×D₁₁	1	1	yes	characteristic, complemented
57	ΓL(1,𝔽₈)	—	2	1	no	—
58	ΓL(1,𝔽₈)	C₁₃×D₃₃	1	1	yes	characteristic, direct product factor
59	C₃×D₃	—	77	1	no	Hall
60	C₂×C₇	—	33	1	no	Hall
61	C₁₃	ΓL(1,𝔽₈)×D₃₃	1	1	yes	characteristic, direct product factor, Sylow
62	C₁₁	D₃×C₁₃×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Sylow
63	C₃²	—	7	1	no	Sylow
64	C₇	C₃×C₁₃×D₃₃	1	1	yes	characteristic, complemented, Sylow
65	C₂×C₃	—	231	1	no	—
66	D₃	—	11	1	no	—
67	C₃	—	7	1	no	—
68	C₃	—	14	1	no	—
69	C₃	C₁₃×ΓL(1,𝔽₈)×D₁₁	1	1	yes	characteristic, complemented
70	C₂	—	33	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₁₃	ΓL(1,𝔽₈)×D₃₃
commutator, cosocle, hypocenter	C₃×C₇×C₁₁	C₂×C₃×C₁₃
Frattini	C₁	G
Fitting, socle	C₃×C₇×C₁₁×C₁₃	C₂×C₃
LCS	G ▷ C₃×C₇×C₁₁
UCS	C₁ ◁ C₁₃

Complements

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

N	H ≅ G/N	type
1. C₃×C₁₁×C₁₃×ΓL(1,𝔽₈)	C₂	semidirect
3. C₇×C₁₃×D₃₃	C₃	semidirect
5. C₁₁×C₁₃×ΓL(1,𝔽₈)	D₃	semidirect
6. C₃×C₇×C₁₁×C₁₃	C₂×C₃	semidirect
10. ΓL(1,𝔽₈)×D₃₃	C₁₃	direct (N × H)
12. C₇×C₁₁×C₁₃	C₃×D₃	semidirect
13. C₁₃×D₃₃	ΓL(1,𝔽₈)	direct (N × H)
15. C₃×C₁₃×ΓL(1,𝔽₈)	D₁₁	semidirect
16. C₃×C₁₁×ΓL(1,𝔽₈)	C₂×C₁₃	semidirect
19. C₇×D₃₃	C₃×C₁₃	semidirect
23. C₃×C₁₁×C₁₃	C₂×ΓL(1,𝔽₈)	semidirect
25. C₃×C₇×C₁₃	C₃×D₁₁	semidirect
27. C₁₃×ΓL(1,𝔽₈)	D₃₃	direct (N × H)
29. C₃×C₇×C₁₁	C₂×C₃×C₁₃	semidirect
31. C₁₁×ΓL(1,𝔽₈)	D₃×C₁₃	semidirect
35. C₁₁×C₁₃	D₃×ΓL(1,𝔽₈)	semidirect
39. C₇×C₁₃	C₃×D₃₃	semidirect
42. C₇×C₁₁	C₃×D₃×C₁₃	semidirect
43. D₃₃	C₁₃×ΓL(1,𝔽₈)	direct (N × H)
45. C₃×ΓL(1,𝔽₈)	C₁₃×D₁₁	semidirect
50. C₃×C₁₃	ΓL(1,𝔽₈)×D₁₁	semidirect
53. C₃×C₁₁	C₂×C₁₃×ΓL(1,𝔽₈)	semidirect
56. C₃×C₇	C₃×C₁₃×D₁₁	semidirect
58. ΓL(1,𝔽₈)	C₁₃×D₃₃	direct (N × H)
61. C₁₃	ΓL(1,𝔽₈)×D₃₃	direct (N × H)
62. C₁₁	D₃×C₁₃×ΓL(1,𝔽₈)	semidirect
64. C₇	C₃×C₁₃×D₃₃	semidirect
69. C₃	C₁₃×ΓL(1,𝔽₈)×D₁₁	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₁₁×C₁₃×ΓL(1,𝔽₈)	2
C₁₃×ΓL(1,𝔽₈)×D₁₁	3
C₇×C₁₃×D₃₃	3
C₃×C₁₃×D₃₃	7
D₃×C₁₃×ΓL(1,𝔽₈)	11
ΓL(1,𝔽₈)×D₃₃	13

Subgroups — C13×ΓL(1,𝔽8)×D33 #31825063

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×ΓL(1,𝔽₈)×D₃₃ #31825063