Subgroups — C₅×C₁₁×H(5,𝔽₂) #13183459

Order 1760.

332 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×C₁₁×#41	C₂	1	6	yes	maximal, complemented
2	C₂×C₅×D₄×C₁₁	C₂	1	9	yes	maximal, complemented
3	C₂³×C₅×C₁₁	C₂²	1	6	yes	complemented
4	C₅×Q₈×C₁₁	C₂²	1	2	yes	complemented
5	C₅×D₄×C₁₁	C₂²	1	18	yes	complemented
6	C₂×C₄×C₅×C₁₁	C₂²	1	9	yes	complemented
7	C₁₁×H(5,𝔽₂)	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
8	C₂²×C₅×C₁₁	—	4	6	no	—
9	C₂²×C₅×C₁₁	C₂³	1	9	yes	—
10	C₄×C₅×C₁₁	C₂³	1	6	yes	—
11	C₁₁×#41	C₂×C₅	1	6	yes	complemented
12	C₂×D₄×C₁₁	C₂×C₅	1	9	yes	complemented
13	C₅×H(5,𝔽₂)	C₁₁	1	1	yes	maximal, characteristic, direct product factor, Hall
14	C₂×C₅×C₁₁	—	2	9	no	—
15	C₂×C₅×C₁₁	C₂⁴	1	1	yes	characteristic
16	C₂³×C₁₁	C₂²×C₅	1	6	yes	complemented
17	Q₈×C₁₁	C₂²×C₅	1	2	yes	complemented
18	D₄×C₁₁	C₂²×C₅	1	18	yes	complemented
19	C₂×C₄×C₁₁	C₂²×C₅	1	9	yes	complemented
20	C₅×#41	C₂×C₁₁	1	6	yes	complemented
21	C₂×C₅×D₄	C₂×C₁₁	1	9	yes	complemented
22	C₅×C₁₁	H(5,𝔽₂)	1	1	yes	characteristic, direct product factor, Hall
23	C₂²×C₁₁	—	4	6	no	—
24	C₂²×C₁₁	C₂³×C₅	1	9	yes	—
25	C₄×C₁₁	C₂³×C₅	1	6	yes	—
26	C₂³×C₅	C₂²×C₁₁	1	6	yes	complemented
27	C₅×Q₈	C₂²×C₁₁	1	2	yes	complemented
28	C₅×D₄	C₂²×C₁₁	1	18	yes	complemented
29	C₂×C₄×C₅	C₂²×C₁₁	1	9	yes	complemented
30	H(5,𝔽₂)	C₅×C₁₁	1	1	yes	characteristic, direct product factor, Sylow
31	C₂×C₁₁	—	2	9	no	—
32	C₂×C₁₁	C₂⁴×C₅	1	1	yes	characteristic
33	C₂²×C₅	—	4	6	no	—
34	C₂²×C₅	C₂³×C₁₁	1	9	yes	—
35	C₄×C₅	C₂³×C₁₁	1	6	yes	—
36	#41	C₂×C₅×C₁₁	1	6	yes	complemented
37	C₂×D₄	C₂×C₅×C₁₁	1	9	yes	complemented
38	C₁₁	C₅×H(5,𝔽₂)	1	1	yes	characteristic, direct product factor, Sylow
39	C₂×C₅	—	2	9	no	—
40	C₂×C₅	C₂⁴×C₁₁	1	1	yes	characteristic
41	C₂³	C₂²×C₅×C₁₁	1	6	yes	complemented
42	Q₈	C₂²×C₅×C₁₁	1	2	yes	complemented
43	D₄	C₂²×C₅×C₁₁	1	18	yes	complemented
44	C₂×C₄	C₂²×C₅×C₁₁	1	9	yes	complemented
45	C₅	C₁₁×H(5,𝔽₂)	1	1	yes	characteristic, direct product factor, Sylow
46	C₂²	—	4	6	no	—
47	C₂²	C₂³×C₅×C₁₁	1	9	yes	—
48	C₄	C₂³×C₅×C₁₁	1	6	yes	—
49	C₂	—	2	9	no	—
50	C₂	C₂⁴×C₅×C₁₁	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₅×C₁₁	C₂⁴
commutator, Frattini, cosocle	C₂	C₂⁴×C₅×C₁₁
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅×C₁₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅×C₁₁×#41	C₂	semidirect
2. C₂×C₅×D₄×C₁₁	C₂	semidirect
3. C₂³×C₅×C₁₁	C₂²	semidirect
4. C₅×Q₈×C₁₁	C₂²	semidirect
5. C₅×D₄×C₁₁	C₂²	semidirect
6. C₂×C₄×C₅×C₁₁	C₂²	semidirect
7. C₁₁×H(5,𝔽₂)	C₅	direct (N × H)
11. C₁₁×#41	C₂×C₅	semidirect
12. C₂×D₄×C₁₁	C₂×C₅	semidirect
13. C₅×H(5,𝔽₂)	C₁₁	direct (N × H)
16. C₂³×C₁₁	C₂²×C₅	semidirect
17. Q₈×C₁₁	C₂²×C₅	semidirect
18. D₄×C₁₁	C₂²×C₅	semidirect
19. C₂×C₄×C₁₁	C₂²×C₅	semidirect
20. C₅×#41	C₂×C₁₁	semidirect
21. C₂×C₅×D₄	C₂×C₁₁	semidirect
22. C₅×C₁₁	H(5,𝔽₂)	direct (N × H)
26. C₂³×C₅	C₂²×C₁₁	semidirect
27. C₅×Q₈	C₂²×C₁₁	semidirect
28. C₅×D₄	C₂²×C₁₁	semidirect
29. C₂×C₄×C₅	C₂²×C₁₁	semidirect
30. H(5,𝔽₂)	C₅×C₁₁	direct (N × H)
36. #41	C₂×C₅×C₁₁	semidirect
37. C₂×D₄	C₂×C₅×C₁₁	semidirect
38. C₁₁	C₅×H(5,𝔽₂)	direct (N × H)
41. C₂³	C₂²×C₅×C₁₁	semidirect
42. Q₈	C₂²×C₅×C₁₁	semidirect
43. D₄	C₂²×C₅×C₁₁	semidirect
44. C₂×C₄	C₂²×C₅×C₁₁	semidirect
45. C₅	C₁₁×H(5,𝔽₂)	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅×C₁₁×#41	2
C₂×C₅×D₄×C₁₁	2
C₁₁×H(5,𝔽₂)	5
C₅×H(5,𝔽₂)	11

Subgroups — C5×C11×H(5,𝔽2) #13183459

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×C₁₁×H(5,𝔽₂) #13183459