Subgroups — C₁₁×(C₁₁⋊C₅)⋊Dic₅ #27130849

Order 12100.

68 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×C₁₁×C₁₁⋊C₅	C₂	1	1	yes	maximal, characteristic
2	C₅×C₁₁×C₁₁⋊C₅	C₄	1	1	yes	characteristic, complemented, Hall
3	C₁₁×Dic₅₅	C₅	1	1	yes	maximal, characteristic, complemented
4	C₁₁×Dic₁₁⋊C₅	—	5	1	no	maximal
5	C₂×C₅×C₁₁²	C₂×C₅	1	1	yes	characteristic
6	C₂×C₁₁×C₁₁⋊C₅	—	2	2	no	—
7	C₂×C₁₁×C₁₁⋊C₅	D₅	1	1	yes	characteristic
8	C₅×C₁₁×Dic₅	—	11	1	no	maximal
9	(C₁₁⋊C₅)⋊Dic₅	C₁₁	1	1	yes	maximal, characteristic, direct product factor
10	C₅×C₁₁²	C₄×C₅	1	1	yes	characteristic, complemented
11	C₁₁×C₁₁⋊C₅	—	2	2	no	—
12	C₁₁×C₁₁⋊C₅	Dic₅	1	1	yes	characteristic, complemented
13	C₂×C₅²×C₁₁	—	11	1	no	—
14	C₂×C₅×C₁₁⋊C₅	C₂×C₁₁	1	1	yes	characteristic
15	C₁₁×Dic₁₁	—	5	1	no	Hall
16	C₅²×C₁₁	—	11	1	no	—
17	C₅×C₁₁⋊C₅	C₄×C₁₁	1	1	yes	characteristic, complemented
18	C₂×C₁₁²	C₅×D₅	1	1	yes	characteristic
19	C₄×C₅×C₁₁	—	55	1	no	—
20	Dic₅₅	C₅×C₁₁	1	1	yes	characteristic, complemented
21	C₁₁×Dic₅	—	11	1	no	—
22	Dic₁₁⋊C₅	—	5	1	no	—
23	C₁₁²	C₅×Dic₅	1	1	yes	characteristic, complemented, Sylow
24	C₂×C₅×C₁₁	—	10	1	no	—
25	C₂×C₅×C₁₁	—	11	1	no	—
26	C₂×C₅×C₁₁	—	22	2	no	—
27	C₂×C₅×C₁₁	AGL(1,𝔽₁₁)	1	1	yes	characteristic
28	C₂×C₅×C₁₁	C₂×C₅×C₁₁	1	1	yes	characteristic
29	C₂×C₁₁⋊C₅	—	2	2	no	—
30	C₂×C₁₁⋊C₅	D₅×C₁₁	1	1	yes	characteristic
31	C₅×Dic₅	—	11	1	no	Hall
32	C₅×C₁₁	—	10	1	no	—
33	C₅×C₁₁	—	11	1	no	—
34	C₅×C₁₁	—	22	2	no	—
35	C₅×C₁₁	Dic₁₁⋊C₅	1	1	yes	characteristic, complemented
36	C₅×C₁₁	C₄×C₅×C₁₁	1	1	yes	characteristic, complemented
37	C₁₁⋊C₅	—	2	2	no	—
38	C₁₁⋊C₅	C₁₁×Dic₅	1	1	yes	characteristic, complemented
39	C₂×C₅²	—	11	1	no	—
40	C₄×C₁₁	—	55	1	no	—
41	Dic₁₁	—	5	1	no	—
42	C₅²	—	11	1	no	Sylow
43	C₂×C₁₁	—	10	1	no	—
44	C₂×C₁₁	C₅⋊AGL(1,𝔽₁₁)	1	1	yes	characteristic
45	C₂×C₁₁	C₅×D₅×C₁₁	1	1	yes	characteristic
46	C₄×C₅	—	55	1	no	—
47	Dic₅	—	11	1	no	—
48	C₁₁	—	10	1	no	—
49	C₁₁	(C₁₁⋊C₅)⋊Dic₅	1	1	yes	characteristic, direct product factor
50	C₁₁	C₅×C₁₁×Dic₅	1	1	yes	characteristic, complemented
51	C₂×C₅	—	11	1	no	—
52	C₂×C₅	—	22	2	no	—
53	C₂×C₅	C₁₁×AGL(1,𝔽₁₁)	1	1	yes	characteristic
54	C₅	—	11	1	no	—
55	C₅	—	22	2	no	—
56	C₅	C₁₁×Dic₁₁⋊C₅	1	1	yes	characteristic, complemented
57	C₄	—	55	1	no	Sylow
58	C₂	C₁₁×C₅⋊AGL(1,𝔽₁₁)	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂×C₁₁	C₅⋊AGL(1,𝔽₁₁)
commutator, hypocenter	C₅×C₁₁	C₄×C₅×C₁₁
Frattini	C₂	C₁₁×C₅⋊AGL(1,𝔽₁₁)
Fitting, socle	C₂×C₅×C₁₁²	C₂×C₅
prime radical	C₂×C₅×C₁₁×C₁₁⋊C₅	C₂
cosocle	C₂×C₅×C₁₁	C₂×C₅×C₁₁
LCS	G ▷ C₅×C₁₁
UCS	C₁ ◁ C₂×C₁₁

Complements

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

N	H ≅ G/N	type
2. C₅×C₁₁×C₁₁⋊C₅	C₄	semidirect
3. C₁₁×Dic₅₅	C₅	semidirect
9. (C₁₁⋊C₅)⋊Dic₅	C₁₁	direct (N × H)
10. C₅×C₁₁²	C₄×C₅	semidirect
12. C₁₁×C₁₁⋊C₅	Dic₅	semidirect
17. C₅×C₁₁⋊C₅	C₄×C₁₁	semidirect
20. Dic₅₅	C₅×C₁₁	semidirect
23. C₁₁²	C₅×Dic₅	semidirect
35. C₅×C₁₁	Dic₁₁⋊C₅	semidirect
36. C₅×C₁₁	C₄×C₅×C₁₁	semidirect
38. C₁₁⋊C₅	C₁₁×Dic₅	semidirect
49. C₁₁	(C₁₁⋊C₅)⋊Dic₅	direct (N × H)
50. C₁₁	C₅×C₁₁×Dic₅	semidirect
56. C₅	C₁₁×Dic₁₁⋊C₅	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×C₁₁×C₁₁⋊C₅	2
C₁₁×Dic₅₅	5
C₁₁×Dic₁₁⋊C₅	5
C₅×C₁₁×Dic₅	11
(C₁₁⋊C₅)⋊Dic₅	11

Subgroups — C11×(C11⋊C5)⋊Dic5 #27130849

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₁×(C₁₁⋊C₅)⋊Dic₅ #27130849