Subgroups — C₅×D₅×C₁₁⋊D₁₁ #27130970

Order 12100.

336 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×C₅⋊(C₁₁⋊D₁₁)	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×D₅×C₁₁²	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅²×C₁₁⋊D₁₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅²×C₁₁²	C₂²	1	1	yes	characteristic, complemented, Hall
5	C₂×C₅×C₁₁⋊D₁₁	—	5	1	no	maximal
6	D₅×C₁₁⋊D₁₁	C₅	1	1	yes	maximal, characteristic, direct product factor
7	C₂×C₅×C₁₁²	—	5	1	no	—
8	C₅⋊(C₁₁⋊D₁₁)	C₂×C₅	1	1	yes	characteristic, complemented
9	D₅×C₁₁²	C₂×C₅	1	1	yes	characteristic, complemented
10	C₅×C₁₁⋊D₁₁	—	2	2	no	—
11	C₅×C₁₁⋊D₁₁	—	5	1	no	—
12	C₅×C₁₁⋊D₁₁	D₅	1	1	yes	characteristic, direct product factor
13	C₅×C₁₁⋊D₁₁	C₂×C₅	1	1	yes	characteristic, complemented
14	C₅×D₅×D₁₁	—	11	12	no	maximal
15	C₅×C₁₁²	—	2	2	no	—
16	C₅×C₁₁²	C₂×D₅	1	1	yes	characteristic, complemented
17	C₅×C₁₁²	C₂²×C₅	1	1	yes	characteristic, complemented
18	C₅×D₅₅	—	11	12	no	—
19	C₅×D₅×C₁₁	D₁₁	1	12	yes	complemented
20	C₅²×D₁₁	—	11	12	no	—
21	C₂×C₁₁⋊D₁₁	—	5	1	no	Hall
22	C₅²×C₁₁	C₂×D₁₁	1	12	yes	complemented
23	C₂×C₁₁²	—	5	1	no	—
24	C₁₁⋊D₁₁	—	5	1	no	—
25	C₁₁⋊D₁₁	C₅×D₅	1	1	yes	characteristic, direct product factor
26	C₂×C₅×D₁₁	—	55	12	no	—
27	D₅×D₁₁	—	11	12	no	—
28	C₁₁²	C₂×C₅×D₅	1	1	yes	characteristic, complemented, Sylow
29	C₂×C₅×C₁₁	—	5	12	no	—
30	D₅₅	—	11	12	no	—
31	C₅×D₁₁	—	11	12	no	—
32	C₅×D₁₁	—	11	12	no	—
33	C₅×D₁₁	—	22	24	no	—
34	C₅×D₁₁	—	55	12	no	—
35	D₅×C₁₁	C₅×D₁₁	1	12	yes	complemented
36	C₂×C₅×D₅	—	121	1	no	Hall
37	C₅×C₁₁	—	2	24	no	—
38	C₅×C₁₁	D₅×D₁₁	1	12	yes	complemented
39	C₅×C₁₁	C₂×C₅×D₁₁	1	12	yes	complemented
40	C₂×C₅²	—	121	1	no	—
41	C₅×D₅	—	121	1	no	—
42	C₅×D₅	C₁₁⋊D₁₁	1	1	yes	characteristic, direct product factor
43	C₂×D₁₁	—	55	12	no	—
44	C₅²	C₂×C₁₁⋊D₁₁	1	1	yes	characteristic, complemented, Sylow
45	C₂×C₁₁	—	5	12	no	—
46	D₁₁	—	11	12	no	—
47	D₁₁	—	55	12	no	—
48	C₂²×C₅	—	605	1	no	—
49	C₂×D₅	—	121	1	no	—
50	C₁₁	C₅×D₅×D₁₁	1	12	yes	complemented
51	C₂×C₅	—	5	1	no	—
52	C₂×C₅	—	605	1	no	—
53	C₂×C₅	—	121	1	no	—
54	C₂×C₅	—	121	1	no	—
55	C₂×C₅	—	242	2	no	—
56	D₅	—	121	1	no	—
57	D₅	C₅×C₁₁⋊D₁₁	1	1	yes	characteristic, direct product factor
58	C₅	—	2	2	no	—
59	C₅	D₅×C₁₁⋊D₁₁	1	1	yes	characteristic, direct product factor
60	C₅	C₂×C₅×C₁₁⋊D₁₁	1	1	yes	characteristic, complemented
61	C₂²	—	605	1	no	Sylow
62	C₂	—	5	1	no	—
63	C₂	—	605	1	no	—
64	C₂	—	121	1	no	—

	subgroup	G/H
center, hypercenter	C₅	D₅×C₁₁⋊D₁₁
commutator, cosocle, hypocenter	C₅×C₁₁²	C₂²×C₅
Frattini	C₁	G
Fitting, socle	C₅²×C₁₁²	C₂²
LCS	G ▷ C₅×C₁₁²
UCS	C₁ ◁ C₅

Complements

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

N	H ≅ G/N	type
1. C₅×C₅⋊(C₁₁⋊D₁₁)	C₂	semidirect
2. C₅×D₅×C₁₁²	C₂	semidirect
3. C₅²×C₁₁⋊D₁₁	C₂	semidirect
4. C₅²×C₁₁²	C₂²	semidirect
6. D₅×C₁₁⋊D₁₁	C₅	direct (N × H)
8. C₅⋊(C₁₁⋊D₁₁)	C₂×C₅	semidirect
9. D₅×C₁₁²	C₂×C₅	semidirect
12. C₅×C₁₁⋊D₁₁	D₅	direct (N × H)
13. C₅×C₁₁⋊D₁₁	C₂×C₅	semidirect
16. C₅×C₁₁²	C₂×D₅	semidirect
17. C₅×C₁₁²	C₂²×C₅	semidirect
19. C₅×D₅×C₁₁	D₁₁	semidirect
22. C₅²×C₁₁	C₂×D₁₁	semidirect
25. C₁₁⋊D₁₁	C₅×D₅	direct (N × H)
28. C₁₁²	C₂×C₅×D₅	semidirect
35. D₅×C₁₁	C₅×D₁₁	semidirect
38. C₅×C₁₁	D₅×D₁₁	semidirect
39. C₅×C₁₁	C₂×C₅×D₁₁	semidirect
42. C₅×D₅	C₁₁⋊D₁₁	direct (N × H)
44. C₅²	C₂×C₁₁⋊D₁₁	semidirect
50. C₁₁	C₅×D₅×D₁₁	semidirect
57. D₅	C₅×C₁₁⋊D₁₁	direct (N × H)
59. C₅	D₅×C₁₁⋊D₁₁	direct (N × H)
60. C₅	C₂×C₅×C₁₁⋊D₁₁	semidirect

Maximal subgroups

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

subgroup	index
C₅×C₅⋊(C₁₁⋊D₁₁)	2
C₅×D₅×C₁₁²	2
C₅²×C₁₁⋊D₁₁	2
C₂×C₅×C₁₁⋊D₁₁	5
D₅×C₁₁⋊D₁₁	5
C₅×D₅×D₁₁	11

Subgroups — C5×D5×C11⋊D11 #27130970

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×D₅×C₁₁⋊D₁₁ #27130970