Subgroups — C₅×#14559056 #27130844

Order 12100.

100 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#10597244	C₂	1	1	yes	maximal, characteristic
2	C₅×#10597244	C₄	1	1	yes	characteristic, complemented, Hall
3	C₅×C₁₁×Dic₁₁	C₅	1	1	yes	maximal, characteristic, complemented
4	#14559056	C₅	1	5	yes	maximal, direct product factor
5	C₂×C₅×C₁₁²	C₂×C₅	1	1	yes	characteristic
6	C₂×#10597244	C₂×C₅	1	5	yes	—
7	C₄×C₅×C₁₁⋊C₅	—	11	1	no	maximal
8	C₅×Dic₁₁⋊C₅	—	11	1	no	maximal
9	C₅×C₁₁²	C₄×C₅	1	1	yes	characteristic, complemented
10	#10597244	C₄×C₅	1	5	yes	complemented
11	C₂×C₅×C₁₁⋊C₅	—	11	1	no	—
12	C₂×C₅×C₁₁⋊C₅	—	11	1	no	—
13	C₁₁×Dic₁₁	C₅²	1	1	yes	characteristic, complemented, Hall
14	C₅×C₁₁⋊C₅	—	11	1	no	—
15	C₅×C₁₁⋊C₅	—	11	1	no	—
16	C₂×C₁₁²	C₂×C₅²	1	1	yes	characteristic
17	C₄×C₅×C₁₁	—	11	1	no	—
18	C₅×Dic₁₁	C₁₁⋊C₅	1	1	yes	characteristic, complemented
19	C₄×C₁₁⋊C₅	—	11	5	no	—
20	Dic₁₁⋊C₅	—	11	5	no	—
21	C₁₁²	C₄×C₅²	1	1	yes	characteristic, complemented, Sylow
22	C₂×C₅×C₁₁	—	10	1	no	—
23	C₂×C₅×C₁₁	AGL(1,𝔽₁₁)	1	1	yes	characteristic
24	C₂×C₅×C₁₁	C₂×C₁₁⋊C₅	1	1	yes	characteristic
25	C₂×C₁₁⋊C₅	—	11	5	no	—
26	C₂×C₁₁⋊C₅	—	11	5	no	—
27	C₄×C₅²	—	121	1	no	Hall
28	C₅×C₁₁	—	10	1	no	—
29	C₅×C₁₁	Dic₁₁⋊C₅	1	1	yes	characteristic, complemented
30	C₅×C₁₁	C₄×C₁₁⋊C₅	1	1	yes	characteristic, complemented
31	C₁₁⋊C₅	—	11	5	no	—
32	C₁₁⋊C₅	—	11	5	no	—
33	C₂×C₅²	—	121	1	no	—
34	C₄×C₁₁	—	11	1	no	—
35	Dic₁₁	C₅×C₁₁⋊C₅	1	1	yes	characteristic, complemented
36	C₅²	—	121	1	no	Sylow
37	C₂×C₁₁	—	10	1	no	—
38	C₂×C₁₁	C₅×AGL(1,𝔽₁₁)	1	1	yes	characteristic
39	C₂×C₁₁	C₂×C₅×C₁₁⋊C₅	1	1	yes	characteristic
40	C₄×C₅	—	11	1	no	—
41	C₄×C₅	—	121	5	no	—
42	C₁₁	—	10	1	no	—
43	C₁₁	C₅×Dic₁₁⋊C₅	1	1	yes	characteristic, complemented
44	C₁₁	C₄×C₅×C₁₁⋊C₅	1	1	yes	characteristic, complemented
45	C₂×C₅	—	121	5	no	—
46	C₂×C₅	#11925897	1	1	yes	characteristic
47	C₅	—	121	5	no	—
48	C₅	#14559056	1	1	yes	characteristic, direct product factor
49	C₄	—	11	1	no	Sylow
50	C₂	C₅×#11925897	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂×C₅	#11925897
commutator, hypocenter	C₁₁²	C₄×C₅²
Frattini	C₂	C₅×#11925897
Fitting, socle	C₂×C₅×C₁₁²	C₂×C₅
prime radical	C₂×C₅×#10597244	C₂
cosocle	C₂×C₁₁²	C₂×C₅²
LCS	G ▷ C₁₁²
UCS	C₁ ◁ C₂×C₅

Complements

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

N	H ≅ G/N	type
2. C₅×#10597244	C₄	semidirect
3. C₅×C₁₁×Dic₁₁	C₅	semidirect
4. #14559056	C₅	direct (N × H)
9. C₅×C₁₁²	C₄×C₅	semidirect
10. #10597244	C₄×C₅	semidirect
13. C₁₁×Dic₁₁	C₅²	semidirect
18. C₅×Dic₁₁	C₁₁⋊C₅	semidirect
21. C₁₁²	C₄×C₅²	semidirect
29. C₅×C₁₁	Dic₁₁⋊C₅	semidirect
30. C₅×C₁₁	C₄×C₁₁⋊C₅	semidirect
35. Dic₁₁	C₅×C₁₁⋊C₅	semidirect
43. C₁₁	C₅×Dic₁₁⋊C₅	semidirect
44. C₁₁	C₄×C₅×C₁₁⋊C₅	semidirect
48. C₅	#14559056	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₅×#10597244	2
C₅×C₁₁×Dic₁₁	5
#14559056	5
C₄×C₅×C₁₁⋊C₅	11
C₅×Dic₁₁⋊C₅	11

Subgroups — C5×#14559056 #27130844

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#14559056 #27130844