Subgroups — C₂×C₅×#11925894 #27130936

Order 12100.

180 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, complemented
2	C₅×#11925894	C₂	1	2	yes	maximal, direct product factor
3	C₅×#10597244	C₂²	1	1	yes	characteristic, complemented, Hall
4	C₂×C₅×C₁₁⋊D₁₁	C₅	1	1	yes	maximal, characteristic, complemented
5	C₂×#11925894	C₅	1	5	yes	maximal, direct product factor
6	C₂×C₅×C₁₁²	C₂×C₅	1	1	yes	characteristic, complemented
7	C₅×C₁₁⋊D₁₁	C₂×C₅	1	2	yes	complemented
8	C₂×#10597244	C₂×C₅	1	5	yes	complemented
9	#11925894	C₂×C₅	1	10	yes	direct product factor
10	C₂×C₅×AGL(1,𝔽₁₁)	—	11	2	no	maximal
11	C₅×C₁₁²	C₂²×C₅	1	1	yes	characteristic, complemented
12	#10597244	C₂²×C₅	1	5	yes	complemented
13	C₂×C₅×C₁₁⋊C₅	—	11	2	no	—
14	C₅×AGL(1,𝔽₁₁)	—	11	4	no	—
15	C₂×C₁₁⋊D₁₁	C₅²	1	1	yes	characteristic, complemented, Hall
16	C₅×C₁₁⋊C₅	—	11	2	no	—
17	C₂×C₁₁²	C₂×C₅²	1	1	yes	characteristic, complemented
18	C₁₁⋊D₁₁	C₂×C₅²	1	2	yes	complemented
19	C₂×C₅×D₁₁	—	11	2	no	—
20	C₂×C₅×D₁₁	—	55	2	no	—
21	C₂×AGL(1,𝔽₁₁)	—	11	10	no	—
22	C₁₁²	C₂²×C₅²	1	1	yes	characteristic, complemented, Sylow
23	C₂×C₅×C₁₁	—	5	2	no	—
24	C₂×C₅×C₁₁	AGL(1,𝔽₁₁)	1	2	yes	complemented
25	C₅×D₁₁	—	11	4	no	—
26	C₅×D₁₁	—	55	4	no	—
27	C₂×C₁₁⋊C₅	—	11	10	no	—
28	AGL(1,𝔽₁₁)	—	11	20	no	—
29	C₂²×C₅²	—	121	1	no	Hall
30	C₅×C₁₁	—	5	2	no	—
31	C₅×C₁₁	C₂×AGL(1,𝔽₁₁)	1	2	yes	complemented
32	C₁₁⋊C₅	—	11	10	no	—
33	C₂×C₅²	—	121	2	no	—
34	C₂×C₅²	—	121	1	no	—
35	C₂×D₁₁	—	11	2	no	—
36	C₂×D₁₁	—	55	2	no	—
37	C₅²	—	121	1	no	Sylow
38	C₂×C₁₁	—	5	2	no	—
39	C₂×C₁₁	C₅×AGL(1,𝔽₁₁)	1	2	yes	complemented
40	D₁₁	—	11	4	no	—
41	D₁₁	—	55	4	no	—
42	C₂²×C₅	—	121	5	no	—
43	C₂²×C₅	—	121	1	no	—
44	C₁₁	—	5	2	no	—
45	C₁₁	C₂×C₅×AGL(1,𝔽₁₁)	1	2	yes	complemented
46	C₂×C₅	—	121	10	no	—
47	C₂×C₅	—	121	2	no	—
48	C₂×C₅	—	121	5	no	—
49	C₂×C₅	#11925894	1	1	yes	characteristic, direct product factor
50	C₅	—	121	5	no	—
51	C₅	C₂×#11925894	1	1	yes	characteristic, direct product factor
52	C₂²	—	121	1	no	Sylow
53	C₂	—	121	2	no	—
54	C₂	C₅×#11925894	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂×C₅	#11925894
commutator, cosocle, hypocenter	C₁₁²	C₂²×C₅²
Frattini	C₁	G
Fitting, socle	C₂×C₅×C₁₁²	C₂×C₅
LCS	G ▷ C₁₁²
UCS	C₁ ◁ C₂×C₅

Complements

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

N	H ≅ G/N	type
1. C₂×C₅×#10597244	C₂	semidirect
2. C₅×#11925894	C₂	direct (N × H)
3. C₅×#10597244	C₂²	semidirect
4. C₂×C₅×C₁₁⋊D₁₁	C₅	semidirect
5. C₂×#11925894	C₅	direct (N × H)
6. C₂×C₅×C₁₁²	C₂×C₅	semidirect
7. C₅×C₁₁⋊D₁₁	C₂×C₅	semidirect
8. C₂×#10597244	C₂×C₅	semidirect
9. #11925894	C₂×C₅	direct (N × H)
11. C₅×C₁₁²	C₂²×C₅	semidirect
12. #10597244	C₂²×C₅	semidirect
15. C₂×C₁₁⋊D₁₁	C₅²	semidirect
17. C₂×C₁₁²	C₂×C₅²	semidirect
18. C₁₁⋊D₁₁	C₂×C₅²	semidirect
22. C₁₁²	C₂²×C₅²	semidirect
24. C₂×C₅×C₁₁	AGL(1,𝔽₁₁)	semidirect
31. C₅×C₁₁	C₂×AGL(1,𝔽₁₁)	semidirect
39. C₂×C₁₁	C₅×AGL(1,𝔽₁₁)	semidirect
45. C₁₁	C₂×C₅×AGL(1,𝔽₁₁)	semidirect
49. C₂×C₅	#11925894	direct (N × H)
51. C₅	C₂×#11925894	direct (N × H)
54. C₂	C₅×#11925894	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×#11925894	2
C₂×C₅×C₁₁⋊D₁₁	5
C₂×#11925894	5
C₂×C₅×AGL(1,𝔽₁₁)	11

Subgroups — C2×C5×#11925894 #27130936

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#11925894 #27130936