Subgroups — C₂×C₅⋊#11925893 #27130947

Order 12100.

116 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#10597245	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅⋊#11925893	C₂	1	2	yes	maximal, direct product factor
3	C₅×#10597245	C₂²	1	1	yes	characteristic, complemented, Hall
4	C₂×C₅⋊(C₁₁⋊D₁₁)	C₅	1	1	yes	maximal, characteristic, complemented
5	C₂×#11925893	—	5	1	no	maximal
6	C₂×C₅×C₁₁²	C₂×C₅	1	1	yes	characteristic, complemented
7	C₅⋊(C₁₁⋊D₁₁)	C₂×C₅	1	2	yes	complemented
8	C₂×#10597245	—	2	2	no	—
9	C₂×#10597245	D₅	1	1	yes	characteristic, complemented
10	#11925893	—	5	2	no	—
11	C₂×C₅⋊AGL(1,𝔽₁₁)	—	11	2	no	maximal
12	C₅×C₁₁²	C₂²×C₅	1	1	yes	characteristic, complemented
13	#10597245	—	2	2	no	—
14	#10597245	C₂×D₅	1	1	yes	characteristic, complemented
15	C₂×C₅×C₁₁⋊C₅	—	11	2	no	—
16	C₅⋊AGL(1,𝔽₁₁)	—	11	4	no	—
17	C₂×C₁₁⋊D₁₁	—	5	1	no	Hall
18	C₅×C₁₁⋊C₅	—	11	2	no	—
19	C₂×C₁₁²	C₅×D₅	1	1	yes	characteristic, complemented
20	C₁₁⋊D₁₁	—	5	2	no	—
21	C₂×D₅₅	—	11	2	no	—
22	C₂×D₅₅	—	55	2	no	—
23	C₂×AGL(1,𝔽₁₁)	—	55	2	no	—
24	C₁₁²	C₂×C₅×D₅	1	1	yes	characteristic, complemented, Sylow
25	C₂×C₅×C₁₁	—	5	2	no	—
26	C₂×C₅×C₁₁	AGL(1,𝔽₁₁)	1	2	yes	complemented
27	D₅₅	—	11	4	no	—
28	D₅₅	—	55	4	no	—
29	C₂×C₁₁⋊C₅	—	11	2	no	—
30	C₂×C₁₁⋊C₅	—	22	4	no	—
31	AGL(1,𝔽₁₁)	—	55	4	no	—
32	C₂×C₅×D₅	—	121	1	no	Hall
33	C₅×C₁₁	—	5	2	no	—
34	C₅×C₁₁	C₂×AGL(1,𝔽₁₁)	1	2	yes	complemented
35	C₁₁⋊C₅	—	11	2	no	—
36	C₁₁⋊C₅	—	22	4	no	—
37	C₂×C₅²	—	121	1	no	—
38	C₅×D₅	—	121	2	no	—
39	C₂×D₁₁	—	275	2	no	—
40	C₂×D₁₁	—	55	2	no	—
41	C₅²	—	121	1	no	Sylow
42	C₂×C₁₁	—	5	2	no	—
43	C₂×C₁₁	C₅⋊AGL(1,𝔽₁₁)	1	2	yes	complemented
44	D₁₁	—	275	4	no	—
45	D₁₁	—	55	4	no	—
46	C₂²×C₅	—	605	1	no	—
47	C₂×D₅	—	121	1	no	—
48	C₁₁	—	5	2	no	—
49	C₁₁	C₂×C₅⋊AGL(1,𝔽₁₁)	1	2	yes	complemented
50	C₂×C₅	—	605	2	no	—
51	C₂×C₅	—	121	1	no	—
52	C₂×C₅	—	242	2	no	—
53	C₂×C₅	#11925893	1	1	yes	characteristic, complemented
54	D₅	—	121	2	no	—
55	C₅	—	121	1	no	—
56	C₅	—	242	2	no	—
57	C₅	C₂×#11925893	1	1	yes	characteristic, complemented
58	C₂²	—	605	1	no	Sylow
59	C₂	—	605	2	no	—
60	C₂	C₅⋊#11925893	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂	C₅⋊#11925893
commutator, cosocle, hypocenter	C₅×C₁₁²	C₂²×C₅
Frattini	C₁	G
Fitting, socle	C₂×C₅×C₁₁²	C₂×C₅
LCS	G ▷ C₅×C₁₁²
UCS	C₁ ◁ C₂

Complements

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

N	H ≅ G/N	type
1. C₂×C₅×#10597245	C₂	semidirect
2. C₅⋊#11925893	C₂	direct (N × H)
3. C₅×#10597245	C₂²	semidirect
4. C₂×C₅⋊(C₁₁⋊D₁₁)	C₅	semidirect
6. C₂×C₅×C₁₁²	C₂×C₅	semidirect
7. C₅⋊(C₁₁⋊D₁₁)	C₂×C₅	semidirect
9. C₂×#10597245	D₅	semidirect
12. C₅×C₁₁²	C₂²×C₅	semidirect
14. #10597245	C₂×D₅	semidirect
19. C₂×C₁₁²	C₅×D₅	semidirect
24. C₁₁²	C₂×C₅×D₅	semidirect
26. C₂×C₅×C₁₁	AGL(1,𝔽₁₁)	semidirect
34. C₅×C₁₁	C₂×AGL(1,𝔽₁₁)	semidirect
43. C₂×C₁₁	C₅⋊AGL(1,𝔽₁₁)	semidirect
49. C₁₁	C₂×C₅⋊AGL(1,𝔽₁₁)	semidirect
53. C₂×C₅	#11925893	semidirect
57. C₅	C₂×#11925893	semidirect
60. C₂	C₅⋊#11925893	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×#10597245	2
C₅⋊#11925893	2
C₂×C₅⋊(C₁₁⋊D₁₁)	5
C₂×#11925893	5
C₂×C₅⋊AGL(1,𝔽₁₁)	11

Subgroups — C2×C5⋊#11925893 #27130947

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅⋊#11925893 #27130947