Subgroups — D₁₁₁⋊(C₅⋊D₅) #26012918

Order 11100.

160 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₇×C₃⋊(C₅⋊D₅)	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅²×D₁₁₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₅⋊D₁₈₅	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅⋊D₅×D₃₇	—	3	1	no	maximal, Hall
5	C₃×C₅²×C₃₇	C₂²	1	1	yes	characteristic, complemented, Hall
6	D₁₅⋊D₃₇	—	5	6	no	maximal
7	C₅⋊D₁₈₅	D₃	1	1	yes	characteristic, complemented
8	C₃₇×C₅⋊D₅	—	3	1	no	—
9	C₅²×D₃₇	—	3	1	no	—
10	D₁₅×C₃₇	—	5	6	no	—
11	C₅×D₁₁₁	D₅	1	6	yes	complemented
12	C₃×D₁₈₅	—	5	6	no	—
13	C₅²×C₃₇	C₂×D₃	1	1	yes	characteristic, complemented, Hall
14	D₅×D₃₇	—	15	6	no	—
15	C₃×C₅×C₃₇	C₂×D₅	1	6	yes	complemented
16	D₃×D₃₇	—	25	1	no	Hall
17	D₁₈₅	—	5	6	no	—
18	C₅×D₃₇	—	3	6	no	—
19	D₅×C₃₇	—	15	6	no	—
20	D₃×C₅⋊D₅	—	37	1	no	maximal, Hall
21	D₁₁₁	C₅⋊D₅	1	1	yes	characteristic, complemented
22	C₃×D₃₇	—	25	1	no	—
23	D₃×C₃₇	—	25	1	no	—
24	C₅×C₃₇	D₃×D₅	1	6	yes	complemented
25	C₃⋊(C₅⋊D₅)	D₃₇	1	1	yes	characteristic, complemented
26	C₅²×D₃	—	37	1	no	—
27	C₃×C₅⋊D₅	—	37	1	no	—
28	C₂×D₃₇	—	75	1	no	Hall
29	C₃×C₃₇	C₂×C₅⋊D₅	1	1	yes	characteristic, complemented, Hall
30	C₂×C₅⋊D₅	—	111	1	no	Hall
31	C₃×C₅²	C₂×D₃₇	1	1	yes	characteristic, complemented, Hall
32	C₂×C₃₇	—	75	1	no	—
33	D₃₇	—	3	1	no	—
34	D₃₇	—	25	1	no	—
35	D₃×D₅	—	185	6	no	—
36	C₂×C₅²	—	111	1	no	—
37	C₅⋊D₅	—	3	1	no	—
38	C₅⋊D₅	—	37	1	no	—
39	C₃₇	D₃×C₅⋊D₅	1	1	yes	characteristic, complemented, Sylow
40	D₁₅	—	5	6	no	—
41	C₃×D₅	—	185	6	no	—
42	C₅×D₃	—	37	6	no	—
43	C₅²	D₃×D₃₇	1	1	yes	characteristic, complemented, Sylow
44	C₂×D₅	—	555	6	no	—
45	C₃×C₅	D₅×D₃₇	1	6	yes	complemented
46	C₂×D₃	—	925	1	no	Hall
47	C₂×C₅	—	111	6	no	—
48	D₅	—	15	6	no	—
49	D₅	—	185	6	no	—
50	C₂×C₃	—	925	1	no	—
51	D₃	—	25	1	no	—
52	D₃	—	37	1	no	—
53	C₅	D₁₅⋊D₃₇	1	6	yes	complemented
54	C₂²	—	2775	1	no	Sylow
55	C₃	C₅⋊D₅×D₃₇	1	1	yes	characteristic, complemented, Sylow
56	C₂	—	75	1	no	—
57	C₂	—	111	1	no	—
58	C₂	—	925	1	no	—

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃×C₅²×C₃₇	C₂²
LCS	G ▷ C₃×C₅²×C₃₇
UCS	C₁

Complements

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

N	H ≅ G/N	type
1. C₃₇×C₃⋊(C₅⋊D₅)	C₂	semidirect
2. C₅²×D₁₁₁	C₂	semidirect
3. C₃×C₅⋊D₁₈₅	C₂	semidirect
5. C₃×C₅²×C₃₇	C₂²	semidirect
7. C₅⋊D₁₈₅	D₃	semidirect
11. C₅×D₁₁₁	D₅	semidirect
13. C₅²×C₃₇	C₂×D₃	semidirect
15. C₃×C₅×C₃₇	C₂×D₅	semidirect
21. D₁₁₁	C₅⋊D₅	semidirect
24. C₅×C₃₇	D₃×D₅	semidirect
25. C₃⋊(C₅⋊D₅)	D₃₇	semidirect
29. C₃×C₃₇	C₂×C₅⋊D₅	semidirect
31. C₃×C₅²	C₂×D₃₇	semidirect
39. C₃₇	D₃×C₅⋊D₅	semidirect
43. C₅²	D₃×D₃₇	semidirect
45. C₃×C₅	D₅×D₃₇	semidirect
53. C₅	D₁₅⋊D₃₇	semidirect
55. C₃	C₅⋊D₅×D₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₃₇×C₃⋊(C₅⋊D₅)	2
C₅²×D₁₁₁	2
C₃×C₅⋊D₁₈₅	2
C₅⋊D₅×D₃₇	3
D₁₅⋊D₃₇	5
D₃×C₅⋊D₅	37

Subgroups — D111⋊(C5⋊D5) #26012918

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₁₁₁⋊(C₅⋊D₅) #26012918