Subgroups — C₃×C₅×C₁₁⋊Dic₅₅ #49137320

Order 36300.

448 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅²×C₁₁²	C₂	1	1	yes	maximal, characteristic
2	C₅×C₁₁⋊Dic₅₅	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
3	C₃×C₅²×C₁₁²	C₄	1	1	yes	characteristic, complemented, Hall
4	C₃×C₁₁⋊Dic₅₅	C₅	1	1	yes	maximal, characteristic, direct product factor
5	C₃×C₅×C₁₁⋊Dic₁₁	—	5	1	no	maximal
6	C₂×C₅²×C₁₁²	C₂×C₃	1	1	yes	characteristic
7	C₂×C₃×C₅×C₁₁²	—	2	2	no	—
8	C₂×C₃×C₅×C₁₁²	D₅	1	1	yes	characteristic
9	C₂×C₃×C₅×C₁₁²	C₂×C₅	1	1	yes	characteristic
10	C₃×C₅×Dic₅₅	—	11	12	no	maximal
11	C₅²×C₁₁²	C₃×C₄	1	1	yes	characteristic, complemented, Hall
12	C₁₁⋊Dic₅₅	C₃×C₅	1	1	yes	characteristic, direct product factor
13	C₅×C₁₁⋊Dic₁₁	—	5	1	no	—
14	C₃×C₅×C₁₁²	—	2	2	no	—
15	C₃×C₅×C₁₁²	Dic₅	1	1	yes	characteristic, complemented
16	C₃×C₅×C₁₁²	C₄×C₅	1	1	yes	characteristic, complemented
17	C₂×C₃×C₅²×C₁₁	D₁₁	1	12	yes	—
18	C₃×C₁₁⋊Dic₁₁	—	5	1	no	Hall
19	C₂×C₅×C₁₁²	—	2	2	no	—
20	C₂×C₅×C₁₁²	C₃×D₅	1	1	yes	characteristic
21	C₂×C₅×C₁₁²	C₂×C₃×C₅	1	1	yes	characteristic
22	C₅×Dic₅₅	—	11	12	no	—
23	C₃×C₅²×C₁₁	Dic₁₁	1	12	yes	complemented
24	C₂×C₃×C₁₁²	C₅×D₅	1	1	yes	characteristic
25	C₃×Dic₅₅	—	11	12	no	—
26	C₃×C₅×Dic₁₁	—	55	12	no	—
27	C₅×C₁₁²	—	2	2	no	—
28	C₅×C₁₁²	C₃×Dic₅	1	1	yes	characteristic, complemented
29	C₅×C₁₁²	C₃×C₄×C₅	1	1	yes	characteristic, complemented
30	C₂×C₅²×C₁₁	C₃×D₁₁	1	12	yes	—
31	C₁₁⋊Dic₁₁	—	5	1	no	Hall
32	C₃×C₁₁²	C₅×Dic₅	1	1	yes	characteristic, complemented, Hall
33	C₂×C₃×C₅×C₁₁	—	2	24	no	—
34	C₂×C₃×C₅×C₁₁	C₅×D₁₁	1	12	yes	—
35	C₂×C₃×C₅×C₁₁	D₅₅	1	12	yes	—
36	C₃×C₅×Dic₅	—	121	1	no	Hall
37	C₅²×C₁₁	C₃×Dic₁₁	1	12	yes	complemented
38	C₂×C₁₁²	C₃×C₅×D₅	1	1	yes	characteristic
39	Dic₅₅	—	11	12	no	—
40	C₅×Dic₁₁	—	55	12	no	—
41	C₃×C₅×C₁₁	—	2	24	no	—
42	C₃×C₅×C₁₁	C₅×Dic₁₁	1	12	yes	complemented
43	C₃×C₅×C₁₁	Dic₅₅	1	12	yes	complemented
44	C₂×C₃×C₅²	C₁₁⋊D₁₁	1	1	yes	characteristic
45	C₃×Dic₁₁	—	55	12	no	—
46	C₁₁²	C₃×C₅×Dic₅	1	1	yes	characteristic, complemented, Sylow
47	C₂×C₅×C₁₁	—	2	24	no	—
48	C₂×C₅×C₁₁	C₃×C₅×D₁₁	1	12	yes	—
49	C₂×C₅×C₁₁	C₃×D₅₅	1	12	yes	—
50	C₅×Dic₅	—	121	1	no	Hall
51	C₃×C₅²	C₁₁⋊Dic₁₁	1	1	yes	characteristic, complemented, Hall
52	C₂×C₃×C₁₁	C₅×D₅₅	1	12	yes	—
53	C₃×C₄×C₅	—	605	1	no	—
54	C₃×Dic₅	—	121	1	no	—
55	C₅×C₁₁	—	2	24	no	—
56	C₅×C₁₁	C₃×C₅×Dic₁₁	1	12	yes	complemented
57	C₅×C₁₁	C₃×Dic₅₅	1	12	yes	complemented
58	C₂×C₅²	C₃×C₁₁⋊D₁₁	1	1	yes	characteristic
59	Dic₁₁	—	55	12	no	—
60	C₃×C₁₁	C₅×Dic₅₅	1	12	yes	complemented
61	C₂×C₃×C₅	—	2	2	no	—
62	C₂×C₃×C₅	C₅×C₁₁⋊D₁₁	1	1	yes	characteristic
63	C₂×C₃×C₅	C₅⋊(C₁₁⋊D₁₁)	1	1	yes	characteristic
64	C₅²	C₃×C₁₁⋊Dic₁₁	1	1	yes	characteristic, complemented, Sylow
65	C₂×C₁₁	C₃×C₅×D₅₅	1	12	yes	—
66	C₄×C₅	—	605	1	no	—
67	Dic₅	—	121	1	no	—
68	C₃×C₅	—	2	2	no	—
69	C₃×C₅	C₅×C₁₁⋊Dic₁₁	1	1	yes	characteristic, complemented
70	C₃×C₅	C₁₁⋊Dic₅₅	1	1	yes	characteristic, direct product factor
71	C₃×C₄	—	605	1	no	Hall
72	C₁₁	C₃×C₅×Dic₅₅	1	12	yes	complemented
73	C₂×C₅	—	2	2	no	—
74	C₂×C₅	C₃×C₅×C₁₁⋊D₁₁	1	1	yes	characteristic
75	C₂×C₅	C₃×C₅⋊(C₁₁⋊D₁₁)	1	1	yes	characteristic
76	C₂×C₃	C₅×C₅⋊(C₁₁⋊D₁₁)	1	1	yes	characteristic
77	C₅	—	2	2	no	—
78	C₅	C₃×C₅×C₁₁⋊Dic₁₁	1	1	yes	characteristic, complemented
79	C₅	C₃×C₁₁⋊Dic₅₅	1	1	yes	characteristic, direct product factor
80	C₄	—	605	1	no	Sylow
81	C₃	C₅×C₁₁⋊Dic₅₅	1	1	yes	characteristic, direct product factor, Sylow
82	C₂	C₃×C₅×C₅⋊(C₁₁⋊D₁₁)	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂×C₃×C₅	C₅⋊(C₁₁⋊D₁₁)
commutator, hypocenter	C₅×C₁₁²	C₃×C₄×C₅
Frattini	C₂	C₃×C₅×C₅⋊(C₁₁⋊D₁₁)
Fitting, socle, prime radical	C₂×C₃×C₅²×C₁₁²	C₂
cosocle	C₂×C₅×C₁₁²	C₂×C₃×C₅
LCS	G ▷ C₅×C₁₁²
UCS	C₁ ◁ C₂×C₃×C₅

Complements

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

N	H ≅ G/N	type
2. C₅×C₁₁⋊Dic₅₅	C₃	direct (N × H)
3. C₃×C₅²×C₁₁²	C₄	semidirect
4. C₃×C₁₁⋊Dic₅₅	C₅	direct (N × H)
11. C₅²×C₁₁²	C₃×C₄	semidirect
12. C₁₁⋊Dic₅₅	C₃×C₅	direct (N × H)
15. C₃×C₅×C₁₁²	Dic₅	semidirect
16. C₃×C₅×C₁₁²	C₄×C₅	semidirect
23. C₃×C₅²×C₁₁	Dic₁₁	semidirect
28. C₅×C₁₁²	C₃×Dic₅	semidirect
29. C₅×C₁₁²	C₃×C₄×C₅	semidirect
32. C₃×C₁₁²	C₅×Dic₅	semidirect
37. C₅²×C₁₁	C₃×Dic₁₁	semidirect
42. C₃×C₅×C₁₁	C₅×Dic₁₁	semidirect
43. C₃×C₅×C₁₁	Dic₅₅	semidirect
46. C₁₁²	C₃×C₅×Dic₅	semidirect
51. C₃×C₅²	C₁₁⋊Dic₁₁	semidirect
56. C₅×C₁₁	C₃×C₅×Dic₁₁	semidirect
57. C₅×C₁₁	C₃×Dic₅₅	semidirect
60. C₃×C₁₁	C₅×Dic₅₅	semidirect
64. C₅²	C₃×C₁₁⋊Dic₁₁	semidirect
69. C₃×C₅	C₅×C₁₁⋊Dic₁₁	semidirect
70. C₃×C₅	C₁₁⋊Dic₅₅	direct (N × H)
72. C₁₁	C₃×C₅×Dic₅₅	semidirect
78. C₅	C₃×C₅×C₁₁⋊Dic₁₁	semidirect
79. C₅	C₃×C₁₁⋊Dic₅₅	direct (N × H)
81. C₃	C₅×C₁₁⋊Dic₅₅	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×C₃×C₅²×C₁₁²	2
C₅×C₁₁⋊Dic₅₅	3
C₃×C₁₁⋊Dic₅₅	5
C₃×C₅×C₁₁⋊Dic₁₁	5
C₃×C₅×Dic₅₅	11

Subgroups — C3×C5×C11⋊Dic55 #49137320

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×C₁₁⋊Dic₅₅ #49137320