Subgroups — C₂×C₄×C₁₁×AGL(1,𝔽₅) #13183334

Order 1760.

216 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₁×AGL(1,𝔽₅)	C₂	1	2	yes	maximal
2	C₂×C₄×D₅×C₁₁	C₂	1	1	yes	maximal, characteristic
3	C₄×C₁₁×AGL(1,𝔽₅)	C₂	1	4	yes	maximal, direct product factor
4	C₂²×D₅×C₁₁	C₂²	1	1	yes	characteristic
5	C₂×C₁₁×AGL(1,𝔽₅)	C₄	1	8	yes	direct product factor
6	C₂×C₁₁×AGL(1,𝔽₅)	C₂²	1	4	yes	—
7	C₂×C₄×C₅×C₁₁	C₄	1	1	yes	characteristic, complemented
8	C₂×C₁₁×Dic₅	C₄	1	1	yes	characteristic, complemented
9	C₄×D₅×C₁₁	C₄	1	2	yes	complemented
10	C₄×D₅×C₁₁	C₂²	1	2	yes	—
11	C₂×C₄²×C₁₁	—	5	1	no	maximal, Hall
12	C₂²×C₅×C₁₁	C₂×C₄	1	1	yes	characteristic
13	C₂×D₅×C₁₁	C₂×C₄	1	1	yes	characteristic
14	C₂×D₅×C₁₁	C₂×C₄	1	4	yes	—
15	C₂×D₅×C₁₁	C₂³	1	1	yes	characteristic
16	C₁₁×AGL(1,𝔽₅)	C₂×C₄	1	8	yes	direct product factor
17	C₄×C₅×C₁₁	C₂×C₄	1	2	yes	complemented
18	C₁₁×Dic₅	C₂×C₄	1	2	yes	complemented
19	C₂²×C₄×C₁₁	—	5	2	no	—
20	C₂²×C₄×C₁₁	—	5	1	no	—
21	C₄²×C₁₁	—	5	4	no	—
22	C₂×C₄×AGL(1,𝔽₅)	C₁₁	1	1	yes	maximal, characteristic, direct product factor, Hall
23	C₂×C₅×C₁₁	C₄²	1	2	yes	complemented
24	C₂×C₅×C₁₁	C₂²×C₄	1	1	yes	characteristic
25	D₅×C₁₁	C₄²	1	2	yes	complemented
26	D₅×C₁₁	C₂²×C₄	1	2	yes	—
27	C₂³×C₁₁	—	5	1	no	—
28	C₂×C₄×C₁₁	—	5	8	no	—
29	C₂×C₄×C₁₁	—	5	4	no	—
30	C₂×C₄×C₁₁	—	5	1	no	—
31	C₂×C₄×C₁₁	—	5	2	no	—
32	C₂×C₄×C₁₁	—	5	2	no	—
33	C₂×C₄×C₁₁	AGL(1,𝔽₅)	1	1	yes	characteristic, direct product factor
34	C₂²×AGL(1,𝔽₅)	C₂×C₁₁	1	2	yes	—
35	C₂×C₄×D₅	C₂×C₁₁	1	1	yes	characteristic
36	C₄×AGL(1,𝔽₅)	C₂×C₁₁	1	4	yes	direct product factor
37	C₅×C₁₁	C₂×C₄²	1	1	yes	characteristic, complemented, Hall
38	C₂²×C₁₁	—	5	4	no	—
39	C₂²×C₁₁	—	5	1	no	—
40	C₂²×C₁₁	—	5	1	no	—
41	C₂²×C₁₁	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
42	C₄×C₁₁	—	5	8	no	—
43	C₄×C₁₁	—	5	2	no	—
44	C₄×C₁₁	C₂×AGL(1,𝔽₅)	1	2	yes	direct product factor
45	C₂²×D₅	C₂²×C₁₁	1	1	yes	characteristic
46	C₂×AGL(1,𝔽₅)	C₄×C₁₁	1	8	yes	direct product factor
47	C₂×AGL(1,𝔽₅)	C₂²×C₁₁	1	4	yes	—
48	C₂×C₄×C₅	C₄×C₁₁	1	1	yes	characteristic, complemented
49	C₂×Dic₅	C₄×C₁₁	1	1	yes	characteristic, complemented
50	C₄×D₅	C₄×C₁₁	1	2	yes	complemented
51	C₄×D₅	C₂²×C₁₁	1	2	yes	—
52	C₂×C₄²	—	5	1	no	Sylow
53	C₂×C₁₁	—	5	2	no	—
54	C₂×C₁₁	—	5	2	no	—
55	C₂×C₁₁	C₄×AGL(1,𝔽₅)	1	2	yes	direct product factor
56	C₂×C₁₁	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
57	C₂²×C₅	C₂×C₄×C₁₁	1	1	yes	characteristic
58	C₂×D₅	C₂×C₄×C₁₁	1	1	yes	characteristic
59	C₂×D₅	C₂×C₄×C₁₁	1	4	yes	—
60	C₂×D₅	C₂³×C₁₁	1	1	yes	characteristic
61	AGL(1,𝔽₅)	C₂×C₄×C₁₁	1	8	yes	direct product factor
62	C₄×C₅	C₂×C₄×C₁₁	1	2	yes	complemented
63	Dic₅	C₂×C₄×C₁₁	1	2	yes	complemented
64	C₂²×C₄	—	5	2	no	—
65	C₂²×C₄	—	5	1	no	—
66	C₄²	—	5	4	no	—
67	C₁₁	C₂×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic, direct product factor, Sylow
68	C₂×C₅	C₄²×C₁₁	1	2	yes	complemented
69	C₂×C₅	C₂²×C₄×C₁₁	1	1	yes	characteristic
70	D₅	C₄²×C₁₁	1	2	yes	complemented
71	D₅	C₂²×C₄×C₁₁	1	2	yes	—
72	C₂³	—	5	1	no	—
73	C₂×C₄	—	5	4	no	—
74	C₂×C₄	—	5	8	no	—
75	C₂×C₄	—	5	1	no	—
76	C₂×C₄	—	5	2	no	—
77	C₂×C₄	—	5	2	no	—
78	C₂×C₄	C₁₁×AGL(1,𝔽₅)	1	1	yes	characteristic, direct product factor
79	C₅	C₂×C₄²×C₁₁	1	1	yes	characteristic, complemented, Sylow
80	C₂²	—	5	4	no	—
81	C₂²	—	5	1	no	—
82	C₂²	—	5	1	no	—
83	C₂²	C₂×C₁₁×AGL(1,𝔽₅)	1	1	yes	characteristic
84	C₄	—	5	8	no	—
85	C₄	—	5	2	no	—
86	C₄	C₂×C₁₁×AGL(1,𝔽₅)	1	2	yes	direct product factor
87	C₂	—	5	2	no	—
88	C₂	—	5	2	no	—
89	C₂	C₄×C₁₁×AGL(1,𝔽₅)	1	2	yes	direct product factor
90	C₂	C₂²×C₁₁×AGL(1,𝔽₅)	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂×C₄×C₁₁	AGL(1,𝔽₅)
commutator, hypocenter	C₅	C₂×C₄²×C₁₁
Frattini	C₂	C₂²×C₁₁×AGL(1,𝔽₅)
Fitting	C₂×C₄×C₅×C₁₁	C₄
socle	C₂²×C₅×C₁₁	C₂×C₄
prime radical	C₂²×D₅×C₁₁	C₂²
cosocle	C₂×D₅	C₂³×C₁₁
LCS	G ▷ C₅
UCS	C₁ ◁ C₂×C₄×C₁₁

Complements

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

N	H ≅ G/N	type
3. C₄×C₁₁×AGL(1,𝔽₅)	C₂	direct (N × H)
5. C₂×C₁₁×AGL(1,𝔽₅)	C₄	direct (N × H)
7. C₂×C₄×C₅×C₁₁	C₄	semidirect
8. C₂×C₁₁×Dic₅	C₄	semidirect
9. C₄×D₅×C₁₁	C₄	semidirect
16. C₁₁×AGL(1,𝔽₅)	C₂×C₄	direct (N × H)
17. C₄×C₅×C₁₁	C₂×C₄	semidirect
18. C₁₁×Dic₅	C₂×C₄	semidirect
22. C₂×C₄×AGL(1,𝔽₅)	C₁₁	direct (N × H)
23. C₂×C₅×C₁₁	C₄²	semidirect
25. D₅×C₁₁	C₄²	semidirect
33. C₂×C₄×C₁₁	AGL(1,𝔽₅)	direct (N × H)
36. C₄×AGL(1,𝔽₅)	C₂×C₁₁	direct (N × H)
37. C₅×C₁₁	C₂×C₄²	semidirect
44. C₄×C₁₁	C₂×AGL(1,𝔽₅)	direct (N × H)
46. C₂×AGL(1,𝔽₅)	C₄×C₁₁	direct (N × H)
48. C₂×C₄×C₅	C₄×C₁₁	semidirect
49. C₂×Dic₅	C₄×C₁₁	semidirect
50. C₄×D₅	C₄×C₁₁	semidirect
55. C₂×C₁₁	C₄×AGL(1,𝔽₅)	direct (N × H)
61. AGL(1,𝔽₅)	C₂×C₄×C₁₁	direct (N × H)
62. C₄×C₅	C₂×C₄×C₁₁	semidirect
63. Dic₅	C₂×C₄×C₁₁	semidirect
67. C₁₁	C₂×C₄×AGL(1,𝔽₅)	direct (N × H)
68. C₂×C₅	C₄²×C₁₁	semidirect
70. D₅	C₄²×C₁₁	semidirect
78. C₂×C₄	C₁₁×AGL(1,𝔽₅)	direct (N × H)
79. C₅	C₂×C₄²×C₁₁	semidirect
86. C₄	C₂×C₁₁×AGL(1,𝔽₅)	direct (N × H)
89. C₂	C₄×C₁₁×AGL(1,𝔽₅)	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₁₁×AGL(1,𝔽₅)	2
C₂×C₄×D₅×C₁₁	2
C₄×C₁₁×AGL(1,𝔽₅)	2
C₂×C₄²×C₁₁	5
C₂×C₄×AGL(1,𝔽₅)	11

Subgroups — C2×C4×C11×AGL(1,𝔽5) #13183334

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×C₁₁×AGL(1,𝔽₅) #13183334