Subgroups — C₂²×C₂²⋊AGL(1,𝔽₅) #66269

Order 320.

674 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×D₅	C₂	1	1	yes	maximal, characteristic
2	C₂³×AGL(1,𝔽₅)	C₂	1	2	yes	maximal, complemented
3	C₂×C₂²⋊AGL(1,𝔽₅)	C₂	1	12	yes	maximal, direct product factor
4	C₂⁴×C₅	C₄	1	1	yes	characteristic, complemented
5	C₂³×D₅	—	2	8	no	—
6	C₂³×D₅	C₄	1	6	yes	complemented
7	C₂³×D₅	C₄	1	1	yes	characteristic, complemented
8	C₂³×D₅	C₂²	1	6	yes	—
9	C₂³×D₅	C₂²	1	1	yes	characteristic
10	C₂²×AGL(1,𝔽₅)	—	2	8	no	—
11	C₂²×AGL(1,𝔽₅)	C₂²	1	12	yes	complemented
12	C₂²⋊AGL(1,𝔽₅)	C₂²	1	16	yes	direct product factor
13	C₂²×C₂²⋊C₄	—	5	1	no	maximal, Sylow
14	C₂³×C₅	—	2	4	no	—
15	C₂³×C₅	C₂×C₄	1	1	yes	characteristic
16	C₂³×C₅	C₂×C₄	1	6	yes	complemented
17	C₂²×D₅	—	2	4	no	—
18	C₂²×D₅	—	2	24	no	—
19	C₂²×D₅	—	2	24	no	—
20	C₂²×D₅	C₂×C₄	1	3	yes	—
21	C₂²×D₅	C₂×C₄	1	12	yes	complemented
22	C₂²×D₅	C₂×C₄	1	6	yes	complemented
23	C₂²×D₅	D₄	1	8	yes	—
24	C₂²×D₅	C₂³	1	4	yes	—
25	C₂²×D₅	C₂³	1	3	yes	—
26	C₂×AGL(1,𝔽₅)	—	2	24	no	—
27	C₂×AGL(1,𝔽₅)	C₂³	1	8	yes	complemented
28	C₂⁵	—	5	1	no	—
29	C₂³×C₄	—	5	2	no	—
30	C₂×C₂²⋊C₄	—	5	12	no	—
31	C₂²×C₅	—	2	12	no	—
32	C₂²×C₅	C₂²⋊C₄	1	4	yes	complemented
33	C₂²×C₅	C₂²×C₄	1	3	yes	—
34	C₂²×C₅	C₂²×C₄	1	4	yes	complemented
35	C₂×D₅	—	2	12	no	—
36	C₂×D₅	—	2	24	no	—
37	C₂×D₅	—	2	8	no	—
38	C₂×D₅	C₂²⋊C₄	1	12	yes	complemented
39	C₂×D₅	C₂²×C₄	1	3	yes	—
40	C₂×D₅	C₂²×C₄	1	4	yes	complemented
41	C₂×D₅	C₂×D₄	1	12	yes	—
42	C₂×D₅	C₂⁴	1	1	yes	characteristic
43	AGL(1,𝔽₅)	—	2	8	no	—
44	C₂⁴	—	5	1	no	—
45	C₂⁴	—	5	1	no	—
46	C₂⁴	—	5	6	no	—
47	C₂⁴	—	5	6	no	—
48	C₂⁴	—	10	8	no	—
49	C₂⁴	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
50	C₂²×C₄	—	5	12	no	—
51	C₂²×C₄	—	10	8	no	—
52	C₂²⋊C₄	—	5	16	no	—
53	C₂×C₅	—	2	4	no	—
54	C₂×C₅	C₂×C₂²⋊C₄	1	6	yes	complemented
55	C₂×C₅	C₂³×C₄	1	1	yes	characteristic
56	D₅	—	2	4	no	—
57	D₅	C₂×C₂²⋊C₄	1	6	yes	complemented
58	D₅	C₂²×D₄	1	2	yes	—
59	C₂³	—	2	4	no	—
60	C₂³	—	5	8	no	—
61	C₂³	—	5	3	no	—
62	C₂³	—	5	3	no	—
63	C₂³	—	5	6	no	—
64	C₂³	—	5	4	no	—
65	C₂³	—	5	12	no	—
66	C₂³	—	10	24	no	—
67	C₂³	—	10	24	no	—
68	C₂³	—	10	4	no	—
69	C₂³	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
70	C₂³	C₂×AGL(1,𝔽₅)	1	6	yes	complemented
71	C₂×C₄	—	5	8	no	—
72	C₂×C₄	—	10	24	no	—
73	C₅	C₂²×C₂²⋊C₄	1	1	yes	characteristic, complemented, Sylow
74	C₂²	—	2	12	no	—
75	C₂²	—	5	12	no	—
76	C₂²	—	5	12	no	—
77	C₂²	—	5	1	no	—
78	C₂²	—	5	3	no	—
79	C₂²	—	5	4	no	—
80	C₂²	—	10	8	no	—
81	C₂²	—	10	24	no	—
82	C₂²	—	10	12	no	—
83	C₂²	C₂²⋊AGL(1,𝔽₅)	1	4	yes	direct product factor
84	C₂²	C₂²×AGL(1,𝔽₅)	1	3	yes	—
85	C₂²	C₂²×AGL(1,𝔽₅)	1	4	yes	complemented
86	C₄	—	10	8	no	—
87	C₂	—	2	4	no	—
88	C₂	—	5	2	no	—
89	C₂	—	5	6	no	—
90	C₂	—	10	4	no	—
91	C₂	C₂×C₂²⋊AGL(1,𝔽₅)	1	6	yes	direct product factor
92	C₂	C₂³×AGL(1,𝔽₅)	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×AGL(1,𝔽₅)
commutator	C₂×C₅	C₂³×C₄
Frattini	C₂	C₂³×AGL(1,𝔽₅)
Fitting	C₂⁴×C₅	C₄
socle	C₂³×C₅	C₂×C₄
prime radical	C₂⁴×D₅	C₂
cosocle	C₂×D₅	C₂⁴
hypercenter	C₂⁴	AGL(1,𝔽₅)
hypocenter	C₅	C₂²×C₂²⋊C₄
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₂⁴

Complements

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

N	H ≅ G/N	type
2. C₂³×AGL(1,𝔽₅)	C₂	semidirect
3. C₂×C₂²⋊AGL(1,𝔽₅)	C₂	direct (N × H)
4. C₂⁴×C₅	C₄	semidirect
6. C₂³×D₅	C₄	semidirect
7. C₂³×D₅	C₄	semidirect
11. C₂²×AGL(1,𝔽₅)	C₂²	semidirect
12. C₂²⋊AGL(1,𝔽₅)	C₂²	direct (N × H)
16. C₂³×C₅	C₂×C₄	semidirect
21. C₂²×D₅	C₂×C₄	semidirect
22. C₂²×D₅	C₂×C₄	semidirect
27. C₂×AGL(1,𝔽₅)	C₂³	semidirect
32. C₂²×C₅	C₂²⋊C₄	semidirect
34. C₂²×C₅	C₂²×C₄	semidirect
38. C₂×D₅	C₂²⋊C₄	semidirect
40. C₂×D₅	C₂²×C₄	semidirect
49. C₂⁴	AGL(1,𝔽₅)	semidirect
54. C₂×C₅	C₂×C₂²⋊C₄	semidirect
57. D₅	C₂×C₂²⋊C₄	semidirect
70. C₂³	C₂×AGL(1,𝔽₅)	semidirect
73. C₅	C₂²×C₂²⋊C₄	semidirect
83. C₂²	C₂²⋊AGL(1,𝔽₅)	direct (N × H)
85. C₂²	C₂²×AGL(1,𝔽₅)	semidirect
91. C₂	C₂×C₂²⋊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
C₂⁴×D₅	2
C₂³×AGL(1,𝔽₅)	2
C₂×C₂²⋊AGL(1,𝔽₅)	2
C₂²×C₂²⋊C₄	5

Subgroups — C22×C22⋊AGL(1,𝔽5) #66269

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₂²⋊AGL(1,𝔽₅) #66269