Subgroups — #65800

Order 320.

506 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	C₂×C₂²⋊AGL(1,𝔽₅)	C₂	1	6	yes	maximal, complemented
3	C₂⁴×C₅	C₄	1	1	yes	characteristic, complemented
4	C₂³×D₅	—	2	12	no	—
5	C₂³×D₅	C₄	1	3	yes	complemented
6	C₂³×D₅	C₂²	1	3	yes	—
7	C₂²×AGL(1,𝔽₅)	C₂²	1	4	yes	complemented
8	C₂²⋊AGL(1,𝔽₅)	—	2	12	no	—
9	#379	—	5	1	no	maximal, Sylow
10	C₂³×C₅	—	2	6	no	—
11	C₂³×C₅	C₂×C₄	1	3	yes	—
12	C₂²×D₅	—	2	32	no	—
13	C₂²×D₅	—	2	6	no	—
14	C₂²×D₅	—	2	12	no	—
15	C₂²×D₅	—	2	12	no	—
16	C₂²×D₅	C₂×C₄	1	3	yes	—
17	C₂²×D₅	D₄	1	12	yes	—
18	C₂²×D₅	C₂³	1	1	yes	characteristic
19	C₂×AGL(1,𝔽₅)	—	2	12	no	—
20	C₂⁵	—	5	1	no	—
21	C₂×C₂²⋊C₄	—	5	6	no	—
22	C₂²×C₅	—	2	8	no	—
23	C₂²×C₅	—	2	6	no	—
24	C₂²×C₅	C₂²⋊C₄	1	6	yes	complemented
25	C₂²×C₅	C₂²×C₄	1	1	yes	characteristic
26	C₂×D₅	—	2	24	no	—
27	C₂×D₅	—	2	6	no	—
28	C₂×D₅	—	2	24	no	—
29	C₂×D₅	C₂²⋊C₄	1	6	yes	complemented
30	C₂×D₅	C₂×D₄	1	6	yes	—
31	AGL(1,𝔽₅)	—	4	4	no	—
32	C₂⁴	—	5	3	no	—
33	C₂⁴	—	5	3	no	—
34	C₂⁴	—	10	12	no	—
35	C₂⁴	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
36	C₂²×C₄	—	5	4	no	—
37	C₂²⋊C₄	—	10	12	no	—
38	C₂×C₅	—	2	6	no	—
39	C₂×C₅	C₂×C₂²⋊C₄	1	3	yes	—
40	D₅	—	2	6	no	—
41	D₅	#120	1	4	yes	—
42	C₂³	—	2	6	no	—
43	C₂³	—	5	1	no	—
44	C₂³	—	5	12	no	—
45	C₂³	—	5	3	no	—
46	C₂³	—	10	12	no	—
47	C₂³	—	10	12	no	—
48	C₂³	—	10	32	no	—
49	C₂³	—	10	6	no	—
50	C₂³	C₂×AGL(1,𝔽₅)	1	3	yes	—
51	C₂×C₄	—	10	12	no	—
52	C₅	#379	1	1	yes	characteristic, complemented, Sylow
53	C₂²	—	2	8	no	—
54	C₂²	—	2	6	no	—
55	C₂²	—	5	6	no	—
56	C₂²	—	5	6	no	—
57	C₂²	—	10	24	no	—
58	C₂²	—	10	24	no	—
59	C₂²	—	10	6	no	—
60	C₂²	C₂²⋊AGL(1,𝔽₅)	1	6	yes	complemented
61	C₂²	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
62	C₄	—	20	4	no	—
63	C₂	—	2	6	no	—
64	C₂	—	5	4	no	—
65	C₂	—	10	6	no	—
66	C₂	C₂×C₂²⋊AGL(1,𝔽₅)	1	3	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂²×AGL(1,𝔽₅)
commutator, socle	C₂²×C₅	C₂²×C₄
Fitting	C₂⁴×C₅	C₄
prime radical	C₂⁴×D₅	C₂
cosocle	C₂²×D₅	C₂³
hypercenter	C₂⁴	AGL(1,𝔽₅)
hypocenter	C₅	#379
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂⁴

Complements

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

N	H ≅ G/N	type
2. C₂×C₂²⋊AGL(1,𝔽₅)	C₂	semidirect
3. C₂⁴×C₅	C₄	semidirect
5. C₂³×D₅	C₄	semidirect
7. C₂²×AGL(1,𝔽₅)	C₂²	semidirect
24. C₂²×C₅	C₂²⋊C₄	semidirect
29. C₂×D₅	C₂²⋊C₄	semidirect
35. C₂⁴	AGL(1,𝔽₅)	semidirect
52. C₅	#379	semidirect
60. C₂²	C₂²⋊AGL(1,𝔽₅)	semidirect

Maximal subgroups

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

subgroup	index
C₂⁴×D₅	2
C₂×C₂²⋊AGL(1,𝔽₅)	2
#379	5