Subgroups — #10598739

Order 640.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#65356	C₂	1	1	yes	maximal, characteristic
2	#64853	C₂	1	2	yes	maximal
3	C₅×#125	C₄	1	1	yes	characteristic, complemented
4	#4020	C₂²	1	1	yes	characteristic
5	#3999	—	2	2	no	—
6	#3941	—	2	2	no	—
7	#3935	C₄	1	1	yes	characteristic, complemented
8	#1412	—	5	1	no	maximal, Sylow
9	C₂×D₂₀	—	4	1	no	—
10	C₂×D₂₀	D₄	1	2	yes	—
11	C₂²⋊AGL(1,𝔽₅)	—	4	2	no	—
12	C₅×C₄⋊C₄	—	2	1	no	—
13	C₅×C₄⋊C₄	—	4	1	no	—
14	C₄²×C₅	C₂×C₄	1	1	yes	characteristic
15	C₂×#168	—	2	1	no	—
16	#490	—	5	1	no	—
17	#353	—	5	2	no	—
18	C₂²×D₅	—	2	2	no	—
19	C₂×AGL(1,𝔽₅)	—	8	2	no	—
20	C₂×C₄×C₅	—	2	1	no	—
21	C₂×C₄×C₅	—	4	1	no	—
22	C₂×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
23	D₂₀	—	4	2	no	—
24	D₂₀	—	8	2	no	—
25	#168	—	4	1	no	—
26	#127	—	5	1	no	—
27	#125	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
28	#102	—	10	2	no	—
29	#99	—	10	2	no	—
30	#97	—	5	1	no	—
31	C₂²×C₅	#99	1	1	yes	characteristic
32	C₂×D₅	—	4	2	no	—
33	C₂×D₅	—	8	2	no	—
34	AGL(1,𝔽₅)	—	16	2	no	—
35	C₄×C₅	—	2	1	no	—
36	C₄×C₅	—	4	1	no	—
37	C₄×C₅	—	8	1	no	—
38	C₂×D₄	—	5	2	no	—
39	C₂×D₄	—	20	1	no	—
40	C₂×C₈	—	10	1	no	—
41	C₄⋊C₄	—	2	1	no	—
42	C₄⋊C₄	—	4	1	no	—
43	C₂²⋊C₄	—	20	2	no	—
44	C₄²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
45	C₂×C₅	—	2	1	no	—
46	C₂×C₅	#354	1	1	yes	characteristic
47	D₅	—	8	2	no	—
48	C₂³	—	10	2	no	—
49	D₄	—	20	2	no	—
50	D₄	—	40	2	no	—
51	C₂×C₄	—	2	1	no	—
52	C₂×C₄	—	4	1	no	—
53	C₂×C₄	—	40	2	no	—
54	C₂×C₄	C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
55	C₈	—	20	1	no	—
56	C₅	#1412	1	1	yes	characteristic, complemented, Sylow
57	C₂²	—	20	2	no	—
58	C₂²	—	40	2	no	—
59	C₂²	#4011	1	1	yes	characteristic
60	C₄	—	2	1	no	—
61	C₄	—	4	1	no	—
62	C₄	—	8	1	no	—
63	C₄	—	80	2	no	—
64	C₂	—	2	1	no	—
65	C₂	—	40	2	no	—
66	C₂	#64928	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#64928
commutator	C₄²×C₅	C₂×C₄
Frattini	C₄²	C₂×AGL(1,𝔽₅)
Fitting	C₅×#125	C₄
socle	C₂×C₅	#354
prime radical, cosocle	#4020	C₂²
hypercenter	#125	AGL(1,𝔽₅)
hypocenter	C₅	#1412
LCS	G ▷ C₄²×C₅ ▷ C₂×C₄×C₅ ▷ C₂²×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ C₄² ◁ #125

Complements

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

N	H ≅ G/N	type
3. C₅×#125	C₄	semidirect
7. #3935	C₄	semidirect
27. #125	AGL(1,𝔽₅)	semidirect
56. C₅	#1412	semidirect

Maximal subgroups

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

subgroup	index
#65356	2
#64853	2
#1412	5