Subgroups — #10598743

Order 640.

78 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#65358	C₂	1	1	yes	maximal, characteristic
2	#64855	C₂	1	2	yes	maximal
3	C₅×#125	C₄	1	1	yes	characteristic
4	#4015	C₂²	1	1	yes	characteristic
5	#4005	—	2	2	no	—
6	#3942	—	2	2	no	—
7	#3935	C₄	1	1	yes	characteristic
8	#1413	—	5	1	no	maximal, Sylow
9	C₂×Dic₁₀	D₄	1	2	yes	—
10	#704	—	4	2	no	—
11	C₅×C₄⋊C₄	—	2	1	no	—
12	C₅×C₄⋊C₄	—	4	1	no	—
13	C₄²×C₅	C₂×C₄	1	1	yes	characteristic
14	C₄⋊Dic₅	—	4	1	no	—
15	C₂×#168	—	2	1	no	—
16	#491	—	5	1	no	—
17	#356	—	5	2	no	—
18	C₂×C₄×C₅	—	2	1	no	—
19	C₂×C₄×C₅	—	4	1	no	—
20	C₂×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
21	C₂×Dic₅	—	2	2	no	—
22	Dic₁₀	—	4	2	no	—
23	#170	—	8	2	no	—
24	#168	—	4	1	no	—
25	C₄⋊Q₈	—	5	1	no	—
26	#125	AGL(1,𝔽₅)	1	1	yes	characteristic
27	#103	—	10	2	no	—
28	#101	—	10	2	no	—
29	#97	—	5	1	no	—
30	C₂²×C₅	#99	1	1	yes	characteristic
31	C₄×C₅	—	2	1	no	—
32	C₄×C₅	—	4	1	no	—
33	C₄×C₅	—	8	1	no	—
34	Dic₅	—	4	2	no	—
35	C₂×Q₈	—	5	2	no	—
36	M₁₆	—	20	2	no	—
37	C₂×C₈	—	10	1	no	—
38	C₄⋊C₄	—	2	1	no	—
39	C₄⋊C₄	—	4	1	no	—
40	C₄⋊C₄	—	20	1	no	—
41	C₄²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
42	C₂×C₅	—	2	1	no	—
43	C₂×C₅	#354	1	1	yes	characteristic
44	Q₈	—	20	2	no	—
45	C₂×C₄	—	2	1	no	—
46	C₂×C₄	—	4	1	no	—
47	C₂×C₄	—	10	2	no	—
48	C₂×C₄	C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
49	C₈	—	20	1	no	—
50	C₈	—	40	2	no	—
51	C₅	#1413	1	1	yes	characteristic, complemented, Sylow
52	C₂²	#4011	1	1	yes	characteristic
53	C₄	—	2	1	no	—
54	C₄	—	4	1	no	—
55	C₄	—	8	1	no	—
56	C₄	—	20	2	no	—
57	C₂	—	2	1	no	—
58	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	C₂²×C₅	#99
cosocle	#4015	C₂²
hypercenter	#125	AGL(1,𝔽₅)
hypocenter	C₅	#1413
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
51. C₅	#1413	semidirect

Maximal subgroups

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

subgroup	index
#65358	2
#64855	2
#1413	5