Subgroups — #65353

Order 320.

86 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#125	C₂	1	1	yes	maximal, characteristic
2	#4016	C₂	1	1	yes	maximal, characteristic
3	#3940	C₂	1	2	yes	maximal
4	#3939	C₂	1	2	yes	maximal
5	C₄×#168	C₂	1	1	yes	maximal, characteristic
6	C₅×C₄⋊C₄	—	2	2	no	—
7	C₅×C₄⋊C₄	C₂²	1	2	yes	—
8	C₄²×C₅	C₂²	1	1	yes	characteristic
9	C₄⋊Dic₅	C₂²	1	2	yes	—
10	#683	—	2	2	no	—
11	C₂×#168	C₂²	1	1	yes	characteristic
12	C₂×#168	C₂²	1	1	yes	characteristic
13	#499	—	5	1	no	maximal, Sylow
14	C₂×C₄×C₅	—	2	2	no	—
15	C₂×C₄×C₅	D₄	1	2	yes	—
16	C₂×C₄×C₅	C₂³	1	1	yes	characteristic
17	C₂×Dic₅	—	2	2	no	—
18	#168	Q₈	1	2	yes	—
19	#168	Q₈	1	2	yes	—
20	#125	—	5	1	no	—
21	#125	D₅	1	1	yes	characteristic
22	#107	—	5	2	no	—
23	#106	—	5	2	no	—
24	C₄×C₈	—	5	1	no	—
25	C₂²×C₅	C₂×D₄	1	1	yes	characteristic
26	C₄×C₅	—	2	2	no	—
27	C₄×C₅	—	4	2	no	—
28	C₄×C₅	C₂×Q₈	1	1	yes	characteristic
29	C₄×C₅	C₂×Q₈	1	1	yes	characteristic
30	Dic₅	—	4	2	no	—
31	C₂×C₈	—	5	1	no	—
32	C₂×C₈	—	5	1	no	—
33	C₄⋊C₄	—	2	2	no	—
34	C₄⋊C₄	—	5	2	no	—
35	C₄⋊C₄	—	10	2	no	—
36	C₄⋊C₄	C₂×D₅	1	2	yes	—
37	C₄²	C₂×D₅	1	1	yes	characteristic
38	C₂×C₅	C₄⋊Q₈	1	1	yes	characteristic
39	C₂×C₅	#135	1	2	yes	—
40	C₂×C₄	—	2	2	no	—
41	C₂×C₄	—	10	2	no	—
42	C₂×C₄	C₂²⋊D₅	1	2	yes	—
43	C₂×C₄	C₂²×D₅	1	1	yes	characteristic
44	C₈	—	5	2	no	—
45	C₈	—	5	2	no	—
46	C₅	#499	1	1	yes	characteristic, complemented, Sylow
47	C₂²	C₂×C₂²⋊D₅	1	1	yes	characteristic
48	C₄	—	2	2	no	—
49	C₄	—	4	2	no	—
50	C₄	—	20	2	no	—
51	C₄	Q₈×D₅	1	1	yes	characteristic
52	C₄	Q₈×D₅	1	1	yes	characteristic
53	C₂	#4090	1	1	yes	characteristic
54	C₂	#4096	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₅
commutator, cosocle	C₂×C₄×C₅	C₂³
Frattini	C₂×C₄	C₂²×D₅
Fitting	C₅×#125	C₂
socle, prime radical	C₂²×C₅	C₂×D₄
hypercenter	#125	D₅
hypocenter	C₅	#499
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #125

Complements

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

N	H ≅ G/N	type
46. C₅	#499	semidirect

Maximal subgroups

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

subgroup	index
C₅×#125	2
#4016	2
#3940	2
#3939	2
C₄×#168	2
#499	5