Subgroups — #65288

Order 320.

114 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#117	C₂	1	1	yes	maximal, characteristic
2	C₂×#681	C₂	1	1	yes	maximal, characteristic
3	C₂²×#168	C₂	1	1	yes	maximal, characteristic
4	C₄⋊#168	C₂	1	4	yes	maximal, complemented
5	C₂²×C₄×C₅	C₂²	1	1	yes	characteristic
6	C₅×C₄⋊C₄	C₄	1	2	yes	—
7	C₅×C₂²⋊C₄	C₄	1	2	yes	—
8	C₄²×C₅	C₂²	1	2	yes	—
9	#681	—	2	2	no	—
10	C₂×#168	—	2	2	no	—
11	C₂×#168	C₂²	1	2	yes	—
12	C₂×#168	C₂²	1	2	yes	—
13	#424	—	5	1	no	maximal, Sylow
14	C₂³×C₅	C₂×C₄	1	1	yes	characteristic
15	C₂×C₄×C₅	C₂×C₄	1	4	yes	—
16	C₂×C₄×C₅	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄×C₅	D₄	1	2	yes	—
18	C₂×C₄×C₅	Q₈	1	2	yes	—
19	C₂×C₄×C₅	C₂³	1	1	yes	characteristic
20	#168	—	2	2	no	—
21	#168	—	2	2	no	—
22	C₂×M₁₆	—	5	1	no	—
23	C₂²×C₈	—	5	1	no	—
24	#117	D₅	1	1	yes	characteristic
25	C₄⋊C₈	—	5	4	no	—
26	C₂²×C₅	—	2	2	no	—
27	C₂²×C₅	C₄⋊C₄	1	2	yes	—
28	C₂²×C₅	C₂²×C₄	1	1	yes	characteristic
29	C₄×C₅	—	2	4	no	—
30	C₄×C₅	C₄⋊C₄	1	2	yes	—
31	C₄×C₅	C₂×D₄	1	1	yes	characteristic
32	C₄×C₅	C₂×Q₈	1	1	yes	characteristic
33	C₂²×C₄	C₂×D₅	1	1	yes	characteristic
34	M₁₆	—	10	2	no	—
35	C₂×C₈	—	5	2	no	—
36	C₂×C₈	—	5	2	no	—
37	C₂×C₈	—	10	2	no	—
38	C₄⋊C₄	Dic₅	1	2	yes	—
39	C₂²⋊C₄	Dic₅	1	2	yes	—
40	C₄²	C₂×D₅	1	2	yes	—
41	C₂×C₅	—	2	2	no	—
42	C₂×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₅	#131	1	2	yes	—
44	C₂³	C₂×Dic₅	1	1	yes	characteristic
45	C₂×C₄	Dic₁₀	1	2	yes	—
46	C₂×C₄	D₂₀	1	2	yes	—
47	C₂×C₄	C₂×Dic₅	1	4	yes	—
48	C₂×C₄	C₂×Dic₅	1	1	yes	characteristic
49	C₂×C₄	C₂²×D₅	1	1	yes	characteristic
50	C₈	—	10	2	no	—
51	C₈	—	10	2	no	—
52	C₅	#424	1	1	yes	characteristic, complemented, Sylow
53	C₂²	—	2	2	no	—
54	C₂²	C₄⋊Dic₅	1	2	yes	—
55	C₂²	C₂²×Dic₅	1	1	yes	characteristic
56	C₄	—	2	4	no	—
57	C₄	C₄⋊Dic₅	1	2	yes	—
58	C₄	C₂×Dic₁₀	1	1	yes	characteristic
59	C₄	C₂×D₂₀	1	1	yes	characteristic
60	C₂	—	2	2	no	—
61	C₂	C₂×C₄⋊Dic₅	1	1	yes	characteristic
62	C₂	#4094	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₅
commutator, socle	C₂²×C₅	C₂²×C₄
Fitting	C₅×#117	C₂
prime radical	C₂³×C₅	C₂×C₄
cosocle	C₂×C₄×C₅	C₂³
hypercenter	#117	D₅
hypocenter	C₅	#424
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ #117

Complements

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

N	H ≅ G/N	type
4. C₄⋊#168	C₂	semidirect
52. C₅	#424	semidirect

Maximal subgroups

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

subgroup	index
C₅×#117	2
C₂×#681	2
C₂²×#168	2
C₄⋊#168	2
#424	5