Subgroups — #64701

Order 320.

64 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#4015	C₂	1	1	yes	maximal, characteristic
2	C₅×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₄⋊#168	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₁₀	C₄	1	2	yes	—
5	C₂×C₅×C₈	—	2	1	no	—
6	C₄²×C₅	C₂²	1	1	yes	characteristic
7	C₄⋊Dic₅	—	2	1	no	—
8	C₄⋊Dic₅	—	2	1	no	—
9	C₂×#168	—	2	1	no	—
10	#333	—	5	1	no	maximal, Sylow
11	C₂×C₄×C₅	C₂×C₄	1	1	yes	characteristic
12	C₂×C₄×C₅	D₄	1	1	yes	characteristic
13	C₂×C₄×C₅	D₄	1	1	yes	characteristic
14	C₂×Dic₅	—	2	2	no	—
15	Dic₁₀	—	4	2	no	—
16	C₅×C₈	—	4	1	no	—
17	#168	—	4	1	no	—
18	C₄⋊Q₈	—	5	1	no	—
19	C₄⋊C₈	—	5	1	no	—
20	C₄⋊C₈	D₅	1	1	yes	characteristic
21	C₂²×C₅	C₂²⋊C₄	1	1	yes	characteristic
22	C₄×C₅	—	2	1	no	—
23	C₄×C₅	SD₁₆	1	1	yes	characteristic
24	C₄×C₅	SD₁₆	1	1	yes	characteristic
25	C₄×C₅	Q₁₆	1	1	yes	characteristic
26	C₄×C₅	Q₁₆	1	1	yes	characteristic
27	Dic₅	—	4	2	no	—
28	C₂×Q₈	—	5	2	no	—
29	C₂×C₈	—	2	1	no	—
30	C₂×C₈	—	10	1	no	—
31	C₄⋊C₄	—	10	1	no	—
32	C₄⋊C₄	—	10	1	no	—
33	C₄²	C₂×D₅	1	1	yes	characteristic
34	C₂×C₅	#100	1	1	yes	characteristic
35	C₂×C₅	#103	1	1	yes	characteristic
36	C₂×C₅	#103	1	1	yes	characteristic
37	Q₈	—	20	2	no	—
38	C₂×C₄	—	10	2	no	—
39	C₂×C₄	C₄×D₅	1	1	yes	characteristic
40	C₂×C₄	D₂₀	1	1	yes	characteristic
41	C₂×C₄	C₂²⋊D₅	1	1	yes	characteristic
42	C₈	—	4	1	no	—
43	C₈	—	20	1	no	—
44	C₅	#333	1	1	yes	characteristic, complemented, Sylow
45	C₂²	#685	1	1	yes	characteristic
46	C₄	—	2	1	no	—
47	C₄	—	20	2	no	—
48	C₄	#677	1	1	yes	characteristic
49	C₄	Dic₂₀	1	1	yes	characteristic
50	C₄	#687	1	1	yes	characteristic
51	C₄	#689	1	1	yes	characteristic
52	C₂	#3942	1	1	yes	characteristic
53	C₂	#3948	1	1	yes	characteristic
54	C₂	#3955	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#685
commutator	C₂×C₄×C₅	C₂×C₄
Frattini	C₄²	C₂×D₅
Fitting	C₅×C₄⋊C₈	C₂
socle, prime radical	C₂²×C₅	C₂²⋊C₄
cosocle	C₄²×C₅	C₂²
hypercenter	C₄⋊C₈	D₅
hypocenter	C₅	#333
LCS	G ▷ C₂×C₄×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
44. C₅	#333	semidirect

Maximal subgroups

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

subgroup	index
#4015	2
C₅×C₄⋊C₈	2
C₄⋊#168	2
#333	5