Subgroups — #64744

Order 320.

118 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄²×C₅	C₂	1	1	yes	maximal, characteristic
2	C₂²×#168	C₂	1	2	yes	maximal
3	C₂²×C₄×C₅	C₄	1	2	yes	—
4	C₂²×C₄×C₅	C₂²	1	1	yes	characteristic
5	C₄²×C₅	—	2	2	no	—
6	C₂×#168	—	2	2	no	—
7	C₂×#168	—	2	2	no	—
8	C₂×#168	C₄	1	4	yes	complemented
9	#336	—	5	1	no	maximal, Sylow
10	C₂³×C₅	C₂×C₄	1	1	yes	characteristic
11	C₂×C₄×C₅	—	2	2	no	—
12	C₂×C₄×C₅	—	2	2	no	—
13	C₂×C₄×C₅	C₈	1	4	yes	complemented
14	C₂×C₄×C₅	C₂×C₄	1	2	yes	—
15	C₂×C₄×C₅	D₄	1	1	yes	characteristic
16	C₂×C₄×C₅	D₄	1	2	yes	—
17	C₂×C₄×C₅	Q₈	1	1	yes	characteristic
18	#168	—	2	4	no	—
19	C₂²×C₈	—	5	2	no	—
20	C₂×C₄²	D₅	1	1	yes	characteristic
21	C₂²×C₅	C₄²	1	1	yes	characteristic
22	C₂²×C₅	C₂²⋊C₄	1	1	yes	characteristic
23	C₂²×C₅	C₄⋊C₄	1	1	yes	characteristic
24	C₂²×C₅	C₂×C₈	1	2	yes	—
25	C₂²×C₅	M₁₆	1	2	yes	—
26	C₄×C₅	—	2	4	no	—
27	C₄×C₅	C₂²⋊C₄	1	2	yes	—
28	C₄×C₅	C₄⋊C₄	1	2	yes	—
29	C₂²×C₄	Dic₅	1	2	yes	—
30	C₂²×C₄	C₂×D₅	1	1	yes	characteristic
31	C₂×C₈	—	5	4	no	—
32	C₂×C₈	—	10	2	no	—
33	C₂×C₈	—	10	2	no	—
34	C₄²	—	2	2	no	—
35	C₂×C₅	#95	1	1	yes	characteristic
36	C₂×C₅	C₄×C₈	1	1	yes	characteristic
37	C₂×C₅	#97	1	1	yes	characteristic
38	C₂×C₅	C₂²⋊C₈	1	2	yes	—
39	C₂×C₅	C₄⋊C₈	1	2	yes	—
40	C₂³	C₂×Dic₅	1	1	yes	characteristic
41	C₂×C₄	—	2	2	no	—
42	C₂×C₄	—	2	2	no	—
43	C₂×C₄	#168	1	4	yes	complemented
44	C₂×C₄	Dic₁₀	1	1	yes	characteristic
45	C₂×C₄	C₄×D₅	1	2	yes	—
46	C₂×C₄	D₂₀	1	1	yes	characteristic
47	C₂×C₄	C₂²⋊D₅	1	2	yes	—
48	C₈	—	10	4	no	—
49	C₅	#336	1	1	yes	characteristic, complemented, Sylow
50	C₂²	C₂×#168	1	2	yes	—
51	C₂²	#681	1	2	yes	—
52	C₂²	C₄×Dic₅	1	1	yes	characteristic
53	C₂²	C₄⋊Dic₅	1	1	yes	characteristic
54	C₂²	C₂²⋊Dic₅	1	1	yes	characteristic
55	C₄	—	2	4	no	—
56	C₄	#683	1	2	yes	—
57	C₄	#685	1	2	yes	—
58	C₂	C₄×#168	1	1	yes	characteristic
59	C₂	#3935	1	1	yes	characteristic
60	C₂	C₄⋊#168	1	2	yes	—
61	C₂	C₂²⋊#168	1	2	yes	—
62	C₂	#3963	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂×D₅
commutator	C₂×C₅	C₄×C₈
Fitting	C₂×C₄²×C₅	C₂
socle, prime radical	C₂³×C₅	C₂×C₄
cosocle	C₂²×C₄×C₅	C₂²
hypercenter	C₂×C₄²	D₅
hypocenter	C₅	#336
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
8. C₂×#168	C₄	semidirect
13. C₂×C₄×C₅	C₈	semidirect
43. C₂×C₄	#168	semidirect
49. C₅	#336	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄²×C₅	2
C₂²×#168	2
#336	5