Subgroups — #64743

Order 320.

110 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	#4072	C₂	1	1	yes	maximal, characteristic
3	C₂×#681	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₄×C₅	—	2	1	no	—
5	C₂²×C₄×C₅	C₂²	1	1	yes	characteristic
6	C₄²×C₅	—	2	1	no	—
7	C₄²×C₅	C₄	1	2	yes	complemented
8	C₂²⋊Dic₅	—	2	1	no	—
9	C₄⋊Dic₅	C₄	1	2	yes	complemented
10	C₄×Dic₅	—	2	1	no	—
11	#681	—	2	1	no	—
12	#681	C₄	1	2	yes	complemented
13	C₂×#168	—	2	1	no	—
14	#339	—	5	1	no	maximal, Sylow
15	C₂³×C₅	D₄	1	1	yes	characteristic
16	C₂×C₄×C₅	—	2	4	no	—
17	C₂×C₄×C₅	—	2	2	no	—
18	C₂×C₄×C₅	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×C₅	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×C₅	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×C₅	D₄	1	1	yes	characteristic
22	C₂×C₄×C₅	D₄	1	1	yes	characteristic
23	C₂×C₄×C₅	Q₈	1	1	yes	characteristic
24	C₂×Dic₅	—	2	2	no	—
25	#168	—	2	2	no	—
26	C₂×M₁₆	—	5	1	no	—
27	#117	—	5	1	no	—
28	C₂×C₄²	D₅	1	1	yes	characteristic
29	C₂²×C₅	—	2	2	no	—
30	C₂²×C₅	C₂²⋊C₄	1	1	yes	characteristic
31	C₂²×C₅	C₂²⋊C₄	1	1	yes	characteristic
32	C₂²×C₅	C₄⋊C₄	1	1	yes	characteristic
33	C₄×C₅	—	2	4	no	—
34	C₄×C₅	C₄²	1	1	yes	characteristic
35	C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
36	C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
37	C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
38	Dic₅	—	4	2	no	—
39	C₂²×C₄	—	2	1	no	—
40	C₂²×C₄	C₂×D₅	1	1	yes	characteristic
41	M₁₆	—	5	2	no	—
42	M₁₆	—	10	1	no	—
43	C₂×C₈	—	10	1	no	—
44	C₄⋊C₄	—	5	2	no	—
45	C₂²⋊C₄	—	10	1	no	—
46	C₄²	—	2	1	no	—
47	C₄²	—	10	1	no	—
48	C₄²	Dic₅	1	2	yes	complemented
49	C₂×C₅	—	2	1	no	—
50	C₂×C₅	—	2	1	no	—
51	C₂×C₅	#95	1	1	yes	characteristic
52	C₂×C₅	#104	1	2	yes	—
53	C₂³	C₂²⋊D₅	1	1	yes	characteristic
54	C₂×C₄	—	2	4	no	—
55	C₂×C₄	—	2	2	no	—
56	C₂×C₄	—	10	2	no	—
57	C₂×C₄	Dic₁₀	1	1	yes	characteristic
58	C₂×C₄	C₄×D₅	1	1	yes	characteristic
59	C₂×C₄	C₄×D₅	1	1	yes	characteristic
60	C₂×C₄	D₂₀	1	1	yes	characteristic
61	C₂×C₄	C₂×Dic₅	1	1	yes	characteristic
62	C₂×C₄	C₂²⋊D₅	1	1	yes	characteristic
63	C₈	—	10	2	no	—
64	C₅	#339	1	1	yes	characteristic, complemented, Sylow
65	C₂²	—	2	2	no	—
66	C₂²	#683	1	1	yes	characteristic
67	C₂²	#685	1	1	yes	characteristic
68	C₂²	C₂²⋊Dic₅	1	1	yes	characteristic
69	C₄	—	2	4	no	—
70	C₄	—	20	2	no	—
71	C₄	C₄×Dic₅	1	1	yes	characteristic
72	C₄	#683	1	1	yes	characteristic
73	C₄	C₄⋊Dic₅	1	1	yes	characteristic
74	C₄	#685	1	1	yes	characteristic
75	C₂	—	2	1	no	—
76	C₂	—	2	1	no	—
77	C₂	#3937	1	2	yes	—
78	C₂	#3963	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₂₀
commutator	C₄×C₅	C₄²
Frattini	C₂²×C₄	C₂×D₅
Fitting	C₂×C₄²×C₅	C₂
socle	C₂²×C₅	C₂²⋊C₄
prime radical	C₂³×C₅	D₄
cosocle	C₂²×C₄×C₅	C₂²
hypercenter	C₂×C₄²	D₅
hypocenter	C₅	#339
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
7. C₄²×C₅	C₄	semidirect
9. C₄⋊Dic₅	C₄	semidirect
12. #681	C₄	semidirect
48. C₄²	Dic₅	semidirect
64. C₅	#339	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
#4072	2
C₂×#681	2
#339	5