Subgroups — #64776

Order 320.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×M₁₆	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	C₂²	1	1	yes	characteristic
5	C₅×M₁₆	—	2	1	no	—
6	C₅×M₁₆	—	2	1	no	—
7	C₂×C₅×C₈	C₄	1	2	yes	complemented
8	C₂²⋊Dic₅	C₄	1	2	yes	—
9	C₄⋊Dic₅	—	2	1	no	—
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	#344	—	5	1	no	maximal, Sylow
15	C₂³×C₅	C₂×C₄	1	1	yes	characteristic
16	C₂×C₄×C₅	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄×C₅	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×C₅	D₄	1	1	yes	characteristic
19	C₂×C₄×C₅	D₄	1	1	yes	characteristic
20	C₂×C₄×C₅	D₄	1	1	yes	characteristic
21	C₂×C₄×C₅	Q₈	1	1	yes	characteristic
22	C₂×Dic₅	—	2	2	no	—
23	C₅×C₈	—	2	2	no	—
24	#168	—	2	2	no	—
25	C₂×M₁₆	—	5	1	no	—
26	C₂×M₁₆	D₅	1	1	yes	characteristic
27	#117	—	5	1	no	—
28	C₂²×C₅	—	4	1	no	—
29	C₂²×C₅	C₄²	1	1	yes	characteristic
30	C₂²×C₅	C₂²⋊C₄	1	1	yes	characteristic
31	C₂²×C₅	C₄⋊C₄	1	1	yes	characteristic
32	C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
33	C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
34	C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
35	C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
36	Dic₅	—	4	2	no	—
37	C₂²×C₄	C₂×D₅	1	1	yes	characteristic
38	M₁₆	—	2	1	no	—
39	M₁₆	—	2	1	no	—
40	M₁₆	—	5	2	no	—
41	M₁₆	—	10	1	no	—
42	C₂×C₈	—	10	1	no	—
43	C₂×C₈	Dic₅	1	2	yes	complemented
44	C₄⋊C₄	—	10	1	no	—
45	C₂²⋊C₄	—	5	2	no	—
46	C₄²	—	10	1	no	—
47	C₂×C₅	—	2	1	no	—
48	C₂×C₅	—	2	1	no	—
49	C₂×C₅	—	2	1	no	—
50	C₂×C₅	#95	1	1	yes	characteristic
51	C₂³	C₄×D₅	1	1	yes	characteristic
52	C₂×C₄	—	10	2	no	—
53	C₂×C₄	Dic₁₀	1	1	yes	characteristic
54	C₂×C₄	C₄×D₅	1	1	yes	characteristic
55	C₂×C₄	D₂₀	1	1	yes	characteristic
56	C₂×C₄	C₂×Dic₅	1	1	yes	characteristic
57	C₂×C₄	C₂²⋊D₅	1	1	yes	characteristic
58	C₂×C₄	C₂²⋊D₅	1	1	yes	characteristic
59	C₈	—	2	2	no	—
60	C₈	—	10	2	no	—
61	C₅	#344	1	1	yes	characteristic, complemented, Sylow
62	C₂²	—	4	1	no	—
63	C₂²	C₄×Dic₅	1	1	yes	characteristic
64	C₂²	#683	1	1	yes	characteristic
65	C₂²	#685	1	1	yes	characteristic
66	C₄	—	20	2	no	—
67	C₄	#683	1	1	yes	characteristic
68	C₄	C₄⋊Dic₅	1	1	yes	characteristic
69	C₄	#685	1	1	yes	characteristic
70	C₄	C₂²⋊Dic₅	1	1	yes	characteristic
71	C₂	—	2	1	no	—
72	C₂	—	2	1	no	—
73	C₂	—	2	1	no	—
74	C₂	#3963	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂²⋊Dic₅
commutator	C₂²×C₅	C₄²
Frattini	C₂²×C₄	C₂×D₅
Fitting	C₂×C₅×M₁₆	C₂
socle	C₂×C₅	#95
prime radical	C₂³×C₅	C₂×C₄
cosocle	C₂²×C₄×C₅	C₂²
hypercenter	C₂×M₁₆	D₅
hypocenter	C₅	#344
LCS	G ▷ C₂²×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×M₁₆

Complements

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

N	H ≅ G/N	type
7. C₂×C₅×C₈	C₄	semidirect
12. #681	C₄	semidirect
43. C₂×C₈	Dic₅	semidirect
61. C₅	#344	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×M₁₆	2
#4072	2
C₂×#681	2
#344	5