Subgroups — #64727

Order 320.

66 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×D₅	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#672	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₄×D₅	C₄	1	1	yes	characteristic
5	C₂×C₅×C₈	C₂²	1	1	yes	characteristic
6	C₂×#168	C₄	1	1	yes	characteristic
7	C₈×D₅	—	2	1	no	—
8	C₈×D₅	—	2	1	no	—
9	C₅×C₁₆	—	2	1	no	—
10	#672	—	2	1	no	—
11	C₂²⋊C₁₆	—	5	1	no	maximal, Sylow
12	C₂²×D₅	C₈	1	1	yes	characteristic
13	C₂×C₄×C₅	C₂×C₄	1	1	yes	characteristic
14	C₂×Dic₅	C₈	1	1	yes	characteristic
15	C₄×D₅	—	2	1	no	—
16	C₄×D₅	—	2	1	no	—
17	C₅×C₈	D₄	1	1	yes	characteristic
18	C₅×C₈	D₄	1	1	yes	characteristic
19	#168	—	2	1	no	—
20	C₂²×C₈	—	5	1	no	—
21	C₂×C₁₆	—	5	1	no	—
22	C₂×C₁₆	D₅	1	1	yes	characteristic, complemented
23	C₂²×C₅	C₂×C₈	1	1	yes	characteristic
24	C₂×D₅	—	2	1	no	—
25	C₂×D₅	—	2	1	no	—
26	C₂×D₅	C₁₆	1	2	yes	complemented
27	C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
28	C₄×C₅	M₁₆	1	1	yes	characteristic
29	Dic₅	—	2	1	no	—
30	C₂²×C₄	—	5	1	no	—
31	C₂×C₈	—	5	1	no	—
32	C₂×C₈	—	10	1	no	—
33	C₂×C₈	—	10	1	no	—
34	C₂×C₈	C₂×D₅	1	1	yes	characteristic
35	C₁₆	—	2	1	no	—
36	C₁₆	—	10	1	no	—
37	C₂×C₅	C₂²⋊C₈	1	1	yes	characteristic
38	C₂×C₅	C₂×C₁₆	1	1	yes	characteristic
39	C₂×C₅	M₃₂	1	1	yes	characteristic
40	D₅	—	2	2	no	—
41	C₂³	—	5	1	no	—
42	C₂×C₄	—	5	1	no	—
43	C₂×C₄	—	10	1	no	—
44	C₂×C₄	—	10	1	no	—
45	C₂×C₄	C₄×D₅	1	1	yes	characteristic
46	C₈	—	10	1	no	—
47	C₈	D₂₀	1	1	yes	characteristic
48	C₈	C₂²⋊D₅	1	1	yes	characteristic
49	C₅	C₂²⋊C₁₆	1	1	yes	characteristic, complemented, Sylow
50	C₂²	—	5	2	no	—
51	C₂²	—	10	1	no	—
52	C₂²	—	10	1	no	—
53	C₂²	C₈×D₅	1	1	yes	characteristic
54	C₄	—	10	1	no	—
55	C₄	#676	1	1	yes	characteristic
56	C₄	#685	1	1	yes	characteristic
57	C₂	—	10	2	no	—
58	C₂	D₅×C₁₆	1	1	yes	characteristic
59	C₂	#3930	1	1	yes	characteristic
60	C₂	#3952	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂×D₅
commutator	C₂×C₅	C₂×C₁₆
Fitting	C₂×C₅×C₁₆	C₂
socle	C₂²×C₅	C₂×C₈
prime radical	C₂²×D₅	C₈
cosocle	C₂×C₅×C₈	C₂²
hypercenter	C₂×C₁₆	D₅
hypocenter	C₅	C₂²⋊C₁₆
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₁₆

Complements

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

N	H ≅ G/N	type
2. C₂×C₅×C₁₆	C₂	semidirect
3. C₂×#672	C₂	semidirect
22. C₂×C₁₆	D₅	semidirect
26. C₂×D₅	C₁₆	semidirect
49. C₅	C₂²⋊C₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×D₅	2
C₂×C₅×C₁₆	2
C₂×#672	2
C₂²⋊C₁₆	5