Subgroups — #65162

Order 320.

110 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₂₀	C₂	1	1	yes	maximal, characteristic
2	#4033	C₂	1	2	yes	maximal
3	C₅×#107	C₂	1	1	yes	maximal, characteristic
4	C₈×Dic₅	C₂	1	1	yes	maximal, characteristic
5	#3942	C₂	1	2	yes	maximal
6	C₂×Dic₁₀	C₂²	1	2	yes	—
7	C₂×C₅×C₈	C₂²	1	1	yes	characteristic
8	C₅×C₄⋊C₄	C₂²	1	2	yes	—
9	#683	—	2	2	no	—
10	C₄×Dic₅	—	2	2	no	—
11	C₄×Dic₅	C₂²	1	1	yes	characteristic
12	C₂×#168	C₂²	1	1	yes	characteristic
13	Dic₂₀	C₄	1	4	yes	complemented
14	C₄×Q₁₆	—	5	1	no	maximal, Sylow
15	C₂×C₄×C₅	—	2	2	no	—
16	C₂×C₄×C₅	C₂³	1	1	yes	characteristic
17	C₂×Dic₅	—	2	2	no	—
18	C₂×Dic₅	D₄	1	1	yes	characteristic
19	C₂×Dic₅	D₄	1	1	yes	characteristic
20	Dic₁₀	—	2	2	no	—
21	Dic₁₀	C₂×C₄	1	4	yes	—
22	C₅×C₈	C₂×C₄	1	2	yes	—
23	#168	—	2	1	no	—
24	C₂×Q₁₆	—	5	1	no	—
25	C₄×Q₈	—	5	2	no	—
26	#107	D₅	1	1	yes	characteristic
27	#103	—	5	2	no	—
28	C₄×C₈	—	5	1	no	—
29	C₂²×C₅	C₂×D₄	1	1	yes	characteristic
30	C₄×C₅	—	4	2	no	—
31	C₄×C₅	C₂²×C₄	1	1	yes	characteristic
32	C₄×C₅	#41	1	1	yes	characteristic
33	Dic₅	—	2	4	no	—
34	Dic₅	—	2	1	no	—
35	Dic₅	Q₁₆	1	2	yes	complemented
36	C₂×Q₈	—	5	2	no	—
37	Q₁₆	—	5	4	no	—
38	C₂×C₈	—	5	1	no	—
39	C₂×C₈	C₂×D₅	1	1	yes	characteristic
40	C₄⋊C₄	—	10	2	no	—
41	C₄⋊C₄	C₂×D₅	1	2	yes	—
42	C₄²	—	5	1	no	—
43	C₄²	—	10	2	no	—
44	C₂×C₅	C₄×D₄	1	1	yes	characteristic
45	C₂×C₅	C₂×Q₁₆	1	1	yes	characteristic
46	C₂×C₅	#135	1	1	yes	characteristic
47	Q₈	—	5	4	no	—
48	Q₈	—	10	2	no	—
49	C₂×C₄	—	2	2	no	—
50	C₂×C₄	—	5	1	no	—
51	C₂×C₄	—	5	1	no	—
52	C₂×C₄	—	10	2	no	—
53	C₂×C₄	C₂²×D₅	1	1	yes	characteristic
54	C₈	—	10	1	no	—
55	C₈	C₄×D₅	1	2	yes	—
56	C₅	C₄×Q₁₆	1	1	yes	characteristic, complemented, Sylow
57	C₂²	D₄×D₅	1	1	yes	characteristic
58	C₄	—	4	2	no	—
59	C₄	—	5	2	no	—
60	C₄	—	10	4	no	—
61	C₄	—	10	1	no	—
62	C₄	C₂×C₄×D₅	1	1	yes	characteristic
63	C₄	#713	1	1	yes	characteristic
64	C₂	#4039	1	1	yes	characteristic
65	C₂	#4058	1	1	yes	characteristic
66	C₂	D₅×Q₁₆	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₅
commutator	C₄×C₅	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₅
Fitting	C₅×#107	C₂
socle, prime radical	C₂²×C₅	C₂×D₄
cosocle	C₂×C₄×C₅	C₂³
hypercenter	#107	D₅
hypocenter	C₅	C₄×Q₁₆
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #107

Complements

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

N	H ≅ G/N	type
13. Dic₂₀	C₄	semidirect
35. Dic₅	Q₁₆	semidirect
56. C₅	C₄×Q₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₂₀	2
#4033	2
C₅×#107	2
C₈×Dic₅	2
#3942	2
C₄×Q₁₆	5