Subgroups — #4040

Order 160.

78 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₂₀	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×D₅	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
4	#685	C₂	1	1	yes	maximal, characteristic, complemented
5	#685	C₂	1	1	yes	maximal, characteristic, complemented
6	#685	C₂	1	1	yes	maximal, characteristic, complemented
7	#683	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₅	C₂²	1	1	yes	characteristic
9	C₂²×D₅	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₅	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₅	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₅	C₂²	1	1	yes	characteristic
13	C₂×Dic₅	C₂²	1	1	yes	characteristic
14	C₂×Dic₅	C₂²	1	1	yes	characteristic
15	D₂₀	—	2	2	no	—
16	C₄×D₅	—	2	2	no	—
17	#123	—	5	1	no	maximal, Sylow
18	C₂²×C₅	C₂³	1	1	yes	characteristic
19	C₂×D₅	—	2	1	no	—
20	C₂×D₅	—	2	1	no	—
21	C₂×D₅	—	2	1	no	—
22	C₂×D₅	—	2	1	no	—
23	C₂×D₅	—	2	1	no	—
24	C₂×D₅	D₄	1	2	yes	—
25	C₄×C₅	—	2	1	no	—
26	C₄×C₅	—	2	1	no	—
27	C₄×C₅	—	2	1	no	—
28	Dic₅	—	2	1	no	—
29	Dic₅	—	2	1	no	—
30	C₂×D₄	—	5	1	no	—
31	C₂²×C₄	—	5	1	no	—
32	C₄⋊C₄	—	5	1	no	—
33	C₄⋊C₄	D₅	1	1	yes	characteristic, complemented
34	C₂²⋊C₄	—	5	1	no	—
35	C₂²⋊C₄	—	5	1	no	—
36	C₂²⋊C₄	—	5	1	no	—
37	C₂×C₅	C₂×D₄	1	1	yes	characteristic
38	C₂×C₅	#41	1	1	yes	characteristic
39	C₂×C₅	#41	1	1	yes	characteristic
40	D₅	—	2	2	no	—
41	D₅	—	4	1	no	—
42	C₂³	—	5	1	no	—
43	C₂³	—	5	1	no	—
44	D₄	—	10	2	no	—
45	C₂×C₄	—	5	1	no	—
46	C₂×C₄	—	5	1	no	—
47	C₂×C₄	—	10	2	no	—
48	C₂×C₄	C₂×D₅	1	1	yes	characteristic
49	C₂×C₄	C₂×D₅	1	1	yes	characteristic
50	C₂×C₄	C₂×D₅	1	1	yes	characteristic
51	C₅	#123	1	1	yes	characteristic, complemented, Sylow
52	C₂²	—	5	2	no	—
53	C₂²	—	10	1	no	—
54	C₂²	—	10	1	no	—
55	C₂²	—	10	1	no	—
56	C₂²	—	10	1	no	—
57	C₂²	—	10	1	no	—
58	C₂²	C₂²×D₅	1	1	yes	characteristic
59	C₄	—	2	1	no	—
60	C₄	—	2	1	no	—
61	C₄	—	2	1	no	—
62	C₄	—	10	1	no	—
63	C₄	—	10	1	no	—
64	C₂	—	10	2	no	—
65	C₂	—	20	1	no	—
66	C₂	#709	1	1	yes	characteristic
67	C₂	D₄×D₅	1	1	yes	characteristic
68	C₂	#713	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₅
commutator, socle, cosocle	C₂²×C₅	C₂³
Fitting	C₅×C₄⋊C₄	C₂
prime radical	C₂×D₂₀	C₂
hypercenter	C₄⋊C₄	D₅
hypocenter	C₅	#123
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₄⋊C₄

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×D₅	C₂	semidirect
3. C₅×C₄⋊C₄	C₂	semidirect
4. #685	C₂	semidirect
5. #685	C₂	semidirect
6. #685	C₂	semidirect
7. #683	C₂	semidirect
33. C₄⋊C₄	D₅	semidirect
51. C₅	#123	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₂₀	2
C₂×C₄×D₅	2
C₅×C₄⋊C₄	2
#685	2
#683	2
#123	5