Subgroups — #4027

Order 160.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂²⋊D₅	C₂	1	1	yes	maximal, characteristic
2	C₂²×Dic₅	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄×D₅	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
5	#685	C₂	1	1	yes	maximal, characteristic, complemented
6	#683	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×Dic₅	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂³×C₅	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	C₂²⋊D₅	C₄	1	4	yes	complemented
13	C₂×Dic₅	—	2	2	no	—
14	C₂×Dic₅	C₂²	1	1	yes	characteristic
15	C₂×Dic₅	C₂²	1	1	yes	characteristic
16	C₂×Dic₅	C₂²	1	1	yes	characteristic
17	C₄×D₅	—	2	2	no	—
18	C₄×D₄	—	5	1	no	maximal, Sylow
19	C₂²×C₅	—	2	1	no	—
20	C₂²×C₅	—	2	1	no	—
21	C₂²×C₅	C₂×C₄	1	2	yes	complemented
22	C₂²×C₅	C₂³	1	1	yes	characteristic
23	C₂×D₅	—	2	1	no	—
24	C₂×D₅	—	2	1	no	—
25	C₂×D₅	C₂×C₄	1	2	yes	complemented
26	C₄×C₅	—	2	1	no	—
27	C₄×C₅	—	2	1	no	—
28	Dic₅	—	2	1	no	—
29	Dic₅	C₂×C₄	1	2	yes	complemented
30	Dic₅	D₄	1	2	yes	complemented
31	C₂×D₄	—	5	1	no	—
32	C₂²×C₄	—	5	1	no	—
33	C₂²×C₄	—	5	1	no	—
34	C₄⋊C₄	—	5	1	no	—
35	C₂²⋊C₄	—	5	1	no	—
36	C₂²⋊C₄	D₅	1	1	yes	characteristic, complemented
37	C₄²	—	5	1	no	—
38	C₂×C₅	—	2	2	no	—
39	C₂×C₅	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₅	C₂×D₄	1	1	yes	characteristic
41	C₂×C₅	#41	1	1	yes	characteristic
42	D₅	—	2	2	no	—
43	C₂³	—	5	1	no	—
44	C₂³	C₂×D₅	1	1	yes	characteristic
45	D₄	—	5	4	no	—
46	C₂×C₄	—	5	1	no	—
47	C₂×C₄	—	5	1	no	—
48	C₂×C₄	—	5	1	no	—
49	C₂×C₄	—	10	2	no	—
50	C₂×C₄	—	10	2	no	—
51	C₂×C₄	C₂×D₅	1	1	yes	characteristic
52	C₂×C₄	C₂×D₅	1	1	yes	characteristic
53	C₅	C₄×D₄	1	1	yes	characteristic, complemented, Sylow
54	C₂²	—	2	1	no	—
55	C₂²	—	2	1	no	—
56	C₂²	—	5	2	no	—
57	C₂²	—	10	1	no	—
58	C₂²	—	10	1	no	—
59	C₂²	C₄×D₅	1	2	yes	complemented
60	C₂²	C₂²×D₅	1	1	yes	characteristic
61	C₄	—	2	1	no	—
62	C₄	—	2	1	no	—
63	C₄	—	5	2	no	—
64	C₄	—	5	2	no	—
65	C₄	—	10	1	no	—
66	C₂	—	2	2	no	—
67	C₂	—	10	2	no	—
68	C₂	C₂×C₄×D₅	1	1	yes	characteristic
69	C₂	D₄×D₅	1	1	yes	characteristic
70	C₂	#711	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. C₂²×Dic₅	C₂	semidirect
3. C₂×C₄×D₅	C₂	semidirect
4. C₅×C₂²⋊C₄	C₂	semidirect
5. #685	C₂	semidirect
6. #683	C₂	semidirect
7. C₄×Dic₅	C₂	semidirect
12. C₂²⋊D₅	C₄	semidirect
21. C₂²×C₅	C₂×C₄	semidirect
25. C₂×D₅	C₂×C₄	semidirect
29. Dic₅	C₂×C₄	semidirect
30. Dic₅	D₄	semidirect
36. C₂²⋊C₄	D₅	semidirect
53. C₅	C₄×D₄	semidirect
59. C₂²	C₄×D₅	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂²⋊D₅	2
C₂²×Dic₅	2
C₂×C₄×D₅	2
C₅×C₂²⋊C₄	2
#685	2
#683	2
C₄×Dic₅	2
C₄×D₄	5