Subgroups — #4044

Order 160.

60 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
2	#685	C₂	1	1	yes	maximal, characteristic
3	#685	C₂	1	1	yes	maximal, characteristic
4	#685	C₂	1	1	yes	maximal, characteristic
5	C₄⋊Dic₅	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	C₂²×D₅	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₅	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×Dic₅	C₂²	1	1	yes	characteristic
13	C₂×Dic₅	C₂²	1	1	yes	characteristic
14	C₂×Dic₅	C₂²	1	1	yes	characteristic
15	#126	—	5	1	no	maximal, Sylow
16	C₂²×C₅	C₂³	1	1	yes	characteristic
17	C₂×D₅	—	2	1	no	—
18	C₂×D₅	—	2	1	no	—
19	C₂×D₅	—	2	1	no	—
20	C₄×C₅	—	2	1	no	—
21	C₄×C₅	—	2	1	no	—
22	C₄×C₅	—	2	1	no	—
23	Dic₅	—	2	1	no	—
24	Dic₅	—	2	1	no	—
25	Dic₅	—	2	1	no	—
26	C₄⋊C₄	—	5	1	no	—
27	C₄⋊C₄	—	5	1	no	—
28	C₄⋊C₄	D₅	1	1	yes	characteristic, complemented
29	C₂²⋊C₄	—	5	1	no	—
30	C₂²⋊C₄	—	5	1	no	—
31	C₂²⋊C₄	—	5	1	no	—
32	C₄²	—	5	1	no	—
33	C₂×C₅	#41	1	1	yes	characteristic
34	C₂×C₅	#41	1	1	yes	characteristic
35	C₂×C₅	#41	1	1	yes	characteristic
36	D₅	—	4	1	no	—
37	C₂³	—	5	1	no	—
38	C₂×C₄	—	5	1	no	—
39	C₂×C₄	—	5	1	no	—
40	C₂×C₄	—	5	1	no	—
41	C₂×C₄	C₂×D₅	1	1	yes	characteristic
42	C₂×C₄	C₂×D₅	1	1	yes	characteristic
43	C₂×C₄	C₂×D₅	1	1	yes	characteristic
44	C₅	#126	1	1	yes	characteristic, complemented, Sylow
45	C₂²	—	10	1	no	—
46	C₂²	—	10	1	no	—
47	C₂²	—	10	1	no	—
48	C₂²	C₂²×D₅	1	1	yes	characteristic
49	C₄	—	2	1	no	—
50	C₄	—	2	1	no	—
51	C₄	—	2	1	no	—
52	C₄	—	10	1	no	—
53	C₄	—	10	1	no	—
54	C₄	—	10	1	no	—
55	C₂	—	20	1	no	—
56	C₂	#709	1	1	yes	characteristic
57	C₂	#711	1	1	yes	characteristic
58	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	C₂²×D₅	C₂²
hypercenter	C₄⋊C₄	D₅
hypocenter	C₅	#126
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
1. C₅×C₄⋊C₄	C₂	semidirect
5. C₄⋊Dic₅	C₂	semidirect
6. #683	C₂	semidirect
7. C₄×Dic₅	C₂	semidirect
28. C₄⋊C₄	D₅	semidirect
44. C₅	#126	semidirect

Maximal subgroups

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

subgroup	index
C₅×C₄⋊C₄	2
#685	2
C₄⋊Dic₅	2
#683	2
C₄×Dic₅	2
#126	5