Subgroups — #65702

Order 320.

134 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₅⋊C₈	C₂	1	2	yes	maximal, complemented
2	#4039	C₂	1	1	yes	maximal, characteristic
3	#4003	C₂	1	2	yes	maximal, complemented
4	C₄⋊#170	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×#170	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₂₀	C₄	1	1	yes	characteristic
7	C₂×C₄×D₅	C₂²	1	2	yes	—
8	C₂×#170	C₂²	1	2	yes	—
9	C₂×#170	C₂²	1	1	yes	characteristic
10	C₂×#170	C₂²	1	1	yes	characteristic
11	D₅⋊C₈	—	2	4	no	—
12	C₅×C₄⋊C₄	C₄	1	1	yes	characteristic
13	#685	C₄	1	2	yes	—
14	C₄×Dic₅	C₂²	1	1	yes	characteristic
15	C₈×D₄	—	5	1	no	maximal, Sylow
16	C₂²×D₅	C₂×C₄	1	2	yes	—
17	C₂×C₄×C₅	C₂×C₄	1	2	yes	—
18	C₂×C₄×C₅	C₂×C₄	1	1	yes	characteristic
19	C₂×Dic₅	C₂×C₄	1	1	yes	characteristic
20	C₂×Dic₅	C₂³	1	1	yes	characteristic
21	D₂₀	C₈	1	4	yes	complemented
22	C₄×D₅	—	2	2	no	—
23	C₄×D₅	—	2	2	no	—
24	#170	—	2	2	no	—
25	#170	—	2	1	no	—
26	#170	D₄	1	2	yes	complemented
27	C₂²×C₈	—	5	2	no	—
28	C₄×D₄	—	5	1	no	—
29	C₄⋊C₈	—	5	1	no	—
30	C₂²⋊C₈	—	5	2	no	—
31	C₄×C₈	—	5	1	no	—
32	C₂²×C₅	C₂²×C₄	1	1	yes	characteristic
33	C₂×D₅	—	2	2	no	—
34	C₂×D₅	—	2	2	no	—
35	C₂×D₅	C₂×C₈	1	4	yes	complemented
36	C₄×C₅	—	2	2	no	—
37	C₄×C₅	C₂×C₈	1	2	yes	complemented
38	Dic₅	—	2	1	no	—
39	Dic₅	C₂×D₄	1	1	yes	characteristic
40	Dic₅	#41	1	1	yes	characteristic
41	C₂×D₄	—	5	1	no	—
42	C₂²×C₄	—	5	2	no	—
43	C₂×C₈	—	5	1	no	—
44	C₂×C₈	—	5	1	no	—
45	C₂×C₈	—	5	2	no	—
46	C₂×C₈	—	10	4	no	—
47	C₄⋊C₄	AGL(1,𝔽₅)	1	1	yes	characteristic
48	C₂²⋊C₄	—	5	2	no	—
49	C₄²	—	5	1	no	—
50	C₂×C₅	C₄×D₄	1	1	yes	characteristic
51	C₂×C₅	C₂²×C₈	1	1	yes	characteristic
52	C₂×C₅	#131	1	1	yes	characteristic
53	D₅	—	2	4	no	—
54	C₂³	—	5	2	no	—
55	D₄	—	5	4	no	—
56	C₂×C₄	—	5	1	no	—
57	C₂×C₄	—	5	1	no	—
58	C₂×C₄	—	10	2	no	—
59	C₂×C₄	—	10	2	no	—
60	C₂×C₄	C₂×AGL(1,𝔽₅)	1	2	yes	—
61	C₂×C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
62	C₈	—	5	2	no	—
63	C₈	—	10	2	no	—
64	C₈	—	10	1	no	—
65	C₅	C₈×D₄	1	1	yes	characteristic, complemented, Sylow
66	C₂²	—	5	4	no	—
67	C₂²	—	10	2	no	—
68	C₂²	—	10	2	no	—
69	C₂²	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
70	C₄	—	2	2	no	—
71	C₄	—	5	1	no	—
72	C₄	—	5	1	no	—
73	C₄	—	10	1	no	—
74	C₄	D₅⋊C₈	1	2	yes	complemented
75	C₂	—	10	4	no	—
76	C₂	C₂×D₅⋊C₈	1	1	yes	characteristic
77	C₂	D₄×AGL(1,𝔽₅)	1	1	yes	characteristic
78	C₂	#4133	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×AGL(1,𝔽₅)
commutator	C₂×C₅	C₂²×C₈
Fitting	C₅×C₄⋊C₄	C₄
socle	C₂²×C₅	C₂²×C₄
prime radical	C₂×D₂₀	C₄
cosocle	C₂×Dic₅	C₂³
hypercenter	C₄⋊C₄	AGL(1,𝔽₅)
hypocenter	C₅	C₈×D₄
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₄⋊C₄

Complements

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

N	H ≅ G/N	type
1. C₂×D₅⋊C₈	C₂	semidirect
3. #4003	C₂	semidirect
4. C₄⋊#170	C₂	semidirect
5. C₄×#170	C₂	semidirect
21. D₂₀	C₈	semidirect
26. #170	D₄	semidirect
35. C₂×D₅	C₂×C₈	semidirect
37. C₄×C₅	C₂×C₈	semidirect
65. C₅	C₈×D₄	semidirect
74. C₄	D₅⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₅⋊C₈	2
#4039	2
#4003	2
C₄⋊#170	2
C₄×#170	2
C₈×D₄	5