Subgroups — #65534

Order 320.

146 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×2₋¹⁺⁴	C₂	1	1	yes	maximal, characteristic, complemented
2	#4095	C₂	1	2	yes	maximal, complemented
3	#4093	C₂	1	1	yes	maximal, characteristic, complemented
4	#3969	C₂	1	2	yes	maximal, complemented
5	#3968	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#41	—	2	2	no	—
7	C₅×#41	—	2	2	no	—
8	C₅×#41	C₂²	1	2	yes	—
9	C₂×C₅×Q₈	—	2	2	no	—
10	C₂×C₅×Q₈	C₂²	1	1	yes	characteristic
11	C₂×D₂₀	C₂²	1	1	yes	characteristic
12	#688	—	2	2	no	—
13	#686	—	2	2	no	—
14	#685	—	2	2	no	—
15	C₄×Dic₅	C₂²	1	1	yes	characteristic
16	#681	C₂²	1	2	yes	—
17	#454	—	5	1	no	maximal, Sylow
18	C₂²×D₅	—	2	1	no	—
19	C₅×Q₈	—	2	2	no	—
20	C₅×Q₈	—	2	2	no	—
21	C₅×Q₈	D₄	1	2	yes	—
22	C₅×D₄	—	2	2	no	—
23	C₅×D₄	—	2	2	no	—
24	C₅×D₄	D₄	1	2	yes	—
25	C₂×C₄×C₅	—	2	4	no	—
26	C₂×C₄×C₅	—	2	2	no	—
27	C₂×C₄×C₅	D₄	1	2	yes	—
28	C₂×C₄×C₅	C₂³	1	1	yes	characteristic
29	C₂×Dic₅	—	2	1	no	—
30	D₂₀	—	2	2	no	—
31	#168	—	2	2	no	—
32	2₋¹⁺⁴	D₅	1	1	yes	characteristic, complemented
33	AGL(1,ℤ/8)	—	5	2	no	—
34	#124	—	5	1	no	—
35	#104	—	5	2	no	—
36	#101	—	5	1	no	—
37	C₂²×C₅	—	2	2	no	—
38	C₂²×C₅	C₂×D₄	1	1	yes	characteristic
39	C₂×D₅	—	4	2	no	—
40	C₂×D₅	—	4	1	no	—
41	C₄×C₅	—	2	2	no	—
42	C₄×C₅	—	2	2	no	—
43	C₄×C₅	C₂×D₄	1	2	yes	—
44	Dic₅	—	4	1	no	—
45	#41	—	2	2	no	—
46	#41	—	2	2	no	—
47	#41	C₂×D₅	1	2	yes	—
48	C₂×Q₈	—	2	2	no	—
49	C₂×Q₈	C₂×D₅	1	1	yes	characteristic
50	C₂×D₄	—	5	1	no	—
51	SD₁₆	—	10	2	no	—
52	D₈	—	10	2	no	—
53	M₁₆	—	5	2	no	—
54	C₂²⋊C₄	—	10	2	no	—
55	C₄²	—	5	1	no	—
56	C₂×C₅	—	2	1	no	—
57	C₂×C₅	—	4	2	no	—
58	C₂×C₅	#120	1	1	yes	characteristic
59	D₅	—	8	1	no	—
60	C₂³	—	10	1	no	—
61	Q₈	—	2	2	no	—
62	Q₈	—	2	2	no	—
63	Q₈	C₂²⋊D₅	1	2	yes	—
64	D₄	—	2	2	no	—
65	D₄	—	2	2	no	—
66	D₄	—	10	2	no	—
67	D₄	C₂²⋊D₅	1	2	yes	—
68	C₂×C₄	—	2	4	no	—
69	C₂×C₄	—	2	2	no	—
70	C₂×C₄	—	10	1	no	—
71	C₂×C₄	C₂²⋊D₅	1	2	yes	—
72	C₂×C₄	C₂²×D₅	1	1	yes	characteristic
73	C₈	—	10	2	no	—
74	C₅	#454	1	1	yes	characteristic, complemented, Sylow
75	C₂²	—	2	2	no	—
76	C₂²	—	20	2	no	—
77	C₂²	—	20	1	no	—
78	C₂²	C₂×C₂²⋊D₅	1	1	yes	characteristic
79	C₄	—	2	2	no	—
80	C₄	—	2	2	no	—
81	C₄	—	20	1	no	—
82	C₄	C₂×C₂²⋊D₅	1	2	yes	—
83	C₂	—	2	1	no	—
84	C₂	—	4	2	no	—
85	C₂	—	40	1	no	—
86	C₂	#4099	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#4099
commutator, cosocle	C₂×C₄×C₅	C₂³
Frattini	C₂×C₄	C₂²×D₅
Fitting	C₅×2₋¹⁺⁴	C₂
socle	C₂×C₅	#120
hypercenter	2₋¹⁺⁴	D₅
hypocenter	C₅	#454
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ 2₋¹⁺⁴

Complements

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

N	H ≅ G/N	type
1. C₅×2₋¹⁺⁴	C₂	semidirect
2. #4095	C₂	semidirect
3. #4093	C₂	semidirect
4. #3969	C₂	semidirect
5. #3968	C₂	semidirect
32. 2₋¹⁺⁴	D₅	semidirect
74. C₅	#454	semidirect

Maximal subgroups

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

subgroup	index
C₅×2₋¹⁺⁴	2
#4095	2
#4093	2
#3969	2
#3968	2
#454	5