Subgroups — #65530

Order 320.

168 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×H(5,𝔽₂)	C₂	1	1	yes	maximal, characteristic, complemented
2	#4095	C₂	1	2	yes	maximal, complemented
3	#4086	C₂	1	1	yes	maximal, characteristic, complemented
4	#3969	C₂	1	2	yes	maximal, complemented
5	#3965	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#41	—	2	2	no	—
7	C₅×#41	C₂²	1	2	yes	—
8	C₂×C₅×D₄	—	2	2	no	—
9	C₂×C₅×D₄	—	2	2	no	—
10	C₂×C₅×D₄	C₂²	1	1	yes	characteristic
11	C₂×C₂²⋊D₅	—	2	2	no	—
12	C₂×D₂₀	C₂²	1	1	yes	characteristic, complemented
13	#688	—	2	2	no	—
14	#686	—	2	2	no	—
15	C₄×Dic₅	C₂²	1	1	yes	characteristic, complemented
16	#681	C₂²	1	2	yes	complemented
17	#453	—	5	1	no	maximal, Sylow
18	C₂³×C₅	—	2	2	no	—
19	C₂³×C₅	D₄	1	2	yes	—
20	C₂²×D₅	—	2	1	no	—
21	C₅×Q₈	D₄	1	2	yes	complemented
22	C₅×D₄	—	2	4	no	—
23	C₅×D₄	—	2	2	no	—
24	C₅×D₄	—	2	2	no	—
25	C₅×D₄	D₄	1	2	yes	complemented
26	C₂×C₄×C₅	—	2	2	no	—
27	C₂×C₄×C₅	—	2	2	no	—
28	C₂×C₄×C₅	C₂³	1	1	yes	characteristic
29	C₂²⋊D₅	—	4	4	no	—
30	C₂×Dic₅	—	2	1	no	—
31	D₂₀	—	2	2	no	—
32	#168	—	2	2	no	—
33	H(5,𝔽₂)	D₅	1	1	yes	characteristic, complemented
34	AGL(1,ℤ/8)	—	5	2	no	—
35	#127	—	5	1	no	—
36	#104	—	5	2	no	—
37	#100	—	5	1	no	—
38	C₂²×C₅	—	2	2	no	—
39	C₂²×C₅	—	2	2	no	—
40	C₂²×C₅	—	4	2	no	—
41	C₂²×C₅	—	8	2	no	—
42	C₂²×C₅	C₂×D₄	1	1	yes	characteristic
43	C₂×D₅	—	4	2	no	—
44	C₂×D₅	—	4	1	no	—
45	C₄×C₅	—	2	2	no	—
46	C₄×C₅	C₂×D₄	1	2	yes	—
47	Dic₅	—	2	2	no	—
48	#41	—	2	2	no	—
49	#41	C₂×D₅	1	2	yes	—
50	C₂×D₄	—	2	2	no	—
51	C₂×D₄	—	2	2	no	—
52	C₂×D₄	—	5	1	no	—
53	C₂×D₄	—	10	2	no	—
54	C₂×D₄	C₂×D₅	1	1	yes	characteristic
55	SD₁₆	—	10	2	no	—
56	D₈	—	10	2	no	—
57	M₁₆	—	5	2	no	—
58	C₄²	—	5	1	no	—
59	C₂×C₅	—	2	1	no	—
60	C₂×C₅	—	4	2	no	—
61	C₂×C₅	—	4	2	no	—
62	C₂×C₅	#120	1	1	yes	characteristic
63	D₅	—	8	1	no	—
64	C₂³	—	2	2	no	—
65	C₂³	—	10	1	no	—
66	C₂³	C₂²⋊D₅	1	2	yes	—
67	Q₈	C₂²⋊D₅	1	2	yes	complemented
68	D₄	—	2	4	no	—
69	D₄	—	2	2	no	—
70	D₄	—	2	2	no	—
71	D₄	—	10	2	no	—
72	D₄	—	20	4	no	—
73	D₄	C₂²⋊D₅	1	2	yes	complemented
74	C₂×C₄	—	2	2	no	—
75	C₂×C₄	—	2	2	no	—
76	C₂×C₄	—	10	1	no	—
77	C₂×C₄	C₂²×D₅	1	1	yes	characteristic
78	C₈	—	10	2	no	—
79	C₅	#453	1	1	yes	characteristic, complemented, Sylow
80	C₂²	—	2	2	no	—
81	C₂²	—	2	2	no	—
82	C₂²	—	4	2	no	—
83	C₂²	—	8	2	no	—
84	C₂²	—	20	2	no	—
85	C₂²	—	20	1	no	—
86	C₂²	C₂×C₂²⋊D₅	1	1	yes	characteristic
87	C₄	—	2	2	no	—
88	C₄	—	10	2	no	—
89	C₄	C₂×C₂²⋊D₅	1	2	yes	—
90	C₂	—	2	1	no	—
91	C₂	—	4	2	no	—
92	C₂	—	4	2	no	—
93	C₂	—	40	1	no	—
94	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×H(5,𝔽₂)	C₂
socle	C₂×C₅	#120
hypercenter	H(5,𝔽₂)	D₅
hypocenter	C₅	#453
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ H(5,𝔽₂)

Complements

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

N	H ≅ G/N	type
1. C₅×H(5,𝔽₂)	C₂	semidirect
2. #4095	C₂	semidirect
3. #4086	C₂	semidirect
4. #3969	C₂	semidirect
5. #3965	C₂	semidirect
12. C₂×D₂₀	C₂²	semidirect
15. C₄×Dic₅	C₂²	semidirect
16. #681	C₂²	semidirect
21. C₅×Q₈	D₄	semidirect
25. C₅×D₄	D₄	semidirect
33. H(5,𝔽₂)	D₅	semidirect
67. Q₈	C₂²⋊D₅	semidirect
73. D₄	C₂²⋊D₅	semidirect
79. C₅	#453	semidirect

Maximal subgroups

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

subgroup	index
C₅×H(5,𝔽₂)	2
#4095	2
#4086	2
#3969	2
#3965	2
#453	5