Subgroups — #10603858

Order 640.

220 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#3995	C₂	1	4	yes	maximal
2	#65681	C₂	1	2	yes	maximal
3	C₄×C₈×D₅	C₂	1	1	yes	maximal, characteristic
4	C₂×#700	C₂²	1	4	yes	—
5	C₂×C₈×D₅	C₂²	1	2	yes	—
6	C₄²×D₅	C₂²	1	1	yes	characteristic
7	C₄⋊#170	—	2	4	no	—
8	#3995	—	2	8	no	—
9	C₄×C₅×C₈	C₄	1	1	yes	characteristic
10	C₈×Dic₅	C₄	1	2	yes	—
11	C₄×#168	C₄	1	1	yes	characteristic
12	#1848	—	5	1	no	maximal, Sylow
13	C₂×C₄×D₅	Q₈	1	2	yes	—
14	C₂×C₄×D₅	C₂³	1	1	yes	characteristic
15	C₂×#170	—	2	4	no	—
16	#700	—	2	8	no	—
17	C₂×C₅×C₈	C₂×C₄	1	2	yes	—
18	C₄²×C₅	C₂×C₄	1	1	yes	characteristic
19	C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
20	C₄×Dic₅	D₄	1	2	yes	—
21	C₂×#168	C₂×C₄	1	2	yes	—
22	C₈×D₅	D₄	1	4	yes	—
23	C₈×D₅	Q₈	1	4	yes	—
24	C₂×#108	—	5	4	no	—
25	#423	—	5	2	no	—
26	C₂×C₄×C₈	—	5	1	no	—
27	C₂²×D₅	C₂×Q₈	1	1	yes	characteristic
28	C₂×C₄×C₅	C₄⋊C₄	1	2	yes	—
29	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
30	C₂×Dic₅	C₄⋊C₄	1	2	yes	—
31	C₂×Dic₅	C₂×D₄	1	1	yes	characteristic
32	C₄×D₅	—	2	4	no	—
33	C₄×D₅	C₂×D₄	1	2	yes	—
34	C₄×D₅	C₂×Q₈	1	2	yes	—
35	#170	—	4	4	no	—
36	C₅×C₈	C₄⋊C₄	1	4	yes	—
37	#168	C₄⋊C₄	1	4	yes	—
38	C₂×M₁₆	—	5	4	no	—
39	C₂²×C₈	—	5	2	no	—
40	C₂×C₄²	—	5	1	no	—
41	#108	—	10	8	no	—
42	C₄⋊C₈	—	10	4	no	—
43	C₄×C₈	—	5	2	no	—
44	C₄×C₈	—	5	1	no	—
45	C₄×C₈	AGL(1,𝔽₅)	1	1	yes	characteristic
46	C₂²×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
47	C₂×D₅	—	2	2	no	—
48	C₂×D₅	C₄⋊Q₈	1	2	yes	—
49	C₄×C₅	#108	1	4	yes	complemented
50	C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
51	Dic₅	—	2	2	no	—
52	Dic₅	#127	1	1	yes	characteristic
53	Dic₅	C₄⋊Q₈	1	1	yes	characteristic
54	C₂²×C₄	—	5	1	no	—
55	C₂²×C₄	—	5	2	no	—
56	M₁₆	—	10	8	no	—
57	C₂×C₈	—	5	2	no	—
58	C₂×C₈	—	5	4	no	—
59	C₂×C₈	—	5	4	no	—
60	C₂×C₈	—	10	4	no	—
61	C₂×C₈	C₂×AGL(1,𝔽₅)	1	2	yes	—
62	C₄²	—	5	2	no	—
63	C₄²	—	5	1	no	—
64	C₄²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
65	C₂×C₅	#384	1	1	yes	characteristic
66	C₂×C₅	C₂×#108	1	2	yes	—
67	D₅	—	2	2	no	—
68	C₂³	—	5	1	no	—
69	C₂×C₄	—	5	1	no	—
70	C₂×C₄	—	5	2	no	—
71	C₂×C₄	—	5	2	no	—
72	C₂×C₄	—	5	2	no	—
73	C₂×C₄	—	10	4	no	—
74	C₂×C₄	C₄⋊AGL(1,𝔽₅)	1	2	yes	—
75	C₂×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
76	C₈	—	5	4	no	—
77	C₈	—	20	4	no	—
78	C₈	C₄⋊AGL(1,𝔽₅)	1	4	yes	—
79	C₅	#1848	1	1	yes	characteristic, complemented, Sylow
80	C₂²	—	5	2	no	—
81	C₂²	—	10	2	no	—
82	C₂²	C₂×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
83	C₄	—	5	1	no	—
84	C₄	—	5	1	no	—
85	C₄	—	10	2	no	—
86	C₄	#3995	1	4	yes	complemented
87	C₄	C₂×C₄⋊AGL(1,𝔽₅)	1	2	yes	—
88	C₂	—	10	2	no	—
89	C₂	#65688	1	1	yes	characteristic
90	C₂	C₂×#3995	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×C₄⋊AGL(1,𝔽₅)
commutator	C₂×C₄×C₅	C₂²×C₄
Frattini	C₂×C₄	C₂²×AGL(1,𝔽₅)
Fitting	C₄×C₅×C₈	C₄
socle	C₂²×C₅	C₂×C₄⋊C₄
prime radical	C₂²×D₅	C₂×Q₈
cosocle	C₂×C₄×D₅	C₂³
hypercenter	C₄×C₈	AGL(1,𝔽₅)
hypocenter	C₅	#1848
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
49. C₄×C₅	#108	semidirect
79. C₅	#1848	semidirect
86. C₄	#3995	semidirect

Maximal subgroups

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

subgroup	index
C₂×#3995	2
#65681	2
C₄×C₈×D₅	2
#1848	5