Subgroups — #10604024

Order 640.

134 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#65119	C₂	1	1	yes	maximal, characteristic
2	#64888	C₂	1	2	yes	maximal
3	#64885	C₂	1	2	yes	maximal
4	C₄⋊#674	C₂	1	1	yes	maximal, characteristic
5	C₄×#674	C₂	1	1	yes	maximal, characteristic
6	C₄×Dic₁₀	C₄	1	1	yes	characteristic
7	C₂×#674	C₂²	1	1	yes	characteristic
8	C₂×#674	C₂²	1	2	yes	—
9	C₂×#674	C₂²	1	1	yes	characteristic
10	C₅×C₄⋊C₈	C₄	1	1	yes	characteristic
11	#3946	C₄	1	2	yes	—
12	C₈×Dic₅	C₂²	1	2	yes	—
13	C₄×#168	C₂²	1	1	yes	characteristic
14	Q₈×C₁₆	—	5	1	no	maximal, Sylow
15	C₂×Dic₁₀	C₈	1	1	yes	characteristic
16	C₂×C₅×C₈	C₂×C₄	1	2	yes	—
17	C₄²×C₅	C₂×C₄	1	1	yes	characteristic
18	C₄⋊Dic₅	C₈	1	1	yes	characteristic
19	#683	C₈	1	2	yes	—
20	C₄×Dic₅	C₂×C₄	1	2	yes	—
21	C₂×#168	C₂×C₄	1	1	yes	characteristic
22	C₂×#168	C₂³	1	1	yes	characteristic
23	#674	—	2	1	no	—
24	#674	—	2	2	no	—
25	#674	Q₈	1	2	yes	complemented
26	C₈×Q₈	—	5	1	no	—
27	C₄⋊C₁₆	—	5	1	no	—
28	C₄⋊C₁₆	—	5	2	no	—
29	C₄×C₁₆	—	5	1	no	—
30	C₄×C₁₆	—	5	2	no	—
31	C₂×C₄×C₅	C₂×C₈	1	1	yes	characteristic
32	C₂×C₄×C₅	C₂×C₈	1	1	yes	characteristic
33	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
34	C₂×Dic₅	C₂×C₈	1	2	yes	—
35	C₂×Dic₅	C₂×C₈	1	2	yes	—
36	Dic₁₀	C₁₆	1	4	yes	complemented
37	C₅×C₈	—	2	2	no	—
38	#168	—	2	1	no	—
39	#168	C₂×Q₈	1	1	yes	characteristic
40	#168	#41	1	1	yes	characteristic
41	C₄×Q₈	—	5	1	no	—
42	C₂×C₁₆	—	5	1	no	—
43	C₂×C₁₆	—	5	1	no	—
44	C₂×C₁₆	—	5	2	no	—
45	C₄⋊C₈	—	5	2	no	—
46	C₄⋊C₈	AGL(1,𝔽₅)	1	1	yes	characteristic
47	C₄×C₈	—	5	2	no	—
48	C₄×C₈	—	5	1	no	—
49	C₂²×C₅	C₂²×C₈	1	1	yes	characteristic
50	C₄×C₅	—	2	1	no	—
51	C₄×C₅	C₂×C₁₆	1	2	yes	—
52	C₄×C₅	C₄×Q₈	1	1	yes	characteristic
53	C₄×C₅	#131	1	1	yes	characteristic
54	Dic₅	—	2	2	no	—
55	Dic₅	C₂×C₁₆	1	4	yes	—
56	C₂×Q₈	—	5	1	no	—
57	C₂×C₈	—	5	1	no	—
58	C₂×C₈	—	5	1	no	—
59	C₂×C₈	C₂×AGL(1,𝔽₅)	1	2	yes	—
60	C₄⋊C₄	—	5	2	no	—
61	C₄⋊C₄	—	5	1	no	—
62	C₄²	—	5	2	no	—
63	C₄²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
64	C₁₆	—	5	2	no	—
65	C₁₆	—	10	1	no	—
66	C₁₆	—	10	2	no	—
67	C₂×C₅	C₈×Q₈	1	1	yes	characteristic
68	C₂×C₅	C₂²×C₁₆	1	1	yes	characteristic
69	C₂×C₅	#504	1	1	yes	characteristic
70	Q₈	—	5	4	no	—
71	C₂×C₄	—	5	2	no	—
72	C₂×C₄	—	5	2	no	—
73	C₂×C₄	D₅⋊C₈	1	1	yes	characteristic
74	C₂×C₄	D₅⋊C₈	1	1	yes	characteristic
75	C₂×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
76	C₈	—	2	2	no	—
77	C₈	—	5	1	no	—
78	C₈	—	5	1	no	—
79	C₈	—	10	1	no	—
80	C₅	Q₈×C₁₆	1	1	yes	characteristic, complemented, Sylow
81	C₂²	C₂×D₅⋊C₈	1	1	yes	characteristic
82	C₄	—	2	1	no	—
83	C₄	—	5	4	no	—
84	C₄	—	10	2	no	—
85	C₄	D₅⋊C₁₆	1	2	yes	—
86	C₄	#4131	1	1	yes	characteristic
87	C₄	Q₈×AGL(1,𝔽₅)	1	1	yes	characteristic
88	C₂	#65703	1	1	yes	characteristic
89	C₂	C₂×D₅⋊C₁₆	1	1	yes	characteristic
90	C₂	#65725	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×AGL(1,𝔽₅)
commutator	C₂×C₅	C₂²×C₁₆
Fitting	C₅×C₄⋊C₈	C₄
socle, prime radical	C₂²×C₅	C₂²×C₈
cosocle	C₂×#168	C₂³
hypercenter	C₄⋊C₈	AGL(1,𝔽₅)
hypocenter	C₅	Q₈×C₁₆
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
25. #674	Q₈	semidirect
36. Dic₁₀	C₁₆	semidirect
80. C₅	Q₈×C₁₆	semidirect

Maximal subgroups

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

subgroup	index
#65119	2
#64888	2
#64885	2
C₄⋊#674	2
C₄×#674	2
Q₈×C₁₆	5