Subgroups — #10600111

Order 640.

150 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₄₀	C₂	1	1	yes	maximal, characteristic
2	#65162	C₂	1	2	yes	maximal
3	C₅×#366	C₂	1	1	yes	maximal, characteristic
4	C₁₆×Dic₅	C₂	1	1	yes	maximal, characteristic
5	#64712	C₂	1	2	yes	maximal
6	C₂×Dic₂₀	C₂²	1	2	yes	—
7	#4033	—	2	2	no	—
8	C₂×C₅×C₁₆	C₂²	1	1	yes	characteristic
9	C₅×#107	C₂²	1	2	yes	—
10	C₈×Dic₅	C₂²	1	1	yes	characteristic
11	C₂×#672	C₂²	1	1	yes	characteristic
12	#3942	—	2	2	no	—
13	Dic₄₀	C₄	1	4	yes	complemented
14	C₄×Q₃₂	—	5	1	no	maximal, Sylow
15	C₂×Dic₁₀	—	2	2	no	—
16	C₂×C₅×C₈	C₂³	1	1	yes	characteristic
17	C₅×C₄⋊C₄	—	2	2	no	—
18	#683	—	4	2	no	—
19	C₄×Dic₅	—	4	2	no	—
20	C₄×Dic₅	D₄	1	1	yes	characteristic
21	C₂×#168	D₄	1	1	yes	characteristic
22	Dic₂₀	—	2	2	no	—
23	Dic₂₀	C₂×C₄	1	4	yes	—
24	C₅×C₁₆	C₂×C₄	1	2	yes	—
25	#672	—	2	1	no	—
26	C₂×Q₃₂	—	5	1	no	—
27	C₄×Q₁₆	—	5	2	no	—
28	#366	D₅	1	1	yes	characteristic
29	#358	—	5	2	no	—
30	C₄×C₁₆	—	5	1	no	—
31	C₂×C₄×C₅	—	4	2	no	—
32	C₂×C₄×C₅	C₂×D₄	1	1	yes	characteristic
33	C₂×Dic₅	—	4	2	no	—
34	C₂×Dic₅	D₈	1	1	yes	characteristic
35	C₂×Dic₅	D₈	1	1	yes	characteristic
36	Dic₁₀	—	2	4	no	—
37	Dic₁₀	—	4	2	no	—
38	C₅×C₈	C₂²×C₄	1	1	yes	characteristic
39	C₅×C₈	#41	1	1	yes	characteristic
40	#168	—	2	1	no	—
41	C₂×Q₁₆	—	5	2	no	—
42	C₄×Q₈	—	10	2	no	—
43	Q₃₂	—	5	4	no	—
44	C₂×C₁₆	—	5	1	no	—
45	C₂×C₁₆	C₂×D₅	1	1	yes	characteristic
46	#107	C₂×D₅	1	2	yes	—
47	#103	—	10	2	no	—
48	C₄×C₈	—	5	1	no	—
49	C₂²×C₅	C₂×D₈	1	1	yes	characteristic
50	C₄×C₅	—	8	2	no	—
51	C₄×C₅	C₄×D₄	1	1	yes	characteristic
52	C₄×C₅	#135	1	1	yes	characteristic
53	Dic₅	—	2	1	no	—
54	Dic₅	—	4	4	no	—
55	Dic₅	Q₃₂	1	2	yes	complemented
56	C₂×Q₈	—	10	2	no	—
57	Q₁₆	—	5	4	no	—
58	Q₁₆	—	10	2	no	—
59	C₂×C₈	—	5	1	no	—
60	C₂×C₈	C₂²×D₅	1	1	yes	characteristic
61	C₄⋊C₄	—	2	2	no	—
62	C₄⋊C₄	—	20	2	no	—
63	C₄²	—	5	1	no	—
64	C₄²	—	20	2	no	—
65	C₁₆	—	10	1	no	—
66	C₁₆	C₄×D₅	1	2	yes	—
67	C₂×C₅	C₄×D₈	1	1	yes	characteristic
68	C₂×C₅	C₂×Q₃₂	1	1	yes	characteristic
69	C₂×C₅	#508	1	1	yes	characteristic
70	Q₈	—	10	4	no	—
71	Q₈	—	20	2	no	—
72	C₂×C₄	—	4	2	no	—
73	C₂×C₄	—	5	1	no	—
74	C₂×C₄	—	5	1	no	—
75	C₂×C₄	—	20	2	no	—
76	C₂×C₄	D₄×D₅	1	1	yes	characteristic
77	C₈	—	10	1	no	—
78	C₈	C₂×C₄×D₅	1	1	yes	characteristic
79	C₈	#713	1	1	yes	characteristic
80	C₅	C₄×Q₃₂	1	1	yes	characteristic, complemented, Sylow
81	C₂²	D₅×D₈	1	1	yes	characteristic
82	C₄	—	5	2	no	—
83	C₄	—	8	2	no	—
84	C₄	—	10	1	no	—
85	C₄	—	20	4	no	—
86	C₄	#4039	1	1	yes	characteristic
87	C₄	#4065	1	1	yes	characteristic
88	C₂	#65161	1	1	yes	characteristic
89	C₂	#65201	1	1	yes	characteristic
90	C₂	D₅×Q₃₂	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₅×D₈
commutator	C₅×C₈	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₅
Fitting	C₅×#366	C₂
socle, prime radical	C₂²×C₅	C₂×D₈
cosocle	C₂×C₅×C₈	C₂³
hypercenter	#366	D₅
hypocenter	C₅	C₄×Q₃₂
LCS	G ▷ C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #366

Complements

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

N	H ≅ G/N	type
13. Dic₄₀	C₄	semidirect
55. Dic₅	Q₃₂	semidirect
80. C₅	C₄×Q₃₂	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₄₀	2
#65162	2
C₅×#366	2
C₁₆×Dic₅	2
#64712	2
C₄×Q₃₂	5