Subgroups — #10599875

Order 640.

148 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	C₄×Dic₂₀	C₂	1	2	yes	maximal
3	C₅×#346	C₂	1	1	yes	maximal, characteristic
4	#64725	C₂	1	1	yes	maximal, characteristic
5	#64723	C₂	1	2	yes	maximal
6	C₂×Dic₂₀	C₂²	1	2	yes	—
7	C₄×Dic₁₀	—	2	2	no	—
8	C₂×C₅×C₁₆	C₂²	1	1	yes	characteristic
9	C₂×C₅×C₁₆	C₂²	1	1	yes	characteristic
10	C₄×C₅×C₈	C₂²	1	1	yes	characteristic
11	#3950	C₂²	1	2	yes	—
12	#3948	—	2	2	no	—
13	Dic₄₀	C₄	1	4	yes	complemented
14	#2176	—	5	1	no	maximal, Sylow
15	C₂×Dic₁₀	—	2	2	no	—
16	C₂×C₅×C₈	D₄	1	1	yes	characteristic
17	C₂×C₅×C₈	C₂³	1	1	yes	characteristic
18	C₄²×C₅	D₄	1	1	yes	characteristic
19	C₄⋊Dic₅	—	2	2	no	—
20	#683	—	4	2	no	—
21	C₄×Dic₅	—	4	2	no	—
22	Dic₂₀	—	2	2	no	—
23	Dic₂₀	C₂×C₄	1	4	yes	—
24	C₅×C₁₆	—	2	1	no	—
25	C₅×C₁₆	C₂×C₄	1	2	yes	—
26	C₂×Q₃₂	—	5	1	no	—
27	C₄×Q₁₆	—	5	2	no	—
28	#367	—	5	1	no	—
29	#358	—	5	2	no	—
30	#346	D₅	1	1	yes	characteristic
31	C₂×C₄×C₅	D₈	1	2	yes	—
32	C₂×C₄×C₅	C₂×D₄	1	1	yes	characteristic
33	C₂×Dic₅	—	4	2	no	—
34	C₂×Dic₅	—	4	2	no	—
35	Dic₁₀	—	2	4	no	—
36	Dic₁₀	—	4	2	no	—
37	C₅×C₈	—	2	1	no	—
38	C₅×C₈	C₂²×C₄	1	1	yes	characteristic
39	C₅×C₈	#41	1	1	yes	characteristic
40	C₂×Q₁₆	—	5	2	no	—
41	C₄×Q₈	—	10	2	no	—
42	Q₃₂	—	5	4	no	—
43	C₂×C₁₆	C₂×D₅	1	1	yes	characteristic
44	C₂×C₁₆	C₂×D₅	1	1	yes	characteristic
45	#107	—	5	2	no	—
46	#103	—	10	2	no	—
47	C₄×C₈	C₂×D₅	1	1	yes	characteristic
48	C₂²×C₅	C₂×D₈	1	1	yes	characteristic
49	C₄×C₅	—	2	2	no	—
50	C₄×C₅	C₄×D₄	1	1	yes	characteristic
51	C₄×C₅	#135	1	1	yes	characteristic
52	Dic₅	—	4	4	no	—
53	Dic₅	—	8	2	no	—
54	C₂×Q₈	—	10	2	no	—
55	Q₁₆	—	5	4	no	—
56	Q₁₆	—	10	2	no	—
57	C₂×C₈	D₂₀	1	1	yes	characteristic
58	C₂×C₈	C₂²×D₅	1	1	yes	characteristic
59	C₄⋊C₄	—	10	2	no	—
60	C₄⋊C₄	—	20	2	no	—
61	C₄²	—	20	2	no	—
62	C₄²	D₂₀	1	1	yes	characteristic
63	C₁₆	—	2	1	no	—
64	C₁₆	C₄×D₅	1	2	yes	—
65	C₂×C₅	C₄×D₈	1	1	yes	characteristic
66	C₂×C₅	#510	1	2	yes	—
67	Q₈	—	10	4	no	—
68	Q₈	—	20	2	no	—
69	C₂×C₄	—	20	2	no	—
70	C₂×C₄	—	20	2	no	—
71	C₂×C₄	D₄₀	1	2	yes	—
72	C₂×C₄	C₂×D₂₀	1	1	yes	characteristic
73	C₈	—	2	1	no	—
74	C₈	C₂×C₄×D₅	1	1	yes	characteristic
75	C₈	#709	1	1	yes	characteristic
76	C₅	#2176	1	1	yes	characteristic, complemented, Sylow
77	C₂²	C₂×D₄₀	1	1	yes	characteristic
78	C₄	—	2	2	no	—
79	C₄	—	20	4	no	—
80	C₄	—	40	2	no	—
81	C₄	C₄×D₂₀	1	1	yes	characteristic
82	C₄	#4050	1	1	yes	characteristic
83	C₂	C₄×D₄₀	1	1	yes	characteristic
84	C₂	#65198	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₄₀
commutator	C₅×C₈	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₅
Fitting	C₅×#346	C₂
socle, prime radical	C₂²×C₅	C₂×D₈
cosocle	C₂×C₅×C₈	C₂³
hypercenter	#346	D₅
hypocenter	C₅	#2176
LCS	G ▷ C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #346

Complements

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

N	H ≅ G/N	type
13. Dic₄₀	C₄	semidirect
76. C₅	#2176	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
C₄×Dic₂₀	2
C₅×#346	2
#64725	2
#64723	2
#2176	5