Subgroups — #10602222

Order 640.

140 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×M₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂×#3951	C₂	1	2	yes	maximal
3	#64726	C₂	1	4	yes	maximal, complemented
4	C₂²×C₅×C₈	C₂²	1	1	yes	characteristic
5	C₂×#681	—	2	2	no	—
6	C₅×M₃₂	C₄	1	4	yes	—
7	C₂×C₅×C₁₆	C₂²	1	2	yes	—
8	#3951	—	2	2	no	—
9	#3951	C₂²	1	4	yes	—
10	#2161	—	5	1	no	maximal, Sylow
11	C₂²×C₄×C₅	D₄	1	1	yes	characteristic
12	C₂×C₅×C₈	C₂×C₄	1	2	yes	—
13	C₂×C₅×C₈	D₄	1	1	yes	characteristic
14	C₂×C₅×C₈	Q₈	1	2	yes	—
15	C₂×C₅×C₈	C₂³	1	1	yes	characteristic
16	#681	—	2	4	no	—
17	#681	—	4	2	no	—
18	C₂×#168	—	4	2	no	—
19	C₅×C₁₆	C₂×C₄	1	4	yes	—
20	C₂×M₃₂	D₅	1	1	yes	characteristic
21	C₂×#108	—	5	2	no	—
22	#368	—	5	4	no	—
23	C₂³×C₅	Q₁₆	1	1	yes	characteristic
24	C₂×C₄×C₅	C₄⋊C₄	1	2	yes	—
25	C₂×C₄×C₅	D₈	1	2	yes	—
26	C₂×C₄×C₅	Q₁₆	1	1	yes	characteristic
27	C₂×C₄×C₅	C₂×D₄	1	1	yes	characteristic
28	C₅×C₈	C₄⋊C₄	1	2	yes	—
29	C₅×C₈	C₂²×C₄	1	1	yes	characteristic
30	C₅×C₈	C₂×Q₈	1	1	yes	characteristic
31	#168	—	4	4	no	—
32	C₂×M₁₆	—	10	2	no	—
33	C₂²×C₈	C₂×D₅	1	1	yes	characteristic
34	M₃₂	Dic₅	1	4	yes	—
35	C₂×C₁₆	C₂×D₅	1	2	yes	—
36	#108	—	5	4	no	—
37	#108	—	10	2	no	—
38	C₂²×C₅	—	4	1	no	—
39	C₂²×C₅	#107	1	2	yes	—
40	C₂²×C₅	C₂×Q₁₆	1	1	yes	characteristic
41	C₄×C₅	#107	1	2	yes	—
42	C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
43	C₄×C₅	C₂×D₈	1	1	yes	characteristic
44	C₂²×C₄	D₂₀	1	1	yes	characteristic
45	M₁₆	—	10	4	no	—
46	M₁₆	—	20	2	no	—
47	C₂×C₈	—	20	2	no	—
48	C₂×C₈	Dic₁₀	1	2	yes	—
49	C₂×C₈	D₂₀	1	1	yes	characteristic
50	C₂×C₈	C₂×Dic₅	1	2	yes	—
51	C₂×C₈	C₂²×D₅	1	1	yes	characteristic
52	C₁₆	C₂×Dic₅	1	4	yes	—
53	C₂×C₅	—	2	2	no	—
54	C₂×C₅	—	2	1	no	—
55	C₂×C₅	C₂×#107	1	1	yes	characteristic
56	C₂³	Dic₂₀	1	1	yes	characteristic
57	C₂×C₄	D₄₀	1	2	yes	—
58	C₂×C₄	Dic₂₀	1	1	yes	characteristic
59	C₂×C₄	C₄⋊Dic₅	1	2	yes	—
60	C₂×C₄	C₂×D₂₀	1	1	yes	characteristic
61	C₈	—	20	4	no	—
62	C₈	C₄⋊Dic₅	1	2	yes	—
63	C₈	C₂×Dic₁₀	1	1	yes	characteristic
64	C₈	C₂²×Dic₅	1	1	yes	characteristic
65	C₅	#2161	1	1	yes	characteristic, complemented, Sylow
66	C₂²	—	4	1	no	—
67	C₂²	#3950	1	2	yes	—
68	C₂²	C₂×Dic₂₀	1	1	yes	characteristic
69	C₄	#3950	1	2	yes	—
70	C₄	C₂×D₄₀	1	1	yes	characteristic
71	C₄	C₂×C₄⋊Dic₅	1	1	yes	characteristic
72	C₂	—	2	2	no	—
73	C₂	—	2	1	no	—
74	C₂	C₂×#3950	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×D₄₀
commutator	C₅×C₈	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₅
Fitting	C₂×C₅×M₃₂	C₂
socle	C₂×C₅	C₂×#107
prime radical	C₂³×C₅	Q₁₆
cosocle	C₂×C₅×C₈	C₂³
hypercenter	C₂×M₃₂	D₅
hypocenter	C₅	#2161
LCS	G ▷ C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×M₃₂

Complements

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

N	H ≅ G/N	type
3. #64726	C₂	semidirect
65. C₅	#2161	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×M₃₂	2
C₂×#3951	2
#64726	2
#2161	5