Subgroups — #10604111

Order 640.

106 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#65196	C₂	1	1	yes	maximal, characteristic
2	#64877	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#3928	C₂	1	1	yes	maximal, characteristic, complemented
4	#64842	C₂	1	2	yes	maximal, complemented
5	D₅⋊C₃₂	C₂	1	2	yes	maximal, complemented
6	#4047	C₄	1	1	yes	characteristic
7	C₅×M₃₂	C₄	1	1	yes	characteristic
8	C₂×#672	C₂²	1	1	yes	characteristic
9	#3930	C₄	1	2	yes	—
10	D₅×C₁₆	C₂²	1	2	yes	—
11	#3928	C₂²	1	1	yes	characteristic
12	#3928	C₂²	1	2	yes	—
13	#3928	C₂²	1	1	yes	characteristic
14	#2258	—	5	1	no	maximal, Sylow
15	#709	C₈	1	1	yes	characteristic
16	C₂×C₅×C₈	C₂×C₄	1	1	yes	characteristic
17	#681	C₈	1	1	yes	characteristic
18	#676	C₈	1	2	yes	—
19	C₈×D₅	C₂×C₄	1	2	yes	—
20	C₅×C₁₆	C₂×C₄	1	2	yes	—
21	#672	C₂×C₄	1	1	yes	characteristic
22	#672	C₂³	1	1	yes	characteristic
23	#504	—	5	1	no	—
24	M₆₄	—	5	2	no	—
25	M₆₄	—	5	1	no	—
26	C₂×C₃₂	—	5	2	no	—
27	C₂×C₃₂	—	5	1	no	—
28	C₂×C₄×C₅	C₂×C₈	1	1	yes	characteristic
29	C₂²⋊D₅	C₁₆	1	2	yes	—
30	D₂₀	C₁₆	1	1	yes	characteristic
31	C₄×D₅	C₂×C₈	1	2	yes	—
32	Dic₁₀	C₁₆	1	1	yes	characteristic
33	C₅×C₈	C₂×C₈	1	1	yes	characteristic
34	C₅×C₈	C₂²×C₄	1	1	yes	characteristic
35	#168	C₂×C₈	1	2	yes	—
36	#131	—	5	1	no	—
37	M₃₂	—	5	2	no	—
38	M₃₂	AGL(1,𝔽₅)	1	1	yes	characteristic
39	C₂×C₁₆	—	5	1	no	—
40	C₂×C₁₆	—	5	2	no	—
41	C₃₂	—	5	1	no	—
42	C₃₂	—	5	2	no	—
43	C₃₂	—	5	1	no	—
44	C₂²×C₅	C₂×C₁₆	1	1	yes	characteristic
45	C₂×D₅	C₂×C₁₆	1	2	yes	—
46	C₄×C₅	C₂×C₁₆	1	1	yes	characteristic
47	C₄×C₅	C₂²×C₈	1	1	yes	characteristic
48	Dic₅	C₂×C₁₆	1	2	yes	—
49	#41	—	5	1	no	—
50	M₁₆	—	5	1	no	—
51	M₁₆	—	5	2	no	—
52	C₂×C₈	—	5	2	no	—
53	C₂×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
54	C₁₆	—	5	1	no	—
55	C₁₆	—	5	1	no	—
56	C₁₆	C₂×AGL(1,𝔽₅)	1	2	yes	—
57	C₂×C₅	—	2	1	no	—
58	C₂×C₅	C₂²×C₁₆	1	1	yes	characteristic
59	D₅	—	2	2	no	—
60	Q₈	—	5	1	no	—
61	D₄	—	5	2	no	—
62	D₄	—	5	1	no	—
63	C₂×C₄	—	5	2	no	—
64	C₂×C₄	D₅⋊C₈	1	1	yes	characteristic
65	C₈	—	5	2	no	—
66	C₈	D₅⋊C₈	1	1	yes	characteristic
67	C₈	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
68	C₅	#2258	1	1	yes	characteristic, complemented, Sylow
69	C₂²	—	5	2	no	—
70	C₂²	D₅⋊C₁₆	1	1	yes	characteristic
71	C₄	—	5	2	no	—
72	C₄	D₅⋊C₁₆	1	1	yes	characteristic
73	C₄	C₂×D₅⋊C₈	1	1	yes	characteristic
74	C₂	—	2	1	no	—
75	C₂	—	10	2	no	—
76	C₂	C₂×D₅⋊C₁₆	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₈	C₂²×AGL(1,𝔽₅)
commutator, socle	C₂×C₅	C₂²×C₁₆
Fitting	C₅×M₃₂	C₄
prime radical	#709	C₈
cosocle	#672	C₂³
hypercenter	M₃₂	AGL(1,𝔽₅)
hypocenter	C₅	#2258
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₈ ◁ M₃₂

Complements

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

N	H ≅ G/N	type
2. #64877	C₂	semidirect
3. C₂×#3928	C₂	semidirect
4. #64842	C₂	semidirect
5. D₅⋊C₃₂	C₂	semidirect
68. C₅	#2258	semidirect

Maximal subgroups

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

subgroup	index
#65196	2
#64877	2
C₂×#3928	2
#64842	2
D₅⋊C₃₂	2
#2258	5