Subgroups — #10603811

Order 640.

196 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#65678	C₂	1	2	yes	maximal
2	C₄×C₈×D₅	C₂	1	1	yes	maximal, characteristic
3	#170⋊C₈	C₂	1	4	yes	maximal, complemented
4	C₂×C₈×D₅	C₄	1	2	yes	—
5	C₄²×D₅	C₂²	1	1	yes	characteristic
6	#4004	C₄	1	4	yes	—
7	#4003	C₄	1	4	yes	—
8	C₄×#170	C₂²	1	4	yes	—
9	C₄×C₅×C₈	C₄	1	1	yes	characteristic
10	C₈×Dic₅	C₂²	1	2	yes	—
11	C₄×#168	C₄	1	1	yes	characteristic
12	#1452	—	5	1	no	maximal, Sylow
13	C₂×C₄×D₅	C₂×C₄	1	2	yes	—
14	C₂×C₄×D₅	C₂×C₄	1	1	yes	characteristic
15	C₂×#170	C₂×C₄	1	8	yes	—
16	C₂×C₅×C₈	C₂×C₄	1	2	yes	—
17	C₄²×C₅	C₂×C₄	1	1	yes	characteristic
18	C₄×Dic₅	C₂×C₄	1	2	yes	—
19	C₄×Dic₅	C₂³	1	1	yes	characteristic
20	C₂×#168	C₂×C₄	1	2	yes	—
21	C₈×D₅	—	2	4	no	—
22	#431	—	5	2	no	—
23	C₂×C₄×C₈	—	5	1	no	—
24	#322	—	5	4	no	—
25	C₂²×D₅	C₄²	1	1	yes	characteristic
26	C₂×C₄×C₅	C₄²	1	2	yes	—
27	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
28	C₂×Dic₅	C₄²	1	1	yes	characteristic
29	C₂×Dic₅	C₂²×C₄	1	2	yes	—
30	C₄×D₅	—	2	4	no	—
31	C₄×D₅	M₁₆	1	4	yes	—
32	#170	—	2	8	no	—
33	C₅×C₈	—	2	2	no	—
34	#168	—	2	2	no	—
35	C₂²×C₈	—	5	2	no	—
36	C₂×C₄²	—	5	1	no	—
37	C₄⋊C₈	—	5	4	no	—
38	C₂²⋊C₈	—	5	4	no	—
39	C₄×C₈	—	5	4	no	—
40	C₄×C₈	—	5	2	no	—
41	C₄×C₈	—	5	1	no	—
42	C₄×C₈	AGL(1,𝔽₅)	1	1	yes	characteristic
43	C₂²×C₅	C₂×C₄²	1	1	yes	characteristic
44	C₂×D₅	—	2	2	no	—
45	C₂×D₅	#97	1	2	yes	—
46	C₄×C₅	—	2	2	no	—
47	C₄×C₅	C₂×M₁₆	1	2	yes	—
48	Dic₅	#97	1	2	yes	—
49	Dic₅	#131	1	4	yes	—
50	C₂²×C₄	—	5	1	no	—
51	C₂²×C₄	—	5	2	no	—
52	C₂×C₈	—	5	8	no	—
53	C₂×C₈	—	5	2	no	—
54	C₂×C₈	—	10	4	no	—
55	C₂×C₈	C₂×AGL(1,𝔽₅)	1	2	yes	—
56	C₄²	—	5	2	no	—
57	C₄²	—	5	1	no	—
58	C₄²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
59	C₂×C₅	C₂×#97	1	1	yes	characteristic
60	C₂×C₅	#405	1	2	yes	—
61	D₅	—	2	2	no	—
62	C₂³	—	5	1	no	—
63	C₂×C₄	—	5	2	no	—
64	C₂×C₄	—	5	1	no	—
65	C₂×C₄	—	5	4	no	—
66	C₂×C₄	—	10	4	no	—
67	C₂×C₄	C₄×AGL(1,𝔽₅)	1	2	yes	—
68	C₂×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
69	C₈	—	2	2	no	—
70	C₈	—	10	8	no	—
71	C₈	—	10	2	no	—
72	C₅	#1452	1	1	yes	characteristic, complemented, Sylow
73	C₂²	—	5	2	no	—
74	C₂²	—	10	2	no	—
75	C₂²	C₂×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
76	C₄	—	2	2	no	—
77	C₄	—	5	2	no	—
78	C₄	—	5	4	no	—
79	C₄	D₅⋊M₁₆	1	2	yes	—
80	C₂	—	10	2	no	—
81	C₂	#65676	1	1	yes	characteristic
82	C₂	#65718	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×AGL(1,𝔽₅)
commutator, socle	C₂²×C₅	C₂×C₄²
Fitting	C₄×C₅×C₈	C₄
prime radical	C₂²×D₅	C₄²
cosocle	C₄×Dic₅	C₂³
hypercenter	C₄×C₈	AGL(1,𝔽₅)
hypocenter	C₅	#1452
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
3. #170⋊C₈	C₂	semidirect
72. C₅	#1452	semidirect

Maximal subgroups

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

subgroup	index
#65678	2
C₄×C₈×D₅	2
#170⋊C₈	2
#1452	5