Subgroups — #10604689

Order 640.

196 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#65748	C₂	1	2	yes	maximal
2	C₂×C₈×Dic₅	C₂	1	1	yes	maximal, characteristic
3	#170⋊C₈	C₂	1	4	yes	maximal, complemented
4	C₂²×C₅×C₈	C₄	1	1	yes	characteristic
5	C₂×C₄×Dic₅	C₂²	1	1	yes	characteristic
6	C₂²×#168	C₄	1	1	yes	characteristic
7	C₂²⋊#170	C₄	1	4	yes	—
8	C₄⋊#170	C₄	1	4	yes	—
9	C₄×#170	C₂²	1	4	yes	—
10	C₈×Dic₅	C₄	1	2	yes	—
11	C₈×Dic₅	C₂²	1	2	yes	—
12	#1452	—	5	1	no	maximal, Sylow
13	C₂²×C₄×C₅	C₂×C₄	1	1	yes	characteristic
14	C₂²×Dic₅	C₂×C₄	1	2	yes	—
15	C₂×#170	C₂×C₄	1	8	yes	—
16	C₂×C₅×C₈	—	2	2	no	—
17	C₂×C₅×C₈	C₂×C₄	1	2	yes	—
18	C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
19	C₄×Dic₅	C₂×C₄	1	2	yes	—
20	C₄×Dic₅	C₂³	1	1	yes	characteristic
21	C₂×#168	—	2	2	no	—
22	C₂×#168	C₂×C₄	1	2	yes	—
23	#431	—	5	2	no	—
24	C₂×C₄×C₈	—	5	1	no	—
25	#322	—	5	4	no	—
26	C₂³×C₅	C₄²	1	1	yes	characteristic
27	C₂×C₄×C₅	C₄²	1	1	yes	characteristic
28	C₂×C₄×C₅	M₁₆	1	2	yes	—
29	C₂×C₄×C₅	M₁₆	1	2	yes	—
30	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
31	C₂×Dic₅	—	2	4	no	—
32	C₂×Dic₅	C₄²	1	2	yes	—
33	C₂×Dic₅	C₂²×C₄	1	2	yes	—
34	#170	—	2	8	no	—
35	C₅×C₈	—	2	2	no	—
36	#168	—	2	2	no	—
37	C₂²×C₈	—	5	1	no	—
38	C₂²×C₈	AGL(1,𝔽₅)	1	1	yes	characteristic
39	C₂×C₄²	—	5	1	no	—
40	C₄⋊C₈	—	5	4	no	—
41	C₂²⋊C₈	—	5	4	no	—
42	C₄×C₈	—	5	4	no	—
43	C₄×C₈	—	5	2	no	—
44	C₄×C₈	—	5	2	no	—
45	C₂²×C₅	—	2	2	no	—
46	C₂²×C₅	#97	1	2	yes	—
47	C₂²×C₅	C₂×C₄²	1	1	yes	characteristic
48	C₄×C₅	#97	1	2	yes	—
49	C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
50	C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
51	Dic₅	—	2	2	no	—
52	Dic₅	#131	1	4	yes	—
53	C₂²×C₄	—	5	2	no	—
54	C₂²×C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
55	C₂×C₈	—	2	2	no	—
56	C₂×C₈	—	5	8	no	—
57	C₂×C₈	—	5	2	no	—
58	C₂×C₈	—	10	2	no	—
59	C₂×C₈	C₂×AGL(1,𝔽₅)	1	2	yes	—
60	C₄²	—	5	2	no	—
61	C₄²	—	5	1	no	—
62	C₄²	—	5	1	no	—
63	C₂×C₅	—	2	2	no	—
64	C₂×C₅	C₂×#97	1	1	yes	characteristic
65	C₂×C₅	#405	1	2	yes	—
66	C₂³	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
67	C₂×C₄	—	5	2	no	—
68	C₂×C₄	—	5	2	no	—
69	C₂×C₄	—	10	4	no	—
70	C₂×C₄	#700	1	2	yes	—
71	C₂×C₄	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
72	C₂×C₄	#704	1	2	yes	—
73	C₂×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
74	C₈	—	2	2	no	—
75	C₈	—	10	8	no	—
76	C₈	—	10	2	no	—
77	C₅	#1452	1	1	yes	characteristic, complemented, Sylow
78	C₂²	—	2	2	no	—
79	C₂²	#4002	1	2	yes	—
80	C₂²	C₂×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
81	C₄	—	5	4	no	—
82	C₄	—	10	2	no	—
83	C₄	#4002	1	2	yes	—
84	C₄	C₂×#700	1	1	yes	characteristic
85	C₄	C₂×#704	1	1	yes	characteristic
86	C₂	—	2	2	no	—
87	C₂	#65718	1	2	yes	—
88	C₂	C₂×#4002	1	1	yes	characteristic

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

Complements

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

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

Maximal subgroups

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

subgroup	index
#65748	2
C₂×C₈×Dic₅	2
#170⋊C₈	2
#1452	5