Subgroups — #10604708

Order 640.

172 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#3998	C₂	1	2	yes	maximal
2	C₂×C₈×Dic₅	C₂	1	1	yes	maximal, characteristic
3	#64888	C₂	1	4	yes	maximal, complemented
4	C₂²×C₅×C₈	C₄	1	1	yes	characteristic
5	C₂×C₄×Dic₅	C₄	1	1	yes	characteristic
6	C₂²×#168	C₂²	1	1	yes	characteristic
7	#3998	—	2	4	no	—
8	C₂×#674	C₂²	1	4	yes	—
9	C₈×Dic₅	C₄	1	2	yes	—
10	C₈×Dic₅	C₂²	1	2	yes	—
11	#2150	—	5	1	no	maximal, Sylow
12	C₂²×C₄×C₅	C₂×C₄	1	1	yes	characteristic
13	C₂²×Dic₅	C₈	1	2	yes	—
14	C₂×C₅×C₈	—	2	2	no	—
15	C₂×C₅×C₈	C₂×C₄	1	2	yes	—
16	C₄×Dic₅	C₈	1	1	yes	characteristic
17	C₄×Dic₅	C₈	1	1	yes	characteristic
18	C₄×Dic₅	C₂×C₄	1	2	yes	—
19	C₂×#168	C₂×C₄	1	1	yes	characteristic
20	C₂×#168	D₄	1	2	yes	—
21	C₂×#168	Q₈	1	2	yes	—
22	C₂×#168	C₂³	1	1	yes	characteristic
23	#674	—	2	4	no	—
24	C₂×M₃₂	—	5	2	no	—
25	C₂×C₄×C₈	—	5	1	no	—
26	C₄⋊C₁₆	—	5	4	no	—
27	C₂³×C₅	C₂×C₈	1	1	yes	characteristic
28	C₂×C₄×C₅	C₄⋊C₄	1	2	yes	—
29	C₂×C₄×C₅	C₂×C₈	1	1	yes	characteristic
30	C₂×C₄×C₅	M₁₆	1	2	yes	—
31	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
32	C₂×Dic₅	—	2	4	no	—
33	C₂×Dic₅	C₂×C₈	1	2	yes	—
34	C₂×Dic₅	C₂×C₈	1	2	yes	—
35	C₅×C₈	—	2	2	no	—
36	#168	C₄⋊C₄	1	2	yes	—
37	#168	C₂×D₄	1	1	yes	characteristic
38	#168	C₂×Q₈	1	1	yes	characteristic
39	C₂²×C₈	—	5	1	no	—
40	C₂²×C₈	AGL(1,𝔽₅)	1	1	yes	characteristic
41	C₂×C₄²	—	5	1	no	—
42	M₃₂	—	10	4	no	—
43	C₂×C₁₆	—	5	4	no	—
44	C₄×C₈	—	5	2	no	—
45	C₄×C₈	—	5	2	no	—
46	C₂²×C₅	—	2	2	no	—
47	C₂²×C₅	C₄⋊C₈	1	2	yes	—
48	C₂²×C₅	C₂²×C₈	1	1	yes	characteristic
49	C₄×C₅	C₄⋊C₈	1	2	yes	—
50	C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
51	C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
52	Dic₅	—	2	2	no	—
53	Dic₅	M₃₂	1	4	yes	complemented
54	C₂²×C₄	—	5	2	no	—
55	C₂²×C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
56	C₂×C₈	—	2	2	no	—
57	C₂×C₈	—	5	2	no	—
58	C₂×C₈	—	5	2	no	—
59	C₂×C₈	—	5	1	no	—
60	C₂×C₈	—	5	1	no	—
61	C₂×C₈	C₂×AGL(1,𝔽₅)	1	2	yes	—
62	C₄²	—	5	1	no	—
63	C₄²	—	5	1	no	—
64	C₄²	—	5	2	no	—
65	C₁₆	—	10	4	no	—
66	C₂×C₅	—	2	2	no	—
67	C₂×C₅	C₂×C₄⋊C₈	1	1	yes	characteristic
68	C₂×C₅	C₂×M₃₂	1	2	yes	—
69	C₂³	D₅⋊C₈	1	1	yes	characteristic
70	C₂×C₄	—	5	2	no	—
71	C₂×C₄	—	5	2	no	—
72	C₂×C₄	—	10	4	no	—
73	C₂×C₄	D₅⋊C₈	1	1	yes	characteristic
74	C₂×C₄	C₄⋊AGL(1,𝔽₅)	1	2	yes	—
75	C₂×C₄	#704	1	2	yes	—
76	C₂×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
77	C₈	—	2	2	no	—
78	C₈	—	5	1	no	—
79	C₈	—	5	2	no	—
80	C₈	—	5	1	no	—
81	C₅	#2150	1	1	yes	characteristic, complemented, Sylow
82	C₂²	—	2	2	no	—
83	C₂²	#4004	1	2	yes	—
84	C₂²	C₂×D₅⋊C₈	1	1	yes	characteristic
85	C₄	—	5	4	no	—
86	C₄	—	10	2	no	—
87	C₄	#4004	1	2	yes	—
88	C₄	C₂×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
89	C₄	C₂×#704	1	1	yes	characteristic
90	C₂	—	2	2	no	—
91	C₂	D₅⋊M₃₂	1	2	yes	—
92	C₂	C₂×#4004	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×AGL(1,𝔽₅)
commutator, socle	C₂²×C₅	C₂²×C₈
Fitting	C₂²×C₅×C₈	C₄
prime radical	C₂³×C₅	C₂×C₈
cosocle	C₂×#168	C₂³
hypercenter	C₂²×C₈	AGL(1,𝔽₅)
hypocenter	C₅	#2150
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. #64888	C₂	semidirect
53. Dic₅	M₃₂	semidirect
81. C₅	#2150	semidirect

Maximal subgroups

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

subgroup	index
C₂×#3998	2
C₂×C₈×Dic₅	2
#64888	2
#2150	5