Subgroups — #10600074

Order 640.

126 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×Dic₁₀	C₂	1	1	yes	maximal, characteristic
2	C₅×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	#64722	C₂	1	2	yes	maximal
4	#64721	C₂	1	2	yes	maximal
5	C₄×#672	C₂	1	1	yes	maximal, characteristic
6	C₄×Dic₁₀	C₄	1	1	yes	characteristic
7	C₂×C₅×C₁₆	C₂²	1	2	yes	—
8	C₄×C₅×C₈	C₂²	1	1	yes	characteristic
9	#3946	C₄	1	2	yes	—
10	C₈×Dic₅	C₂²	1	2	yes	—
11	C₂×#672	C₂²	1	1	yes	characteristic
12	C₂×#672	C₂²	1	1	yes	characteristic
13	C₄⋊#168	C₄	1	1	yes	characteristic
14	#2183	—	5	1	no	maximal, Sylow
15	C₂×Dic₁₀	C₈	1	1	yes	characteristic
16	C₂×C₅×C₈	C₂×C₄	1	1	yes	characteristic
17	C₂×C₅×C₈	C₂³	1	1	yes	characteristic
18	C₄²×C₅	C₂×C₄	1	1	yes	characteristic
19	C₄⋊Dic₅	C₈	1	1	yes	characteristic
20	#683	C₈	1	2	yes	—
21	C₄×Dic₅	C₂×C₄	1	2	yes	—
22	C₂×#168	C₂×C₄	1	2	yes	—
23	C₅×C₁₆	—	2	2	no	—
24	#672	—	2	1	no	—
25	#672	Q₈	1	2	yes	complemented
26	C₈×Q₈	—	5	1	no	—
27	C₄⋊C₁₆	—	5	2	no	—
28	C₄⋊C₁₆	D₅	1	1	yes	characteristic
29	#346	—	5	2	no	—
30	C₄×C₁₆	—	5	1	no	—
31	C₂×C₄×C₅	C₂×C₈	1	1	yes	characteristic
32	C₂×C₄×C₅	C₂×C₈	1	1	yes	characteristic
33	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
34	C₂×Dic₅	C₂×C₈	1	2	yes	—
35	C₂×Dic₅	C₂×C₈	1	2	yes	—
36	Dic₁₀	—	2	2	no	—
37	C₅×C₈	—	2	1	no	—
38	C₅×C₈	C₂×Q₈	1	1	yes	characteristic
39	C₅×C₈	#41	1	1	yes	characteristic
40	#168	—	2	2	no	—
41	C₄×Q₈	—	5	1	no	—
42	C₂×C₁₆	—	5	1	no	—
43	C₂×C₁₆	—	5	1	no	—
44	C₂×C₁₆	C₂×D₅	1	2	yes	—
45	C₄⋊C₈	—	5	2	no	—
46	C₄⋊C₈	—	5	1	no	—
47	C₄×C₈	—	5	2	no	—
48	C₄×C₈	C₂×D₅	1	1	yes	characteristic
49	C₂²×C₅	C₂²×C₈	1	1	yes	characteristic
50	C₄×C₅	—	2	1	no	—
51	C₄×C₅	M₃₂	1	2	yes	—
52	C₄×C₅	C₄×Q₈	1	1	yes	characteristic
53	C₄×C₅	#131	1	1	yes	characteristic
54	Dic₅	—	2	2	no	—
55	Dic₅	—	2	2	no	—
56	C₂×Q₈	—	5	1	no	—
57	C₂×C₈	—	5	2	no	—
58	C₂×C₈	C₄×D₅	1	1	yes	characteristic
59	C₂×C₈	C₂²×D₅	1	1	yes	characteristic
60	C₄⋊C₄	—	5	2	no	—
61	C₄⋊C₄	—	5	1	no	—
62	C₄²	—	5	2	no	—
63	C₄²	C₄×D₅	1	1	yes	characteristic
64	C₁₆	—	2	2	no	—
65	C₁₆	—	5	2	no	—
66	C₁₆	—	10	1	no	—
67	C₂×C₅	C₈×Q₈	1	1	yes	characteristic
68	C₂×C₅	C₂×M₃₂	1	1	yes	characteristic
69	C₂×C₅	#504	1	1	yes	characteristic
70	Q₈	—	10	2	no	—
71	C₂×C₄	—	5	2	no	—
72	C₂×C₄	—	5	2	no	—
73	C₂×C₄	C₈×D₅	1	1	yes	characteristic
74	C₂×C₄	C₈×D₅	1	1	yes	characteristic
75	C₂×C₄	C₂×C₄×D₅	1	1	yes	characteristic
76	C₈	—	2	1	no	—
77	C₈	—	10	2	no	—
78	C₈	#711	1	1	yes	characteristic
79	C₈	Q₈×D₅	1	1	yes	characteristic
80	C₅	#2183	1	1	yes	characteristic, complemented, Sylow
81	C₂²	C₂×C₈×D₅	1	1	yes	characteristic
82	C₄	—	2	1	no	—
83	C₄	—	10	2	no	—
84	C₄	—	10	2	no	—
85	C₄	#3930	1	2	yes	—
86	C₄	#4033	1	1	yes	characteristic
87	C₄	#4053	1	1	yes	characteristic
88	C₂	#65119	1	1	yes	characteristic
89	C₂	C₂×#3930	1	1	yes	characteristic
90	C₂	#65196	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₅
commutator, socle, prime radical	C₂²×C₅	C₂²×C₈
Fitting	C₅×C₄⋊C₁₆	C₂
cosocle	C₂×C₅×C₈	C₂³
hypercenter	C₄⋊C₁₆	D₅
hypocenter	C₅	#2183
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₈ ◁ C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
25. #672	Q₈	semidirect
80. C₅	#2183	semidirect

Maximal subgroups

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

subgroup	index
C₈×Dic₁₀	2
C₅×C₄⋊C₁₆	2
#64722	2
#64721	2
C₄×#672	2
#2183	5