Subgroups — #10600657

Order 640.

128 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₅×M₁₆	C₂	1	1	yes	maximal, characteristic
2	#64773	C₂	1	2	yes	maximal
3	#64686	C₂	1	2	yes	maximal, complemented
4	#64683	C₂	1	2	yes	maximal, complemented
5	C₂×C₅×M₁₆	C₄	1	1	yes	characteristic
6	C₂×C₅×M₁₆	C₂²	1	1	yes	characteristic
7	C₂×C₄²×C₅	C₄	1	1	yes	characteristic
8	C₅×#97	C₄	1	1	yes	characteristic
9	C₅×#97	C₂²	1	1	yes	characteristic
10	C₄×C₅×C₈	C₄	1	1	yes	characteristic
11	C₄×C₅×C₈	C₂²	1	1	yes	characteristic
12	#3944	C₂²	1	4	yes	—
13	#2165	—	5	1	no	maximal, Sylow
14	C₂²×C₄×C₅	—	2	1	no	—
15	C₂²×C₄×C₅	C₂×C₄	1	1	yes	characteristic
16	C₅×M₁₆	—	2	2	no	—
17	C₅×M₁₆	—	2	2	no	—
18	C₂×C₅×C₈	C₂×C₄	1	1	yes	characteristic
19	C₂×C₅×C₈	C₂×C₄	1	1	yes	characteristic
20	C₂×C₅×C₈	C₂×C₄	1	1	yes	characteristic
21	C₂×C₅×C₈	C₂³	1	1	yes	characteristic
22	C₄²×C₅	—	2	1	no	—
23	C₄²×C₅	C₂×C₄	1	1	yes	characteristic
24	C₄²×C₅	C₂×C₄	1	1	yes	characteristic
25	#672	—	2	4	no	—
26	C₄×M₁₆	D₅	1	1	yes	characteristic
27	#364	—	5	2	no	—
28	#349	—	5	2	no	—
29	#347	—	5	2	no	—
30	C₂³×C₅	M₁₆	1	1	yes	characteristic
31	C₂×C₄×C₅	—	2	1	no	—
32	C₂×C₄×C₅	—	2	1	no	—
33	C₂×C₄×C₅	—	2	1	no	—
34	C₂×C₄×C₅	—	2	1	no	—
35	C₂×C₄×C₅	—	4	2	no	—
36	C₂×C₄×C₅	M₁₆	1	1	yes	characteristic
37	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
38	C₅×C₈	—	2	1	no	—
39	C₅×C₈	—	2	1	no	—
40	C₅×C₈	M₁₆	1	2	yes	—
41	C₅×C₈	#41	1	2	yes	—
42	C₂×M₁₆	Dic₅	1	1	yes	characteristic
43	C₂×M₁₆	C₂×D₅	1	1	yes	characteristic
44	C₂×C₄²	Dic₅	1	1	yes	characteristic
45	M₃₂	—	5	4	no	—
46	#97	Dic₅	1	1	yes	characteristic
47	#97	C₂×D₅	1	1	yes	characteristic
48	C₄×C₈	Dic₅	1	1	yes	characteristic
49	C₄×C₈	C₂×D₅	1	1	yes	characteristic
50	C₂²×C₅	—	2	1	no	—
51	C₂²×C₅	—	4	1	no	—
52	C₂²×C₅	C₂×M₁₆	1	1	yes	characteristic
53	C₄×C₅	—	2	2	no	—
54	C₄×C₅	—	2	1	no	—
55	C₄×C₅	—	4	1	no	—
56	C₄×C₅	#117	1	1	yes	characteristic
57	C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
58	C₂²×C₄	—	2	1	no	—
59	C₂²×C₄	C₂×Dic₅	1	1	yes	characteristic
60	M₁₆	—	2	2	no	—
61	M₁₆	—	2	2	no	—
62	C₂×C₈	C₂×Dic₅	1	1	yes	characteristic
63	C₂×C₈	C₂×Dic₅	1	1	yes	characteristic
64	C₂×C₈	C₂×Dic₅	1	1	yes	characteristic
65	C₂×C₈	C₂²×D₅	1	1	yes	characteristic
66	C₄²	—	2	1	no	—
67	C₄²	C₂×Dic₅	1	1	yes	characteristic
68	C₄²	C₂×Dic₅	1	1	yes	characteristic
69	C₁₆	—	10	4	no	—
70	C₂×C₅	—	2	1	no	—
71	C₂×C₅	—	4	1	no	—
72	C₂×C₅	#432	1	1	yes	characteristic
73	C₂³	#681	1	1	yes	characteristic
74	C₂×C₄	—	2	1	no	—
75	C₂×C₄	—	2	1	no	—
76	C₂×C₄	—	2	1	no	—
77	C₂×C₄	—	2	1	no	—
78	C₂×C₄	—	4	2	no	—
79	C₂×C₄	#681	1	1	yes	characteristic
80	C₂×C₄	C₂²×Dic₅	1	1	yes	characteristic
81	C₈	—	2	1	no	—
82	C₈	—	2	1	no	—
83	C₈	#681	1	2	yes	—
84	C₈	#709	1	2	yes	—
85	C₅	#2165	1	1	yes	characteristic, complemented, Sylow
86	C₂²	—	2	1	no	—
87	C₂²	—	4	1	no	—
88	C₂²	C₂×#681	1	1	yes	characteristic
89	C₄	—	2	2	no	—
90	C₄	—	2	1	no	—
91	C₄	—	4	1	no	—
92	C₄	C₂×#681	1	1	yes	characteristic
93	C₄	#4072	1	1	yes	characteristic
94	C₂	—	2	1	no	—
95	C₂	—	4	1	no	—
96	C₂	#65215	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#4072
commutator	C₂×C₄×C₅	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₅
Fitting	C₄×C₅×M₁₆	C₂
socle	C₂×C₅	#432
prime radical	C₂³×C₅	M₁₆
cosocle	C₂×C₅×C₈	C₂³
hypercenter	C₄×M₁₆	D₅
hypocenter	C₅	#2165
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₄ ◁ #97 ◁ C₄×M₁₆

Complements

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

N	H ≅ G/N	type
3. #64686	C₂	semidirect
4. #64683	C₂	semidirect
85. C₅	#2165	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₅×M₁₆	2
#64773	2
#64686	2
#64683	2
#2165	5