Subgroups — #10604752

Order 640.

128 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	M₁₆×Dic₅	C₂	1	1	yes	maximal, characteristic
2	#64912	C₂	1	2	yes	maximal
3	#64890	C₂	1	2	yes	maximal, complemented
4	#64889	C₂	1	2	yes	maximal, complemented
5	C₂×C₅×M₁₆	C₄	1	1	yes	characteristic
6	C₂×C₄×Dic₅	C₄	1	1	yes	characteristic
7	C₂×#681	C₂²	1	1	yes	characteristic
8	#3998	C₂²	1	4	yes	—
9	#3947	C₄	1	1	yes	characteristic
10	#3947	C₂²	1	1	yes	characteristic
11	C₈×Dic₅	C₄	1	1	yes	characteristic
12	C₈×Dic₅	C₂²	1	1	yes	characteristic
13	#2165	—	5	1	no	maximal, Sylow
14	C₂²×C₄×C₅	C₂×C₄	1	1	yes	characteristic
15	C₂²×Dic₅	—	2	1	no	—
16	C₅×M₁₆	—	2	2	no	—
17	C₂×C₅×C₈	C₂×C₄	1	1	yes	characteristic
18	C₂×C₅×C₈	C₂×C₄	1	1	yes	characteristic
19	C₄×Dic₅	—	2	1	no	—
20	C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
21	C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
22	#681	—	2	2	no	—
23	C₂×#168	C₂×C₄	1	1	yes	characteristic
24	C₂×#168	C₂³	1	1	yes	characteristic
25	#674	—	2	4	no	—
26	C₄×M₁₆	—	5	1	no	—
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	M₁₆	1	1	yes	characteristic
34	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
35	C₂×Dic₅	—	2	1	no	—
36	C₂×Dic₅	—	2	1	no	—
37	C₂×Dic₅	—	4	2	no	—
38	C₅×C₈	—	2	1	no	—
39	C₅×C₈	M₁₆	1	2	yes	—
40	#168	—	2	1	no	—
41	#168	#41	1	2	yes	—
42	C₂×M₁₆	—	5	1	no	—
43	C₂×M₁₆	AGL(1,𝔽₅)	1	1	yes	characteristic
44	C₂×C₄²	—	5	1	no	—
45	M₃₂	—	5	4	no	—
46	#97	—	5	1	no	—
47	#97	—	5	1	no	—
48	C₄×C₈	—	5	1	no	—
49	C₄×C₈	—	5	1	no	—
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	1	no	—
54	C₄×C₅	#117	1	1	yes	characteristic
55	C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
56	Dic₅	—	2	2	no	—
57	Dic₅	—	4	1	no	—
58	C₂²×C₄	—	10	1	no	—
59	C₂²×C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
60	M₁₆	—	2	2	no	—
61	M₁₆	—	10	2	no	—
62	C₂×C₈	—	5	1	no	—
63	C₂×C₈	—	5	1	no	—
64	C₂×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
65	C₂×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
66	C₄²	—	5	1	no	—
67	C₄²	—	5	1	no	—
68	C₄²	—	10	1	no	—
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	#700	1	1	yes	characteristic
74	C₂×C₄	—	2	1	no	—
75	C₂×C₄	—	2	1	no	—
76	C₂×C₄	—	10	1	no	—
77	C₂×C₄	—	10	1	no	—
78	C₂×C₄	—	20	2	no	—
79	C₂×C₄	#700	1	1	yes	characteristic
80	C₂×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
81	C₈	—	2	1	no	—
82	C₈	—	5	2	no	—
83	C₈	—	10	1	no	—
84	C₈	#704	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×#700	1	1	yes	characteristic
89	C₄	—	2	1	no	—
90	C₄	—	10	2	no	—
91	C₄	—	20	1	no	—
92	C₄	#4130	1	1	yes	characteristic
93	C₄	C₂×#704	1	1	yes	characteristic
94	C₂	—	2	1	no	—
95	C₂	—	4	1	no	—
96	C₂	#65759	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#4130
commutator	C₂×C₄×C₅	C₂²×C₄
Frattini	C₂×C₄	C₂²×AGL(1,𝔽₅)
Fitting	C₂×C₅×M₁₆	C₄
socle	C₂×C₅	#432
prime radical	C₂³×C₅	M₁₆
cosocle	C₂×#168	C₂³
hypercenter	C₂×M₁₆	AGL(1,𝔽₅)
hypocenter	C₅	#2165
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₂×M₁₆

Complements

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

N	H ≅ G/N	type
3. #64890	C₂	semidirect
4. #64889	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
M₁₆×Dic₅	2
#64912	2
#64890	2
#64889	2
#2165	5