Subgroups — #10600677

Order 640.

160 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#405	C₂	1	1	yes	maximal, characteristic
2	C₂×#3944	C₂	1	2	yes	maximal
3	#64686	C₂	1	4	yes	maximal, complemented
4	C₂×C₅×M₁₆	C₄	1	1	yes	characteristic
5	C₂²×C₅×C₈	C₂²	1	1	yes	characteristic
6	C₅×#117	C₄	1	1	yes	characteristic
7	C₅×#97	C₂²	1	2	yes	—
8	C₄×C₅×C₈	C₄	1	2	yes	—
9	#3944	C₄	1	4	yes	complemented
10	#3944	C₂²	1	4	yes	—
11	C₂×#672	C₄	1	4	yes	complemented
12	#2110	—	5	1	no	maximal, Sylow
13	C₂²×C₄×C₅	C₂×C₄	1	1	yes	characteristic
14	C₅×M₁₆	—	2	2	no	—
15	C₂×C₅×C₈	C₂×C₄	1	2	yes	—
16	C₂×C₅×C₈	C₂×C₄	1	2	yes	—
17	C₂×C₅×C₈	C₂×C₄	1	2	yes	—
18	C₂×C₅×C₈	C₂×C₄	1	1	yes	characteristic
19	C₂×C₅×C₈	C₂³	1	1	yes	characteristic
20	C₅×C₄⋊C₄	—	2	1	no	—
21	C₅×C₂²⋊C₄	—	2	1	no	—
22	C₄²×C₅	C₂×C₄	1	2	yes	—
23	#672	C₂×C₄	1	8	yes	—
24	C₂×M₃₂	—	5	2	no	—
25	#405	D₅	1	1	yes	characteristic
26	#347	—	5	4	no	—
27	C₂³×C₅	M₁₆	1	1	yes	characteristic
28	C₂×C₄×C₅	—	2	2	no	—
29	C₂×C₄×C₅	C₄²	1	2	yes	—
30	C₂×C₄×C₅	M₁₆	1	1	yes	characteristic
31	C₂×C₄×C₅	M₁₆	1	2	yes	—
32	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
33	C₅×C₈	—	2	2	no	—
34	C₅×C₈	C₄²	1	1	yes	characteristic
35	C₅×C₈	C₄²	1	1	yes	characteristic
36	C₅×C₈	C₂²×C₄	1	2	yes	—
37	C₂×M₁₆	Dic₅	1	1	yes	characteristic
38	C₂²×C₈	C₂×D₅	1	1	yes	characteristic
39	#117	Dic₅	1	1	yes	characteristic
40	M₃₂	—	5	4	no	—
41	M₃₂	—	5	4	no	—
42	C₂×C₁₆	—	5	4	no	—
43	#97	C₂×D₅	1	2	yes	—
44	C₄×C₈	Dic₅	1	2	yes	—
45	C₂²×C₅	—	4	1	no	—
46	C₂²×C₅	#97	1	2	yes	—
47	C₂²×C₅	C₂×M₁₆	1	1	yes	characteristic
48	C₄×C₅	—	4	2	no	—
49	C₄×C₅	#97	1	2	yes	—
50	C₄×C₅	C₂×C₄²	1	1	yes	characteristic
51	C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
52	C₂²×C₄	C₂×Dic₅	1	1	yes	characteristic
53	M₁₆	—	2	2	no	—
54	C₂×C₈	C₄×D₅	1	2	yes	—
55	C₂×C₈	C₄×D₅	1	2	yes	—
56	C₂×C₈	C₂×Dic₅	1	2	yes	—
57	C₂×C₈	C₂×Dic₅	1	1	yes	characteristic
58	C₂×C₈	C₂²×D₅	1	1	yes	characteristic
59	C₄⋊C₄	—	2	1	no	—
60	C₂²⋊C₄	—	2	1	no	—
61	C₄²	C₂×Dic₅	1	2	yes	—
62	C₁₆	—	5	8	no	—
63	C₂×C₅	—	2	1	no	—
64	C₂×C₅	—	2	2	no	—
65	C₂×C₅	C₂×#97	1	1	yes	characteristic
66	C₂³	#681	1	1	yes	characteristic
67	C₂×C₄	—	2	2	no	—
68	C₂×C₄	#681	1	1	yes	characteristic
69	C₂×C₄	#681	1	2	yes	—
70	C₂×C₄	C₄×Dic₅	1	2	yes	—
71	C₂×C₄	C₂²×Dic₅	1	1	yes	characteristic
72	C₈	—	2	2	no	—
73	C₈	C₄×Dic₅	1	1	yes	characteristic
74	C₈	C₄×Dic₅	1	1	yes	characteristic
75	C₈	C₂×C₄×D₅	1	2	yes	—
76	C₅	#2110	1	1	yes	characteristic, complemented, Sylow
77	C₂²	—	4	1	no	—
78	C₂²	#3935	1	2	yes	—
79	C₂²	C₂×#681	1	1	yes	characteristic
80	C₄	—	4	2	no	—
81	C₄	#3935	1	2	yes	—
82	C₄	C₂×#681	1	1	yes	characteristic
83	C₄	C₂×C₄×Dic₅	1	1	yes	characteristic
84	C₂	—	2	1	no	—
85	C₂	—	2	2	no	—
86	C₂	C₂×#3935	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₄×Dic₅
commutator	C₄×C₅	C₂×C₄²
Frattini	C₂×C₈	C₂²×D₅
Fitting	C₅×#405	C₂
socle	C₂×C₅	C₂×#97
prime radical	C₂³×C₅	M₁₆
cosocle	C₂×C₅×C₈	C₂³
hypercenter	#405	D₅
hypocenter	C₅	#2110
LCS	G ▷ C₄×C₅ ▷ C₅
UCS	C₁ ◁ C₈ ◁ #405

Complements

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

N	H ≅ G/N	type
3. #64686	C₂	semidirect
9. #3944	C₄	semidirect
11. C₂×#672	C₄	semidirect
76. C₅	#2110	semidirect

Maximal subgroups

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

subgroup	index
C₅×#405	2
C₂×#3944	2
#64686	2
#2110	5