Subgroups — #10600151

Order 640.

144 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#65193	C₂	1	1	yes	maximal, characteristic
2	#65184	C₂	1	2	yes	maximal, complemented
3	C₅×#368	C₂	1	1	yes	maximal, characteristic, complemented
4	#64722	C₂	1	1	yes	maximal, characteristic, complemented
5	#64717	C₂	1	2	yes	maximal, complemented
6	#4053	—	2	2	no	—
7	#4050	C₂²	1	2	yes	—
8	C₂×C₅×C₁₆	C₂²	1	1	yes	characteristic
9	C₅×#108	C₂²	1	2	yes	—
10	#3957	—	2	2	no	—
11	C₈×Dic₅	C₂²	1	1	yes	characteristic
12	C₂×#672	C₂²	1	1	yes	characteristic
13	Dic₄₀	C₄	1	1	yes	characteristic, complemented
14	#3932	C₄	1	2	yes	complemented
15	D₈₀	C₄	1	1	yes	characteristic, complemented
16	#2179	—	5	1	no	maximal, Sylow
17	#709	—	2	2	no	—
18	C₅×M₁₆	—	2	2	no	—
19	C₂×C₅×C₈	C₂³	1	1	yes	characteristic
20	C₄×Dic₅	D₄	1	1	yes	characteristic
21	C₂×#168	D₄	1	1	yes	characteristic
22	Dic₂₀	C₂×C₄	1	2	yes	—
23	D₄₀	C₂×C₄	1	2	yes	—
24	#677	—	2	2	no	—
25	#676	—	4	2	no	—
26	C₈×D₅	—	4	2	no	—
27	C₅×C₁₆	C₂×C₄	1	1	yes	characteristic
28	C₅×C₁₆	C₂×C₄	1	1	yes	characteristic
29	#672	—	2	1	no	—
30	#508	—	5	1	no	—
31	#443	—	5	2	no	—
32	#368	D₅	1	1	yes	characteristic, complemented
33	#359	—	5	2	no	—
34	#346	—	5	1	no	—
35	C₂×C₄×C₅	C₂×D₄	1	1	yes	characteristic
36	C₂²⋊D₅	—	4	2	no	—
37	C₂×Dic₅	—	2	1	no	—
38	D₂₀	—	2	2	no	—
39	C₄×D₅	—	4	2	no	—
40	Dic₁₀	—	2	2	no	—
41	C₅×C₈	—	4	2	no	—
42	C₅×C₈	C₂²×C₄	1	1	yes	characteristic
43	C₅×C₈	#41	1	1	yes	characteristic
44	#168	D₈	1	2	yes	—
45	#135	—	5	2	no	—
46	#131	—	10	2	no	—
47	Q₃₂	—	5	1	no	—
48	SD₃₂	—	5	2	no	—
49	D₁₆	—	5	1	no	—
50	C₂×C₁₆	—	5	1	no	—
51	C₂×C₁₆	C₂×D₅	1	1	yes	characteristic
52	#108	C₂×D₅	1	2	yes	—
53	#104	—	10	2	no	—
54	C₄×C₈	—	5	1	no	—
55	C₂²×C₅	#135	1	1	yes	characteristic
56	C₂×D₅	—	4	2	no	—
57	C₄×C₅	C₄×D₄	1	1	yes	characteristic
58	C₄×C₅	C₂×D₈	1	1	yes	characteristic
59	Dic₅	—	4	1	no	—
60	Dic₅	—	4	2	no	—
61	#41	—	10	2	no	—
62	Q₁₆	—	5	2	no	—
63	SD₁₆	—	10	2	no	—
64	D₈	—	5	2	no	—
65	M₁₆	—	2	2	no	—
66	M₁₆	—	20	2	no	—
67	C₂×C₈	—	5	1	no	—
68	C₂×C₈	—	20	2	no	—
69	C₂×C₈	C₂²×D₅	1	1	yes	characteristic
70	C₄²	—	5	1	no	—
71	C₁₆	—	10	1	no	—
72	C₁₆	C₄×D₅	1	1	yes	characteristic
73	C₁₆	C₄×D₅	1	1	yes	characteristic
74	C₂×C₅	—	2	1	no	—
75	C₂×C₅	C₄×D₈	1	1	yes	characteristic
76	D₅	—	8	2	no	—
77	Q₈	—	10	2	no	—
78	D₄	—	10	2	no	—
79	D₄	—	20	2	no	—
80	C₂×C₄	—	10	1	no	—
81	C₂×C₄	—	20	2	no	—
82	C₂×C₄	D₄×D₅	1	1	yes	characteristic
83	C₈	—	4	2	no	—
84	C₈	—	5	2	no	—
85	C₈	C₂×C₄×D₅	1	1	yes	characteristic
86	C₈	#713	1	1	yes	characteristic
87	C₅	#2179	1	1	yes	characteristic, complemented, Sylow
88	C₂²	—	20	2	no	—
89	C₂²	#4065	1	1	yes	characteristic
90	C₄	—	20	1	no	—
91	C₄	—	20	2	no	—
92	C₄	#4039	1	1	yes	characteristic
93	C₄	D₅×D₈	1	1	yes	characteristic
94	C₂	—	2	1	no	—
95	C₂	—	40	2	no	—
96	C₂	#65161	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₅×D₈
commutator	C₅×C₈	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₅
Fitting	C₅×#368	C₂
socle	C₂×C₅	C₄×D₈
prime radical	#65193	C₂
cosocle	C₂×C₅×C₈	C₂³
hypercenter	#368	D₅
hypocenter	C₅	#2179
LCS	G ▷ C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #368

Complements

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

N	H ≅ G/N	type
2. #65184	C₂	semidirect
3. C₅×#368	C₂	semidirect
4. #64722	C₂	semidirect
5. #64717	C₂	semidirect
13. Dic₄₀	C₄	semidirect
14. #3932	C₄	semidirect
15. D₈₀	C₄	semidirect
32. #368	D₅	semidirect
87. C₅	#2179	semidirect

Maximal subgroups

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

subgroup	index
#65193	2
#65184	2
C₅×#368	2
#64722	2
#64717	2
#2179	5