Subgroups — #13099713

Order 1600.

113 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×#1042	C₂	1	1	yes	maximal, characteristic
2	C₅×D₅⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	D₅×#672	C₂	1	1	yes	maximal, characteristic
4	C₄×#1042	C₄	1	1	yes	characteristic
5	D₅⋊#168	C₄	1	1	yes	characteristic
6	C₅×C₈×D₅	C₂²	1	1	yes	characteristic
7	C₅×#674	C₄	1	2	yes	complemented
8	#87756	C₄	1	1	yes	characteristic, complemented
9	C₅×#672	C₄	1	1	yes	characteristic, complemented
10	#64844	—	5	1	no	maximal
11	#64722	—	5	1	no	maximal
12	C₂×#1042	C₈	1	2	yes	—
13	C₄×C₅×D₅	C₂×C₄	1	1	yes	characteristic
14	#6032	C₈	1	2	yes	—
15	C₅²×C₈	C₂×C₄	1	1	yes	characteristic
16	C₅×#168	C₂×C₄	1	1	yes	characteristic
17	C₈×AGL(1,𝔽₅)	—	5	1	no	—
18	D₅⋊C₁₆	D₅	1	1	yes	characteristic
19	C₂×C₅×C₁₆	—	5	1	no	—
20	C₈×Dic₅	—	5	1	no	—
21	C₂×#672	—	5	1	no	—
22	D₅×C₁₆	—	5	1	no	—
23	C₂×C₅×D₅	C₂×C₈	1	1	yes	characteristic
24	#1042	—	2	2	no	—
25	C₄×C₅²	C₄²	1	1	yes	characteristic
26	C₅×Dic₅	C₂×C₈	1	1	yes	characteristic
27	C₄×AGL(1,𝔽₅)	—	5	1	no	—
28	D₅⋊C₈	—	5	1	no	—
29	C₂×C₅×C₈	—	5	1	no	—
30	C₄×Dic₅	—	5	1	no	—
31	C₂×#168	—	5	1	no	—
32	C₈×D₅	C₂×D₅	1	1	yes	characteristic
33	#674	Dic₅	1	2	yes	complemented
34	C₅×C₁₆	—	5	2	no	—
35	C₅×C₁₆	—	5	1	no	—
36	#672	—	5	1	no	—
37	#672	—	5	1	no	—
38	#672	—	20	1	no	—
39	#672	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
40	#346	—	25	1	no	Sylow
41	C₂×C₅²	C₄×C₈	1	1	yes	characteristic
42	C₅×D₅	M₃₂	1	2	yes	—
43	C₂×AGL(1,𝔽₅)	—	5	2	no	—
44	C₂×C₄×C₅	—	5	1	no	—
45	C₂×Dic₅	—	5	2	no	—
46	C₄×D₅	C₄×D₅	1	1	yes	characteristic
47	#170	—	5	2	no	—
48	C₅×C₈	—	4	1	no	—
49	C₅×C₈	—	5	1	no	—
50	C₅×C₈	C₄×D₅	1	1	yes	characteristic
51	C₅×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
52	#168	—	5	2	no	—
53	#168	C₂×Dic₅	1	1	yes	characteristic
54	C₂×C₁₆	—	5	1	no	—
55	C₂×C₁₆	—	25	1	no	—
56	C₄×C₈	—	25	1	no	—
57	C₅²	#346	1	1	yes	characteristic, complemented, Sylow
58	C₂²×C₅	—	5	1	no	—
59	C₂×D₅	C₈×D₅	1	1	yes	characteristic
60	AGL(1,𝔽₅)	—	10	2	no	—
61	C₄×C₅	—	4	1	no	—
62	C₄×C₅	—	5	1	no	—
63	C₄×C₅	C₄×Dic₅	1	1	yes	characteristic
64	C₄×C₅	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
65	Dic₅	—	10	2	no	—
66	Dic₅	C₈×D₅	1	1	yes	characteristic
67	C₂×C₈	—	5	1	no	—
68	C₂×C₈	—	25	1	no	—
69	C₄²	—	25	1	no	—
70	C₁₆	—	5	2	no	—
71	C₁₆	—	5	1	no	—
72	C₁₆	—	25	1	no	—
73	C₂×C₅	—	4	1	no	—
74	C₂×C₅	—	5	2	no	—
75	C₂×C₅	C₈×Dic₅	1	1	yes	characteristic
76	C₂×C₅	C₈×AGL(1,𝔽₅)	1	1	yes	characteristic
77	D₅	#3930	1	2	yes	—
78	C₂×C₄	—	5	1	no	—
79	C₂×C₄	—	25	2	no	—
80	C₈	—	5	1	no	—
81	C₈	—	25	2	no	—
82	C₈	D₅×AGL(1,𝔽₅)	1	1	yes	characteristic
83	C₅	—	4	1	no	—
84	C₅	#64722	1	1	yes	characteristic, complemented
85	C₅	#64844	1	1	yes	characteristic, complemented
86	C₂²	—	5	1	no	—
87	C₄	—	5	1	no	—
88	C₄	—	50	2	no	—
89	C₄	Dic₅×AGL(1,𝔽₅)	1	1	yes	characteristic
90	C₂	—	5	2	no	—
91	C₂	#11716731	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₈	D₅×AGL(1,𝔽₅)
commutator, socle	C₂×C₅²	C₄×C₈
Fitting	C₅²×C₈	C₂×C₄
prime radical	C₂×C₅×D₅	C₂×C₈
cosocle	C₅×C₈×D₅	C₂²
hypocenter	C₅²	#346
LCS	G ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₈

Complements

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

N	H ≅ G/N	type
7. C₅×#674	C₄	semidirect
8. #87756	C₄	semidirect
9. C₅×#672	C₄	semidirect
33. #674	Dic₅	semidirect
39. #672	AGL(1,𝔽₅)	semidirect
57. C₅²	#346	semidirect
84. C₅	#64722	semidirect
85. C₅	#64844	semidirect

Maximal subgroups

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

subgroup	index
C₈×#1042	2
C₅×D₅⋊C₁₆	2
D₅×#672	2
#64844	5
#64722	5