Subgroups — #13100142

Order 1600.

123 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#87763	C₂	1	2	yes	maximal
2	C₄×C₅×#168	C₂	1	1	yes	maximal, characteristic
3	C₄²×C₅²	C₄	1	1	yes	characteristic
4	C₂×C₅×#168	C₄	1	1	yes	characteristic
5	C₂×C₅×#168	C₂²	1	1	yes	characteristic
6	#87763	C₄	1	4	yes	complemented
7	#64859	—	5	1	no	maximal
8	#64681	—	5	1	no	maximal
9	C₂×C₄×C₅²	C₈	1	2	yes	—
10	C₂×C₄×C₅²	C₂×C₄	1	1	yes	characteristic
11	C₅×#168	C₈	1	2	yes	—
12	C₅×#168	C₂×C₄	1	2	yes	—
13	C₂×#674	—	5	2	no	—
14	C₄×C₅×C₈	—	5	1	no	—
15	C₂×#672	—	5	2	no	—
16	C₄×#168	D₅	1	1	yes	characteristic
17	C₂²×C₅²	C₂×C₈	1	1	yes	characteristic
18	C₄×C₅²	—	2	2	no	—
19	C₄×C₅²	C₄²	1	1	yes	characteristic
20	C₄×C₅²	C₂×C₈	1	1	yes	characteristic
21	C₂×C₅×C₈	—	5	1	no	—
22	C₂×C₅×C₈	—	5	1	no	—
23	C₄²×C₅	—	4	1	no	—
24	C₄²×C₅	Dic₅	1	1	yes	characteristic
25	C₄²×C₅	AGL(1,𝔽₅)	1	1	yes	characteristic
26	C₂×#168	Dic₅	1	1	yes	characteristic
27	C₂×#168	C₂×D₅	1	1	yes	characteristic
28	#674	—	5	4	no	—
29	#672	—	5	4	no	—
30	#346	—	25	1	no	Sylow
31	C₂×C₅²	C₄×C₈	1	1	yes	characteristic
32	C₂×C₅²	M₃₂	1	2	yes	—
33	C₂×C₄×C₅	—	4	2	no	—
34	C₂×C₄×C₅	—	4	1	no	—
35	C₂×C₄×C₅	#168	1	2	yes	—
36	C₂×C₄×C₅	#170	1	2	yes	—
37	C₂×C₄×C₅	C₂×Dic₅	1	1	yes	characteristic
38	C₂×C₄×C₅	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
39	C₅×C₈	—	5	2	no	—
40	C₅×C₈	—	5	2	no	—
41	#168	#168	1	2	yes	—
42	#168	C₄×D₅	1	2	yes	—
43	C₂×C₁₆	—	25	2	no	—
44	C₄×C₈	—	5	1	no	—
45	C₅²	#346	1	1	yes	characteristic, complemented, Sylow
46	C₂²×C₅	—	4	1	no	—
47	C₂²×C₅	C₂×#168	1	1	yes	characteristic
48	C₂²×C₅	C₂×#170	1	1	yes	characteristic
49	C₄×C₅	—	2	2	no	—
50	C₄×C₅	—	2	2	no	—
51	C₄×C₅	—	4	4	no	—
52	C₄×C₅	—	4	1	no	—
53	C₄×C₅	—	4	1	no	—
54	C₄×C₅	C₂×#168	1	1	yes	characteristic
55	C₄×C₅	C₄×Dic₅	1	1	yes	characteristic
56	C₄×C₅	D₅⋊C₈	1	1	yes	characteristic
57	C₄×C₅	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
58	C₂×C₈	—	5	1	no	—
59	C₂×C₈	—	5	1	no	—
60	C₄²	#1042	1	1	yes	characteristic
61	C₁₆	—	25	4	no	—
62	C₂×C₅	—	4	2	no	—
63	C₂×C₅	—	4	1	no	—
64	C₂×C₅	C₄×#168	1	1	yes	characteristic
65	C₂×C₅	#3944	1	2	yes	—
66	C₂×C₅	#3998	1	2	yes	—
67	C₂×C₅	C₄×#170	1	1	yes	characteristic
68	C₂×C₄	#6032	1	2	yes	—
69	C₂×C₄	C₂×#1042	1	1	yes	characteristic
70	C₈	—	5	2	no	—
71	C₈	—	5	2	no	—
72	C₅	—	4	1	no	—
73	C₅	#64681	1	1	yes	characteristic, complemented
74	C₅	#64859	1	1	yes	characteristic, complemented
75	C₂²	C₂×#6032	1	1	yes	characteristic
76	C₄	—	2	2	no	—
77	C₄	D₅⋊#168	1	1	yes	characteristic
78	C₄	C₄×#1042	1	1	yes	characteristic
79	C₂	#11716841	1	2	yes	—
80	C₂	C₄×#6032	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂×#1042
commutator	C₂×C₅²	C₄×C₈
Fitting	C₄²×C₅²	C₄
socle, prime radical	C₂²×C₅²	C₂×C₈
cosocle	C₂×C₅×#168	C₂²
hypercenter	C₄²	#1042
hypocenter	C₅²	#346
LCS	G ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂×C₄ ◁ C₄²

Complements

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

N	H ≅ G/N	type
6. #87763	C₄	semidirect
45. C₅²	#346	semidirect
73. C₅	#64681	semidirect
74. C₅	#64859	semidirect

Maximal subgroups

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

subgroup	index
C₂×#87763	2
C₄×C₅×#168	2
#64859	5
#64681	5