Subgroups — #13100145

Order 1600.

107 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11716841	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	—	2	2	no	—
7	#64861	—	5	1	no	maximal
8	#64683	—	5	1	no	maximal
9	C₂×C₄×C₅²	—	2	1	no	—
10	C₂×C₄×C₅²	C₂×C₄	1	1	yes	characteristic
11	C₅×#168	C₈	1	2	yes	—
12	C₅×#168	D₄	1	1	yes	characteristic
13	C₅×#168	Q₈	1	1	yes	characteristic
14	#3998	—	5	2	no	—
15	C₄×C₅×C₈	—	5	1	no	—
16	#3944	—	5	2	no	—
17	C₄×#168	D₅	1	1	yes	characteristic
18	C₂²×C₅²	M₁₆	1	1	yes	characteristic
19	C₄×C₅²	—	2	2	no	—
20	C₄×C₅²	C₄⋊C₄	1	1	yes	characteristic
21	C₄×C₅²	C₂×C₈	1	1	yes	characteristic
22	C₂×C₅×C₈	—	5	1	no	—
23	C₂×C₅×C₈	—	5	1	no	—
24	C₄²×C₅	—	4	1	no	—
25	C₄²×C₅	Dic₅	1	1	yes	characteristic
26	C₄²×C₅	AGL(1,𝔽₅)	1	1	yes	characteristic
27	C₂×#168	Dic₅	1	1	yes	characteristic
28	C₂×#168	C₂×D₅	1	1	yes	characteristic
29	#674	—	10	2	no	—
30	#672	—	10	2	no	—
31	#364	—	25	1	no	Sylow
32	C₂×C₅²	—	2	1	no	—
33	C₂×C₅²	C₄⋊C₈	1	1	yes	characteristic
34	C₂×C₄×C₅	—	2	1	no	—
35	C₂×C₄×C₅	—	2	1	no	—
36	C₂×C₄×C₅	—	4	2	no	—
37	C₂×C₄×C₅	—	4	1	no	—
38	C₂×C₄×C₅	C₂×Dic₅	1	1	yes	characteristic
39	C₂×C₄×C₅	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
40	C₅×C₈	—	5	2	no	—
41	C₅×C₈	—	5	1	no	—
42	C₅×C₈	—	5	1	no	—
43	#168	#168	1	2	yes	—
44	#168	Dic₁₀	1	1	yes	characteristic
45	#168	D₂₀	1	1	yes	characteristic
46	M₃₂	—	25	2	no	—
47	C₄×C₈	—	5	1	no	—
48	C₅²	#364	1	1	yes	characteristic, complemented, Sylow
49	C₂²×C₅	—	4	1	no	—
50	C₂²×C₅	#681	1	1	yes	characteristic
51	C₂²×C₅	#704	1	1	yes	characteristic
52	C₄×C₅	—	2	2	no	—
53	C₄×C₅	—	2	2	no	—
54	C₄×C₅	—	4	4	no	—
55	C₄×C₅	—	4	1	no	—
56	C₄×C₅	—	4	1	no	—
57	C₄×C₅	C₂×#168	1	1	yes	characteristic
58	C₄×C₅	C₄⋊Dic₅	1	1	yes	characteristic
59	C₄×C₅	D₅⋊C₈	1	1	yes	characteristic
60	C₄×C₅	C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
61	C₂×C₈	—	5	1	no	—
62	C₂×C₈	—	5	1	no	—
63	C₄²	#1042	1	1	yes	characteristic
64	C₁₆	—	50	2	no	—
65	C₂×C₅	—	2	1	no	—
66	C₂×C₅	—	2	1	no	—
67	C₂×C₅	—	4	2	no	—
68	C₂×C₅	—	4	1	no	—
69	C₂×C₅	C₄⋊#168	1	1	yes	characteristic
70	C₂×C₅	#4004	1	1	yes	characteristic
71	C₂×C₄	—	2	1	no	—
72	C₂×C₄	C₂×#1042	1	1	yes	characteristic
73	C₈	—	5	2	no	—
74	C₈	—	5	1	no	—
75	C₈	—	5	1	no	—
76	C₅	—	4	1	no	—
77	C₅	#64683	1	1	yes	characteristic, complemented
78	C₅	#64861	1	1	yes	characteristic, complemented
79	C₂²	#87853	1	1	yes	characteristic
80	C₄	—	2	2	no	—
81	C₄	D₅⋊#168	1	1	yes	characteristic
82	C₄	C₄⋊#1042	1	1	yes	characteristic
83	C₂	—	2	1	no	—
84	C₂	#11716847	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₄⋊#1042
commutator	C₄×C₅²	C₂×C₈
Frattini	C₂×C₄	C₂×#1042
Fitting	C₄²×C₅²	C₄
socle	C₂×C₅²	C₄⋊C₈
prime radical	C₂²×C₅²	M₁₆
cosocle	C₂×C₅×#168	C₂²
hypercenter	C₄²	#1042
hypocenter	C₅²	#364
LCS	G ▷ C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₄²

Complements

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

N	H ≅ G/N	type
48. C₅²	#364	semidirect
77. C₅	#64683	semidirect
78. C₅	#64861	semidirect

Maximal subgroups

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

subgroup	index
#11716841	2
C₄×C₅×#168	2
#64861	5
#64683	5