Subgroups — #13104677

Order 1600.

188 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#4094	C₂	1	1	yes	maximal, characteristic
2	#11716841	C₂	1	3	yes	maximal, complemented
3	C₂×#87763	C₂	1	3	yes	maximal, complemented
4	C₅²×#41	C₄	1	1	yes	characteristic
5	C₅×#681	C₄	1	3	yes	—
6	C₂×C₅×#168	C₂²	1	3	yes	—
7	#87763	C₂²	1	1	yes	characteristic
8	#87763	C₂²	1	3	yes	—
9	#65791	—	5	1	no	maximal
10	#65429	—	5	1	no	maximal
11	C₅²×Q₈	C₈	1	1	yes	characteristic
12	C₅²×D₄	C₈	1	3	yes	—
13	C₂×C₄×C₅²	C₂×C₄	1	3	yes	—
14	C₅×#168	C₂×C₄	1	3	yes	—
15	C₅×#168	C₂³	1	1	yes	characteristic
16	C₅×#131	—	5	1	no	—
17	#4094	D₅	1	1	yes	characteristic
18	#3998	—	5	3	no	—
19	C₂×#674	—	5	3	no	—
20	#3944	—	5	3	no	—
21	C₂×#672	—	5	3	no	—
22	C₂²×C₅²	C₂×C₈	1	3	yes	—
23	C₄×C₅²	C₂×C₈	1	3	yes	—
24	C₄×C₅²	C₂²×C₄	1	1	yes	characteristic
25	C₅×#41	—	4	1	no	—
26	C₅×#41	Dic₅	1	1	yes	characteristic
27	C₅×#41	AGL(1,𝔽₅)	1	1	yes	characteristic
28	C₅×M₁₆	—	5	3	no	—
29	C₂×C₅×C₈	—	5	3	no	—
30	#681	Dic₅	1	3	yes	—
31	C₂×#168	C₂×D₅	1	3	yes	—
32	#674	—	5	1	no	—
33	#674	—	5	3	no	—
34	#672	—	5	1	no	—
35	#672	—	5	3	no	—
36	#504	—	25	1	no	Sylow
37	C₂×C₅²	—	2	3	no	—
38	C₂×C₅²	C₂²×C₈	1	1	yes	characteristic
39	C₅×Q₈	—	4	1	no	—
40	C₅×Q₈	#168	1	1	yes	characteristic
41	C₅×Q₈	#170	1	1	yes	characteristic
42	C₅×D₄	—	4	3	no	—
43	C₅×D₄	#168	1	3	yes	—
44	C₅×D₄	#170	1	3	yes	—
45	C₂×C₄×C₅	—	4	3	no	—
46	C₂×C₄×C₅	C₂×Dic₅	1	3	yes	—
47	C₂×C₄×C₅	C₂×AGL(1,𝔽₅)	1	3	yes	—
48	C₅×C₈	—	5	3	no	—
49	C₅×C₈	—	5	1	no	—
50	#168	C₂×Dic₅	1	3	yes	—
51	#168	C₂²×D₅	1	1	yes	characteristic
52	#131	—	5	1	no	—
53	M₃₂	—	25	3	no	—
54	C₂×C₁₆	—	25	3	no	—
55	C₅²	#504	1	1	yes	characteristic, complemented, Sylow
56	C₂²×C₅	—	4	3	no	—
57	C₂²×C₅	C₂×#168	1	3	yes	—
58	C₂²×C₅	C₂×#170	1	3	yes	—
59	C₄×C₅	—	4	3	no	—
60	C₄×C₅	—	4	1	no	—
61	C₄×C₅	C₂×#168	1	3	yes	—
62	C₄×C₅	C₂×#170	1	3	yes	—
63	C₄×C₅	C₂²×Dic₅	1	1	yes	characteristic
64	C₄×C₅	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
65	#41	#1042	1	1	yes	characteristic
66	M₁₆	—	5	3	no	—
67	C₂×C₈	—	5	3	no	—
68	C₁₆	—	25	1	no	—
69	C₁₆	—	25	3	no	—
70	C₂×C₅	—	2	3	no	—
71	C₂×C₅	—	2	3	no	—
72	C₂×C₅	—	4	1	no	—
73	C₂×C₅	—	8	3	no	—
74	C₂×C₅	C₂²×#168	1	1	yes	characteristic
75	C₂×C₅	C₂²×#170	1	1	yes	characteristic
76	Q₈	#6032	1	1	yes	characteristic
77	D₄	#6032	1	3	yes	—
78	C₂×C₄	C₂×#1042	1	3	yes	—
79	C₈	—	5	1	no	—
80	C₈	—	5	3	no	—
81	C₅	—	4	1	no	—
82	C₅	#65429	1	1	yes	characteristic, complemented
83	C₅	#65791	1	1	yes	characteristic, complemented
84	C₂²	C₂×#6032	1	3	yes	—
85	C₄	C₂×#6032	1	3	yes	—
86	C₄	C₂²×#1042	1	1	yes	characteristic
87	C₂	—	2	3	no	—
88	C₂	C₂²×#6032	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄	C₂²×#1042
commutator, socle	C₂×C₅²	C₂²×C₈
Fitting, prime radical	C₅²×#41	C₄
cosocle	C₅×#168	C₂³
hypercenter	#41	#1042
hypocenter	C₅²	#504
LCS	G ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₄ ◁ #41

Complements

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

N	H ≅ G/N	type
2. #11716841	C₂	semidirect
3. C₂×#87763	C₂	semidirect
55. C₅²	#504	semidirect
82. C₅	#65429	semidirect
83. C₅	#65791	semidirect

Maximal subgroups

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

subgroup	index
C₅×#4094	2
#11716841	2
C₂×#87763	2
#65791	5
#65429	5