Subgroups — #13098780

Order 1600.

126 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11716510	C₂	1	1	yes	maximal, characteristic
2	#11716414	C₂	1	3	yes	maximal, complemented
3	C₂×#87709	C₂	1	3	yes	maximal, complemented
4	#41×C₂₅	C₄	1	1	yes	characteristic
5	#87716	C₄	1	3	yes	—
6	C₂×#6014	C₂²	1	3	yes	—
7	#87709	C₂²	1	1	yes	characteristic
8	#87709	C₂²	1	3	yes	—
9	#65791	—	5	1	no	maximal
10	Q₈×C₂₅	C₈	1	1	yes	characteristic
11	D₄×C₂₅	C₈	1	3	yes	—
12	C₂×C₄×C₂₅	C₂×C₄	1	3	yes	—
13	#6014	C₂×C₄	1	3	yes	—
14	#6014	C₂³	1	1	yes	characteristic
15	#4094	—	5	1	no	—
16	#3998	—	5	3	no	—
17	C₂×#674	—	5	3	no	—
18	C₂²×C₂₅	C₂×C₈	1	3	yes	—
19	C₄×C₂₅	C₂×C₈	1	3	yes	—
20	C₄×C₂₅	C₂²×C₄	1	1	yes	characteristic
21	C₅×#41	AGL(1,𝔽₅)	1	1	yes	characteristic
22	#681	—	5	3	no	—
23	C₂×#168	—	5	3	no	—
24	#674	—	5	1	no	—
25	#674	—	5	3	no	—
26	#504	—	25	1	no	Sylow
27	C₂×C₂₅	—	2	3	no	—
28	C₂×C₂₅	C₂²×C₈	1	1	yes	characteristic
29	C₅×Q₈	#170	1	1	yes	characteristic
30	C₅×D₄	#170	1	3	yes	—
31	C₂×C₄×C₅	C₂×AGL(1,𝔽₅)	1	3	yes	—
32	#168	—	5	3	no	—
33	#168	—	5	1	no	—
34	#131	—	25	1	no	—
35	M₃₂	—	25	3	no	—
36	C₂×C₁₆	—	25	3	no	—
37	C₂₅	#504	1	1	yes	characteristic, complemented, Sylow
38	C₂²×C₅	C₂×#170	1	3	yes	—
39	C₄×C₅	C₂×#170	1	3	yes	—
40	C₄×C₅	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
41	#41	#1035	1	1	yes	characteristic
42	M₁₆	—	25	3	no	—
43	C₂×C₈	—	25	3	no	—
44	C₁₆	—	25	1	no	—
45	C₁₆	—	25	3	no	—
46	C₂×C₅	—	2	3	no	—
47	C₂×C₅	C₂²×#170	1	1	yes	characteristic
48	Q₈	#6016	1	1	yes	characteristic
49	D₄	#6016	1	3	yes	—
50	C₂×C₄	C₂×#1035	1	3	yes	—
51	C₈	—	25	1	no	—
52	C₈	—	25	3	no	—
53	C₅	#65791	1	1	yes	characteristic
54	C₂²	C₂×#6016	1	3	yes	—
55	C₄	C₂×#6016	1	3	yes	—
56	C₄	C₂²×#1035	1	1	yes	characteristic
57	C₂	—	2	3	no	—
58	C₂	C₂²×#6016	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂²×#1035
commutator	C₂×C₂₅	C₂²×C₈
Frattini	C₄×C₅	C₂²×AGL(1,𝔽₅)
Fitting	#41×C₂₅	C₄
socle	C₂×C₅	C₂²×#170
prime radical	C₅×#41	AGL(1,𝔽₅)
cosocle	#6014	C₂³
hypercenter	#41	#1035
hypocenter	C₂₅	#504
LCS	G ▷ C₂×C₂₅ ▷ C₂₅
UCS	C₁ ◁ C₄ ◁ #41

Complements

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

N	H ≅ G/N	type
2. #11716414	C₂	semidirect
3. C₂×#87709	C₂	semidirect
37. C₂₅	#504	semidirect

Maximal subgroups

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

subgroup	index
#11716510	2
#11716414	2
C₂×#87709	2
#65791	5