Subgroups — #13098677

Order 1600.

279 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊#1035	C₂	1	6	yes	maximal
2	C₄²×D₂₅	C₂	1	1	yes	maximal, characteristic
3	C₂²×#1035	C₂²	1	4	yes	—
4	C₂×C₄×D₂₅	C₂²	1	3	yes	—
5	C₄⋊#1035	—	2	12	no	—
6	C₄²×C₂₅	C₄	1	1	yes	characteristic, complemented
7	C₄×Dic₂₅	C₄	1	3	yes	complemented
8	#65688	—	5	1	no	maximal
9	C₂²×D₂₅	C₂³	1	1	yes	characteristic
10	C₂×#1035	—	2	12	no	—
11	C₂×C₄×C₂₅	C₂×C₄	1	3	yes	—
12	C₂×Dic₂₅	C₂×C₄	1	3	yes	—
13	C₄×D₂₅	D₄	1	6	yes	—
14	C₄×D₂₅	Q₈	1	6	yes	—
15	C₂×C₄⋊AGL(1,𝔽₅)	—	5	6	no	—
16	C₄²×D₅	—	5	1	no	—
17	C₂²×C₂₅	C₂²×C₄	1	1	yes	characteristic
18	C₂×D₂₅	C₂×D₄	1	3	yes	—
19	C₂×D₂₅	C₂×Q₈	1	3	yes	—
20	#1035	—	4	4	no	—
21	C₄×C₂₅	C₄⋊C₄	1	6	yes	complemented
22	Dic₂₅	C₄⋊C₄	1	6	yes	complemented
23	C₂²×AGL(1,𝔽₅)	—	5	4	no	—
24	C₂×C₄×D₅	—	5	3	no	—
25	C₄⋊AGL(1,𝔽₅)	—	10	12	no	—
26	C₄²×C₅	AGL(1,𝔽₅)	1	1	yes	characteristic
27	C₄×Dic₅	—	5	3	no	—
28	#384	—	25	1	no	Sylow
29	C₂×C₂₅	C₂×C₄⋊C₄	1	3	yes	—
30	D₂₅	#127	1	1	yes	characteristic
31	D₂₅	C₄⋊Q₈	1	3	yes	—
32	C₂²×D₅	—	5	1	no	—
33	C₂×AGL(1,𝔽₅)	—	10	12	no	—
34	C₂×C₄×C₅	C₂×AGL(1,𝔽₅)	1	3	yes	—
35	C₂×Dic₅	—	5	3	no	—
36	C₄×D₅	—	5	6	no	—
37	C₄×D₅	—	5	6	no	—
38	C₂×C₄⋊C₄	—	25	6	no	—
39	C₂×C₄²	—	25	1	no	—
40	C₂₅	#384	1	1	yes	characteristic, complemented, Sylow
41	C₂²×C₅	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
42	C₂×D₅	—	5	3	no	—
43	C₂×D₅	—	5	3	no	—
44	AGL(1,𝔽₅)	—	20	4	no	—
45	C₄×C₅	C₄⋊AGL(1,𝔽₅)	1	6	yes	—
46	Dic₅	—	5	6	no	—
47	C₂²×C₄	—	25	3	no	—
48	C₂²×C₄	—	25	4	no	—
49	C₄⋊C₄	—	50	12	no	—
50	C₄²	—	25	3	no	—
51	C₄²	#1035	1	1	yes	characteristic, complemented
52	C₂×C₅	C₂×C₄⋊AGL(1,𝔽₅)	1	3	yes	—
53	D₅	—	5	1	no	—
54	D₅	—	5	3	no	—
55	C₂³	—	25	1	no	—
56	C₂×C₄	—	25	3	no	—
57	C₂×C₄	—	25	6	no	—
58	C₂×C₄	—	25	6	no	—
59	C₂×C₄	—	50	12	no	—
60	C₂×C₄	C₂×#1035	1	3	yes	—
61	C₅	#65688	1	1	yes	characteristic
62	C₂²	—	25	3	no	—
63	C₂²	—	25	3	no	—
64	C₂²	C₂²×#1035	1	1	yes	characteristic
65	C₄	—	25	6	no	—
66	C₄	—	100	4	no	—
67	C₄	C₄⋊#1035	1	6	yes	complemented
68	C₂	—	25	1	no	—
69	C₂	—	25	3	no	—
70	C₂	C₂×C₄⋊#1035	1	3	yes	—

	subgroup	G/H
center	C₂²	C₂²×#1035
commutator	C₂²×C₂₅	C₂²×C₄
Frattini, socle	C₂²×C₅	C₂²×AGL(1,𝔽₅)
Fitting	C₄²×C₂₅	C₄
prime radical, cosocle	C₂²×D₂₅	C₂³
hypercenter	C₄²	#1035
hypocenter	C₂₅	#384
LCS	G ▷ C₂²×C₂₅ ▷ C₂₅
UCS	C₁ ◁ C₂² ◁ C₄²

Complements

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

N	H ≅ G/N	type
6. C₄²×C₂₅	C₄	semidirect
7. C₄×Dic₂₅	C₄	semidirect
21. C₄×C₂₅	C₄⋊C₄	semidirect
22. Dic₂₅	C₄⋊C₄	semidirect
40. C₂₅	#384	semidirect
51. C₄²	#1035	semidirect
67. C₄	C₄⋊#1035	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊#1035	2
C₄²×D₂₅	2
#65688	5