Subgroups — #13097749

Order 1600.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₅×#125	C₂	1	1	yes	maximal, characteristic
2	#11716351	C₂	1	2	yes	maximal
3	C₄⋊C₄×C₂₅	—	2	2	no	—
4	C₄⋊C₄×C₂₅	C₄	1	2	yes	—
5	C₄²×C₂₅	C₂²	1	1	yes	characteristic
6	C₂×#6014	—	2	2	no	—
7	#64763	—	5	1	no	maximal
8	C₂×C₄×C₂₅	—	2	2	no	—
9	C₂×C₄×C₂₅	C₂×C₄	1	1	yes	characteristic
10	C₂×C₄×C₂₅	D₄	1	2	yes	—
11	#6014	—	2	4	no	—
12	C₅×#125	D₅	1	1	yes	characteristic
13	#3935	—	5	2	no	—
14	C₂²×C₂₅	C₂²⋊C₄	1	1	yes	characteristic
15	C₄×C₂₅	—	2	1	no	—
16	C₄×C₂₅	—	2	2	no	—
17	C₄×C₂₅	—	4	2	no	—
18	C₅×C₄⋊C₄	—	2	2	no	—
19	C₅×C₄⋊C₄	Dic₅	1	2	yes	—
20	C₄²×C₅	C₂×D₅	1	1	yes	characteristic
21	C₂×#168	—	10	2	no	—
22	#330	—	25	1	no	Sylow
23	C₂×C₂₅	#101	1	1	yes	characteristic
24	C₂×C₂₅	#104	1	2	yes	—
25	C₂×C₄×C₅	—	2	2	no	—
26	C₂×C₄×C₅	C₂×Dic₅	1	1	yes	characteristic
27	C₂×C₄×C₅	C₂²⋊D₅	1	2	yes	—
28	#168	—	10	4	no	—
29	#125	D₂₅	1	1	yes	characteristic
30	#97	—	25	2	no	—
31	C₂₅	#330	1	1	yes	characteristic, complemented, Sylow
32	C₂²×C₅	C₂²⋊Dic₅	1	1	yes	characteristic
33	C₄×C₅	—	2	1	no	—
34	C₄×C₅	—	2	2	no	—
35	C₄×C₅	—	4	2	no	—
36	C₂×C₈	—	50	2	no	—
37	C₄⋊C₄	—	2	2	no	—
38	C₄⋊C₄	Dic₂₅	1	2	yes	—
39	C₄²	C₂×D₂₅	1	1	yes	characteristic
40	C₂×C₅	#3968	1	1	yes	characteristic
41	C₂×C₅	#3969	1	2	yes	—
42	C₂×C₄	—	2	2	no	—
43	C₂×C₄	C₂×Dic₂₅	1	1	yes	characteristic
44	C₂×C₄	C₂²⋊D₂₅	1	2	yes	—
45	C₈	—	50	4	no	—
46	C₅	#64763	1	1	yes	characteristic
47	C₂²	C₂²⋊Dic₂₅	1	1	yes	characteristic
48	C₄	—	2	1	no	—
49	C₄	—	2	2	no	—
50	C₄	—	4	2	no	—
51	C₂	#11716383	1	1	yes	characteristic
52	C₂	#11716385	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²⋊Dic₂₅
commutator	C₂×C₄×C₂₅	C₂×C₄
Frattini	C₄²×C₅	C₂×D₅
Fitting	C₂₅×#125	C₂
socle, prime radical	C₂²×C₅	C₂²⋊Dic₅
cosocle	C₄²×C₂₅	C₂²
hypercenter	#125	D₂₅
hypocenter	C₂₅	#330
LCS	G ▷ C₂×C₄×C₂₅ ▷ C₂²×C₂₅ ▷ C₂₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #125

Complements

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

N	H ≅ G/N	type
31. C₂₅	#330	semidirect

Maximal subgroups

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

subgroup	index
C₂₅×#125	2
#11716351	2
#64763	5