Subgroups — #13097750

Order 1600.

126 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₅×#127	C₂	1	1	yes	maximal, characteristic
2	C₄⋊#6014	C₂	1	2	yes	maximal, complemented
3	C₂×D₄×C₂₅	—	2	2	no	—
4	C₂×D₄×C₂₅	C₄	1	2	yes	—
5	C₄²×C₂₅	C₂²	1	1	yes	characteristic
6	C₂×#6014	—	2	2	no	—
7	#64764	—	5	1	no	maximal
8	C₂³×C₂₅	—	2	2	no	—
9	D₄×C₂₅	—	4	2	no	—
10	D₄×C₂₅	—	4	2	no	—
11	D₄×C₂₅	—	4	2	no	—
12	C₂×C₄×C₂₅	C₂×C₄	1	1	yes	characteristic
13	C₂×C₄×C₂₅	D₄	1	2	yes	—
14	#6014	—	4	2	no	—
15	C₅×#127	D₅	1	1	yes	characteristic
16	C₄⋊#168	—	5	2	no	—
17	C₂²×C₂₅	—	4	2	no	—
18	C₂²×C₂₅	—	4	4	no	—
19	C₂²×C₂₅	C₂²⋊C₄	1	1	yes	characteristic
20	C₄×C₂₅	—	2	1	no	—
21	C₄×C₂₅	D₈	1	2	yes	—
22	C₄×C₂₅	SD₁₆	1	2	yes	—
23	C₂×C₅×D₄	—	2	2	no	—
24	C₂×C₅×D₄	Dic₅	1	2	yes	—
25	C₄²×C₅	C₂×D₅	1	1	yes	characteristic
26	C₂×#168	—	10	2	no	—
27	#331	—	25	1	no	Sylow
28	C₂×C₂₅	—	8	2	no	—
29	C₂×C₂₅	#100	1	1	yes	characteristic
30	C₂×C₂₅	#102	1	2	yes	—
31	C₂³×C₅	—	2	2	no	—
32	C₅×D₄	—	4	2	no	—
33	C₅×D₄	—	4	2	no	—
34	C₅×D₄	—	4	2	no	—
35	C₂×C₄×C₅	C₂×Dic₅	1	1	yes	characteristic
36	C₂×C₄×C₅	C₂²⋊D₅	1	2	yes	—
37	#168	—	20	2	no	—
38	#127	D₂₅	1	1	yes	characteristic
39	C₄⋊C₈	—	25	2	no	—
40	C₂₅	#331	1	1	yes	characteristic, complemented, Sylow
41	C₂²×C₅	—	4	4	no	—
42	C₂²×C₅	—	4	2	no	—
43	C₂²×C₅	C₂²⋊Dic₅	1	1	yes	characteristic
44	C₄×C₅	—	2	1	no	—
45	C₄×C₅	#686	1	2	yes	—
46	C₄×C₅	#687	1	2	yes	—
47	C₂×D₄	—	2	2	no	—
48	C₂×D₄	Dic₂₅	1	2	yes	—
49	C₂×C₈	—	50	2	no	—
50	C₄²	C₂×D₂₅	1	1	yes	characteristic
51	C₂×C₅	—	8	2	no	—
52	C₂×C₅	#3964	1	2	yes	—
53	C₂×C₅	#3965	1	1	yes	characteristic
54	C₂³	—	2	2	no	—
55	D₄	—	4	2	no	—
56	D₄	—	4	2	no	—
57	D₄	—	4	2	no	—
58	C₂×C₄	C₂×Dic₂₅	1	1	yes	characteristic
59	C₂×C₄	C₂²⋊D₂₅	1	2	yes	—
60	C₈	—	100	2	no	—
61	C₅	#64764	1	1	yes	characteristic
62	C₂²	—	4	2	no	—
63	C₂²	—	4	4	no	—
64	C₂²	C₂²⋊Dic₂₅	1	1	yes	characteristic
65	C₄	—	2	1	no	—
66	C₄	#87721	1	2	yes	—
67	C₄	#87722	1	2	yes	—
68	C₂	—	8	2	no	—
69	C₂	#11716380	1	1	yes	characteristic
70	C₂	#11716381	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²⋊Dic₂₅
commutator	C₂×C₄×C₂₅	C₂×C₄
Frattini	C₄²×C₅	C₂×D₅
Fitting	C₂₅×#127	C₂
socle	C₂²×C₅	C₂²⋊Dic₅
prime radical	C₅×#127	D₅
cosocle	C₄²×C₂₅	C₂²
hypercenter	#127	D₂₅
hypocenter	C₂₅	#331
LCS	G ▷ C₂×C₄×C₂₅ ▷ C₂²×C₂₅ ▷ C₂₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #127

Complements

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

N	H ≅ G/N	type
2. C₄⋊#6014	C₂	semidirect
40. C₂₅	#331	semidirect

Maximal subgroups

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

subgroup	index
C₂₅×#127	2
C₄⋊#6014	2
#64764	5